file

CITY UNIVERSITY OF HONG KONGô/Œ⇥'x

Energy-Efficient Heuristics for JobAssignment in Server Farms

�Ÿh4-‹ºÿ˝H˚Ÿ⌃MOLÑ_|✏ó’

Submitted toDepartment of Electronic Engineering

˚PÂ↵˚in Partial Fulfillment of the Requirements

for the Degree of Doctor of PhilosophyÚxZÎxM

by

FU JingÖg

June 2016

Abstract

The rapidly increasing energy consumption in modern server farms has driven

studies to improve their energy efficiency. In this thesis, we analyze the stochastic

job assignment in a server farm comprising servers with various speeds, power

consumptions and buffer sizes. Ultimately, we seek to optimize the energy efficiency

of server farms by controlling the carried load on the networked servers.

We consider two types of assignment policies, with and without jockeying. The

jockeying in a multi-queue system indicates reassignment of incomplete jobs. The

job assignment policies considered here select a server, which has a vacancy, to

accept new arrival jobs. In the jockeying case, the reassignment of jobs that are

being served in the system will also be considered by a job assignment policy. The

energy efficiency of a server farm is defined as the ratio of its job throughput to its

power consumption and represents the useful work per unit cost. The maximization

of the ratio of job throughput to power consumption can be modeled as a (semi)

Markov decision process, and an optimal policy is obtained by conventional dynamic

programming techniques. However, the algorithm for an optimal solution in our

multi-queue systems is constrained by its computational complexity, and is imple-

mented as a baseline policy when comparing the performance of policies that are

computationally feasible in a server farm with tens of thousands of servers.

For the jockeying case, we propose a robust job assignment policy called E*, to

maximize the energy efficiency of a server farm already defined earlier. We model

the server farm as a system of parallel finite-buffer processor-sharing queues with

heterogeneous server speeds and energy consumption rates. We devise E* as an

insensitive policy so that the stationary distribution of the number of jobs in the

system depends on the job size distribution only through its mean. We provide a

rigorous analysis of E* and compare it with a baseline approach, known as most

energy-efficient server first (MEESF), that greedily chooses the most energy-efficient

servers for job assignment. We show that E* has always a higher job throughput

than that of MEESF, and derive realistic conditions under which E* is guaranteed to

outperform MEESF in terms of energy efficiency. Extensive numerical results are

presented to illustrate that E* can improve energy efficiency by up to 100%.

We also study the problem of job assignment in a large-scale realistically-

dimensioned server farm comprising multiple processor-sharing servers with differ-

ent service rates, energy consumption rates, and buffer sizes. Our aim is to optimize

the energy efficiency of such a server farm by effectively controlling carried load

on networked servers. To this end, we propose a job assignment policy, called Most

energy-efficient available server first Accounting for Idle Power (MAIP), which is

both scalable and near optimal. MAIP focuses on reducing the productive power used

to support the processing service rate. Using the framework of semi-Markov decision

process we show that, with exponentially distributed job sizes, MAIP is equivalent

to the well-known Whittle’s index policy. This equivalence and the methodology of

Weber and Weiss enable us to prove that, in server farms where a loss of jobs happens

if and only if all buffers are full, MAIP is asymptotically optimal as the number of

servers tends to infinity under certain conditions associated with the large number of

servers as we have in a real server farm. Through extensive numerical simulations,

we demonstrate the effectiveness of MAIP and its robustness to different job-size

distributions, and observe that significant improvement in energy efficiency can be

achieved by utilizing knowledge of energy consumption rate of idle servers.

iv

CITY UNIVERSITY OF HONG KONG Qualifying Panel and Examination Panel

Surname:

FU

First Name:

Jing

Degree: PhD

College/Department: Department of Electronic Engineering

The Qualifying Panel of the above student is composed of: Supervisor(s) Dr. WONG Wing Ming Eric Department of Electronic Engineering

City University of Hong Kong

Co-Supervisor(s) Prof. ZUKERMAN Moshe Department of Electronic Engineering


Qualifying Panel Member(s) Dr. CHAN Chi Hung Sammy Department of Electronic Engineering


Prof. CHEN Guanrong Department of Electronic Engineering


This thesis has been examined and approved by the following examiners: Dr. CHAN Chi Hung Sammy Department of Electronic Engineering


Dr. YEUNG Kai Hau Alan Department of Electronic Engineering City University of Hong Kong

Dr. WONG Wing Ming Eric Department of Electronic Engineering City University of Hong Kong

Prof. CHAN Shueng Han Department of Computer Science and Engineering Hong Kong University of Science & Technology

Acknowledgements

I extend my sincere thanks to my supervisor, Dr. Eric W. M. Wong, who guided me

into this area with great patience and optimistically walked me through those tough

areas in my Ph. D. study. I offer my heartfelt gratitude to my co-supervisor, Prof.

Moshe Zukerman. I could not finish my study without his invaluable support and

supervision. I express my sincere gratitude to Prof. Bill Moran for his innovative

inspirations and insightful comments, which amazingly broadened my horizons.

I offer my deep gratefulness to Dr. Jun Guo for all his time and efforts in helping

and teaching me, and I have learnt so much from him.

I am so grateful for my families and all my friends who have delivered their great

courages to me for my research and my dream.

Table of contents

List of figures xv

List of tables xvii

Summary of major symbols and acronyms xix

1 Introduction 1

1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.3 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.4 Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.5 Publications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2 Maximization of the Ratio of Long-Run Expected Reward to Long-Run

Expeced Cost 11

2.1 Server Farm Model . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.2 Markov/Semi-Markov Decision Process . . . . . . . . . . . . . . . 13

2.3 Long-Run Average Reward . . . . . . . . . . . . . . . . . . . . . . 15

2.4 Maximization of Average Reward divided by Average Cost . . . . . 16

2.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

3 Job-Assignment Heuristics with Jockeying 21

3.1 Related Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

Table of contents

3.2 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

3.3 Insensitive Conditions with Jockeying . . . . . . . . . . . . . . . . 29

3.3.1 Adaptation of symmetric queue . . . . . . . . . . . . . . . 30

3.4 Optimality for Insensitive Job Assignments . . . . . . . . . . . . . 33

3.4.1 Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

3.4.2 Polynomial Computational Complexity . . . . . . . . . . . 35

3.5 Jockeying Job-Assignment Heuristics . . . . . . . . . . . . . . . . 38

3.5.1 MEESF . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

3.5.2 E⇤ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

3.5.3 Rate-Matching . . . . . . . . . . . . . . . . . . . . . . . . 39

3.5.4 Insensitive MEESF Policy . . . . . . . . . . . . . . . . . . 40

3.5.5 Insensitive E⇤ Policy . . . . . . . . . . . . . . . . . . . . . 48

3.5.6 Approximate E⇤ . . . . . . . . . . . . . . . . . . . . . . . 65

3.6 conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

4 Job Assignment Heuristics without Jockeying 69

4.1 Related Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

4.2 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

4.3 Job Assignment Policy: MAIP . . . . . . . . . . . . . . . . . . . . 81

4.4 Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

4.4.1 Stochastic Process . . . . . . . . . . . . . . . . . . . . . . 83

4.4.2 Whittle’s Index . . . . . . . . . . . . . . . . . . . . . . . . 85

4.4.3 Indexability . . . . . . . . . . . . . . . . . . . . . . . . . . 87

4.4.4 Asymptotic optimality . . . . . . . . . . . . . . . . . . . . 90

4.5 Numerical results . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

4.5.1 Effect of Idle Power . . . . . . . . . . . . . . . . . . . . . 96

4.5.2 Effect of Jockeying Cost . . . . . . . . . . . . . . . . . . . 100

4.5.3 Sensitivity to the Shape of Job-Size Distributions . . . . . . 103

xii

Table of contents

4.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

5 Conclusions 107

References 111

Appendix A Theoretical results for Jockeying Job-Assignments 125

A.1 An Example for Mapping Qf . . . . . . . . . . . . . . . . . . . . . 125

A.2 Pseudo-Code of the Algorithm for an Insensitive Optimal Policy . . 126

A.3 Proof of Proposition 3.1 . . . . . . . . . . . . . . . . . . . . . . . . 127

A.4 Proof of Lemma 3.2 . . . . . . . . . . . . . . . . . . . . . . . . . . 133

A.5 Proof of Lemma 3.3 . . . . . . . . . . . . . . . . . . . . . . . . . . 135

A.6 Proof of Lemma 3.4 . . . . . . . . . . . . . . . . . . . . . . . . . . 135


A.8 Lemmas for Proposition 3.6 . . . . . . . . . . . . . . . . . . . . . 139



Appendix B Theoretical Proofs for Non-Jockeying Job-Assignments 161

B.1 Proof for Proposition 4.1 . . . . . . . . . . . . . . . . . . . . . . . 161



B.4 Consequences of the averaging principle . . . . . . . . . . . . . . . 166

xiii

List of figures

3.1 State transition diagram of the logically-combined queue under any

insensitive jockeying policy f 2Fs. . . . . . . . . . . . . . . . . . 32

3.2 Illustration of optimizing bK for the E* policy. . . . . . . . . . . . . 41

3.3 Energy efficiency of E⇤ and MEESF versus arrival rate. . . . . . . . 45

3.4 Energy efficiency of E⇤ and MEESF versus buffer size. . . . . . . . 46

3.5 Relative difference of energy efficiency of E⇤ to MEESF. . . . . . . 47

3.6 Relative difference of energy efficiency of E⇤ to MEESF. . . . . . . 48

3.7 Relative difference of power consumption of E⇤ to MEESF. . . . . . 49

3.8 Virtual probability of service rate. . . . . . . . . . . . . . . . . . . 61

3.9 Relative difference of energy efficiency of E⇤ to RM. . . . . . . . . 65

3.10 Difference of the value of bK of E⇤ to RM. . . . . . . . . . . . . . . 67

4.1 Performance comparison with respect to normalized system load r .

(a) Relative difference of L MAIP/E MAIP to L MNIP/E MNIP. (b) Job

throughput. (c) Relative difference of E MAIP to E MNIP. . . . . . . . 97

4.2 Performance comparison with respect to number of servers K. (a)

Energy efficiency. (b) Job throughput. (c) Power consumption. . . . 98

4.3 Cumulative distribution of relative difference of L MAIP/E MAIP to

L MNIP/E MNIP. (a) r = 0.4. (b) r = 0.6. (c) r = 0.8. . . . . . . . . 99

xv

List of figures

4.4 Performance comparison in terms of the energy efficiency with

respect to the number of servers K. (a) D = 0. (b) D = 0.0005. (c)

D = 0.01. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

4.5 Performance comparison in terms of the job throughput with respect

to the number of servers K. (a) D = 0. (b) D = 0.0005. (c) D = 0.01. 103

4.6 Cumulative distribution of relative difference of L MAIP,F/E MAIP,F

to L MAIP,Exponential/E MAIP,Exponential; (a) r = 0.4. (b) r = 0.6. (c)

r = 0.8. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

A.1 An example for multi-queue system. . . . . . . . . . . . . . . . . . 126

A.2 An example for multi-queue system. . . . . . . . . . . . . . . . . . 126

A.3 An example for arrival event. (a) Multi-queue system. (b) Logically

combined queue. . . . . . . . . . . . . . . . . . . . . . . . . . . . 127

A.4 An example for jockeying upon arrival event. (a) Multi-queue system.

(b) Logically combined queue. . . . . . . . . . . . . . . . . . . . . 127

A.5 An example for departure event. (a) Multi-queue system. (b) Logi-

cally combined queue. . . . . . . . . . . . . . . . . . . . . . . . . 128

A.6 An example for jockeying upon departure event. (a) Multi-queue

system. (b) Logically combined queue. . . . . . . . . . . . . . . . . 128

xvi

List of tables

3.1 Summary of major symbols . . . . . . . . . . . . . . . . . . . . . . 28

3.2 The running times of the algorithm for I-J-OPT . . . . . . . . . . . 37

3.3 The Counter Example for the Instantaneous Optimality of MEESF . 44

4.1 Summary of Frequently Used Symbols . . . . . . . . . . . . . . . . 80

xvii

Summary of major symbols and

acronyms

Roman Symbols

af

j (n j) indicator for server j in state n j under policy f : server j is tagged if af

j (n j) =

1; untagged if af

j (n j) = 0.

af (t) random variable, representing the action taken at time t under policy f

B j buffer size of server j

e scaler for e-revised reward

g scaler for g-revised reward

j server label

K number of servers

bK parameter for E⇤ policy

n vector of states (n1,n2, . . . ,nK)

n j state of server j representing the number of jobs in server j

R set of real

R j reward rate function for server j: a mapping from N j to R+

xix

Summary of major symbols and acronyms

R j(n j) reward rate of server j in state n j

R+ set of positive real

R vector of reward rate functions (R1,R2, . . . ,RK)

eB j aggregate buffer size of the first j servers

Xf

j (t) random variable, representing the number of jobs in server j at time t under

policy f

Greek Symbols

e j energy consumption rate of server j when it is busy

e

0j energy consumption rate of server j when it is idle

k parameter for extended E⇤ policy

l average arrival rate

µ j service rate of server j

n

⇤j (n j) index value for server j in state n j

F set of all stationary job assignment policy without jockeying

f job assignment policy

FJ set of all stationary job assignment policy with jockeying

g

f (R) long-run average reward of multi-queue system with reward rate function

vector R under policy f

r normalized offered traffic

e

e

bK aggregated energy consumption rate of the virtual server under policy E⇤ with

parameter bK

xx


e

µ

bK aggregated service rate of the virtual server under policy E⇤ with parameter bK

Other Symbols

E f long-run average energy consumption rate (power consumption) under policy

f

K set of servers

L f long-run average throughput (job throughput) under policy f

N set of all state vectors n

N j set of all states of server j

N {0,1}j set of controllable states of server j

N {0}j set of uncontrollable states of server j

R j set of all reward rate functions for server j

Acronyms / Abbreviations

CDF cumulative distribution function

E⇤ E⇤ policy

FIFO first in, first out

I-J-OPT optimal solution among all insensitive jockeying policies

JSQ Join the Shortest Queue policy

MAIP Most energy efficient available server first-Accounting for Idle Power

MEESF Most Energy Efficient Server First policy

MNIP Most energy efficient available server first-Neglecting Idle Power

xxi


PDF probability density function

PS processor sharing

QoS quality of service

RM Rate Matching policy

xxii

Chapter 1

Introduction

1.1 Introduction

Data centers have become essential to the functioning of virtually every sector of a

modern economy [1]. Server farms in data centers are known for their massive power

consumption [2]. The energy efficiency of server farms is an important concern when

considering greenhouse gas emissions. Driven by the green datacenter initiative to

facilitate a low-carbon economy in the information age [3], there are strong incentives

for managing the power consumption of server farms while maintaining acceptable

levels of performance [4].

Also, the data-center industry, with more than 500 thousand data centers world-

wide in 2011 [5], has been driven by dramatically increasing Internet traffic. For

example, the global IP traffic will annually pass 1.1 zettabyte by 2016 [6]. An

estimated 91 billion kWh of electricity was consumed by U.S. data centers in 2013,

and annual consumption is growing. It will achieve $13 billion annually on elec-

tricity bills, and will emit nearly 100 million metric tons of carbon pollution per

year by 2020 [7]. Computing servers do account for the major portion of the energy

consumption of data centers [8] .

1

Introduction

A standard model for a data center is that of several server farms [1], such a

structure being also suitable for other business models including cloud computing [9].

Our aim here is to describe optimal scheduling/dispatching strategies for incoming

requests in order to improve energy efficiency.

This has been a topic of interest for some time, with approaches such as speed

scaling approach that minimizes server cost by controlling server speed(s) [10–16],

right-sizing server pool that powers on/off servers according to system’s workload

[17–19], and resource allocation methodologies that distributes power budget for

energy conservation [20]. Our approaches that are in line with [21, 22] and investigate

the problem of stationary job assignment in a server farm. These approaches provide

a way for optimizing the power consumption of server farms by controlling the

carried load on the networked servers as a function of the (fixed) server speeds and

energy consumption rates. It can also be combined with speed scaling at each server

for local fine-tuning.

Rapid improvements in computer hardware have resulted in server farms with

a range of different computer resources (heterogeneous servers) being deployed in

modern data centers [23]. This heterogeneity significantly complicates the optimiza-

tion, since each server needs to be considered individually. Here we exhibit energy

efficiency improvements by exploiting of this heterogeneity by means of scalable

job-assignment policies that are applicable to real server farms with tens of thousands

of servers.

As defined in [21, 22], we regard the ratio of the long-run average throughput

divided by the expected energy consumption rate as our objective function, hereafter

referred to as the energy efficiency of a server farm. This objective function can be

justified as the amount of useful work (e.g., data rate, throughput) per watt.

In Chapter 2, for later analysis, we build up connections between the expected

cumulative reward and the long-run average reward per unit cost, following the idea

in [21].

2

1.1 Introduction

In Chapters 3 and 4, we analyze the processor sharing (PS) discipline on each

queue, where all jobs on the same queue share the processing capacity and are

served at the same speed. The PS discipline avoids unfair delays for jobs which

face extremely large jobs ahead of them; as a result PS is suitable in modeling web

server farms [24–26] where job-size distributions are of high variability [27]. In

communication systems, broader applications for PS queues have been studied, e.g.,

[28, 29].

In Chapter 3, we focus on job assignment policies that allow jockeying. When

jockeying is permitted, jobs can be reassigned to any server with buffer vacancies

at any time before they are completed. Assignment with jockeying provides more

freedom and are very suitable for server farms where the servers are collocated in a

single physical center and can use e.g., a shared DRAM-storage [30] or flash-storage

[31]. Jockeying policies in general can significantly improve system performance.

In addition, they are scalable in computation and hence make resource optimization

more tractable.

For such a problem, a straightforward heuristic approach is to greedily choose the

most energy-efficient servers for job assignment. It can be shown under certain con-

ditions that this approach, which we call most energy-efficient server first (MEESF)

in Chapter 3, maximizes the ratio of instantaneous job departure rate to instantaneous

energy consumption rate. In general, however, this is not the case. We shall see that

MEESF does not necessarily maximize the ratio of long-run average job departure

rate (i.e., job throughput) to long-run average energy consumption rate (i.e., power

consumption). This observation motivates us to design a more robust heuristic policy

for improving the energy efficiency of the system but which nevertheless achieves a

higher job throughput than can be achieved with MEESF.

In Chapter 4, we study a single server farm model with a large number of

heterogeneous servers where jockeying is not allowed. The jockeying case discussed

in [21, 22] is more appropriate for a localized server farm, in which the cost of

3

Introduction

jockeying actions (e.g. live-migration) is negligible. For server farms with significant

jockeying costs, a simple scalable job-assignment policy without jockeying is more

attractive. Policies in [21, 22] assume, in addition, that power consumption on the

idle servers is negligible; this too is an unrealistic assumption for some computing

environments. Here, we consider a more realistic system in which idle servers do not

necessarily have negligible power consumption [32]. A key feature of our approach

is to model the problem as an instance of the Restless Multi-Armed Bandit Problem

(RMABP) [33] in which, at each epoch, a policy chooses a server to be tagged for

a new job assignment (other servers are said to be untagged). The RMABP has

been proved to be SPACE-hard [34], and this has led to studies in scalable and

near-optimal approximations, such as index policies. An index policy selects a set of

tagged servers at any epoch based on their state-dependent indices.

We assume that a server farm (queueing system) cannot turn away a job if it has

buffer space available. When a farm has some inefficient servers, rejection of some

jobs might save energy if only those servers are available, but this is not permitted.

This might occur, for instance, when a server farm vendor is unable to replace all

older servers simultaneously, but legacy inefficient servers might be needed to meet

a service level agreement. In other words, the constraint on the number of tagged

servers in conventional RMABP has to be replaced by a constraint on the number of

tagged servers in controllable states. To the best of our knowledge, no theoretical

work has been published for the asymptotic optimality of an index policy for a

multi-server system with finite buffer size, where loss of jobs happens if and only

if all servers (queues) are full. Buffering spill-over creates dependencies between

servers, and requires us to postulate uncontrollable states [35].

4

1.2 background

1.2 background

Server farm management has been studied for many years, with applications includ-

ing data centers, cloud computing, large scale storage system, and network resource

allocation. These problems are modeled as multi-server (queue) systems, where each

server has a limited quantities of resources (e.g. bandwidth, memories, positions

for parallel tasks, etc.). The resources used by jobs (tasks) have to be paid, and the

reward from completed jobs then obtained.

Data center vendors will always balance the operation costs, e.g., energy con-

sumption, and Service Level Agreement (SLA), e.g., throughput and response time.

For interactive applications that are sensitive to their response time, some researchers

have considered minimizing the average delay of a server farm model with homoge-

neous servers (queues) [18, 36].

With an eye towards optimizing response time in heterogeneous environments,

experimental work in on/off powering servers has been studied in order to improve

the proportionality when system utilization is low [37]. A group of work proposed

to minimize the sum of weighted delay time and power consumption with geographi-

cally distributed data centers [38], with geographically differed electricity prices and

setup-delay [39]; other work is done with service rates in smaller time granularity

than the decision of the number of servers [40] and with consideration of multi-core

processors [41]. In particular, in [39–42], the authors mapped the optimization

problem as a convex optimization problem, the papers [39–41] assume a convex

relationship between the service rate and the power consumption of the server, and

[40] gave a theoretical bound for the performance deviation of the heuristic policy

from an optimal solution.

Another approach to improve efficiency in server farms is to minimize the energy

consumption with guaranteed average response time. In [43], three-tier web clusters

was considered with different servers in each tier by Mixed Integer Non-Linear

5

Introduction

Programming (MINLP). In [44], methodology was studied with geographically

varying electricity prices based on convex optimization and multi-step resource

usage prediction; and, in [45], scheduling in a multi-server model was analyzed

with no-waiting room in each server. In [46], the authors considered both cases

that optimizes power consumption with bounded response time and that minimizes

response time with given power budget by Pareto optimization.

Other experimental works in the heterogeneous environment with respect to

response time include improving server utilization with forecasting two classes of

workload on each computing infrastructure [47]. Khazaei et al. in [48] analyzed the

case of generally distributed service times for user requests (tasks) in a M/G/m/m+ r

model by assuming a limited maximum number (e.g. three) of departures between

two arrivals.

To better differentiate our proposed scheme with previous work, more detailed

related work will be distributed into corresponding contribution chapters.

1.3 Overview

In Chapter 2, we discuss the general framework for optimizing a long-run average

reward per unit cost by means of SMDP. In Chapters 3 and 4, we analyze the job

assignment policies in server farms with and without jockeying actions, respectively.

1.4 Contributions

In this thesis, we study the energy-efficient job assignment policies in server farms,

aiming at maximization of the energy efficiency (the ratio of the long-run average

throughput to the long-run average power consumption).

In Chapter 2, we introduce the problem for heterogeneous server farms as SMDP

and then introduce the e-revised criteria [21] in a vein of conventional g-revised

6

1.4 Contributions

criteria [49]. The g-revised criteria connects the optimization of expected cumulative

reward to the optimization of the long-run average reward per unit time. The e-

revised criteria connects the optimization of expected cumulative reward to the

optimization of the long-run average reward per unit cost. This cost can be specified

as time, energy consumption and etc.

We then continue our analysis of energy-efficient job assignments in both jockey-

ing and non-jockeying cases.

For the jockeying policies, our main contributions in Chapter 3 are summarized

as follows:

• We prove that the computational complexity of the algorithm proposed in [21]

for an optimal solution is O(K logKB), where K is the number of servers and

B is the total number of positions to allocate jobs in the entire system. For

a system with homogeneous buffer size of each server, the algorithm is of

linear complexity in the buffer size and of quadratic-logarithmic complexity in

the number of servers. For the cubic power functions, the complexity of the

algorithm is quadratically increasing in the number of servers.

• We prove that the optimal insensitive policy in the jockeying case also optimize

the energy efficiency for all jockeying policies with exponentially distributed

job sizes.

• Despite the polynomial complexity mentioned above, the algorithm is not

sufficiently scalable for systems with tens of thousands of servers. According

to our numerical experiments, it takes around five days running time to obtain

an optimal solution for 1,000 servers with 100 buffers for each server. This

motivates our studies on nearly optimal and scalable methodologies

• We propose a robust heuristic policy for stochastic job assignment in a finite-

buffer PS server farm. In this chapter we shall name this heuristic policy as E*

7

Introduction

to reflect our goal of designing a “star” policy that can maximize the energy

efficiency of the system. We demonstrate the effectiveness of E* by comparing

it to the baseline MEESF policy. Unlike MEESF that greedily chooses the

most energy-efficient servers for job assignment, E* aggregates an optimal

number of bK � 2 most energy-efficient servers to form a virtual server. In

our design, E* always gives preference to this virtual server and utilizes its

service capacity in such a way as to guarantee a higher job throughput than

what is achievable with MEESF, and yet improve the energy efficiency of

the system. The decision variable bK provides a degree of freedom for E* to

fine-tune performance.

• We discuss the insight gained from our design of E*. This will lead us to

proposing a simple rule of thumb for determining the optimal bK value that

maximizes the energy efficiency of the system under E*. The resulting policy,

referred to as rate matching (RM), chooses the value of bK so that the aggregate

service rate of the virtual server matches the job arrival rate. We provide

extensive numerical results to demonstrate the effectiveness of RM relative to

E*.

• We devise E* as an insensitive job assignment policy, in that, the stationary

(steady-state) distribution of the number of jobs in the system depends on the

job size distribution only through its mean. This insensitivity property is useful

for enssuring the robustness and predictability of the performance of the server

farm under a wide range of job size distributions.

• We perform a rigorous analysis of E*. In particular, we prove that E* has

always a higher job throughput than MEESF. Under the realistic scenario

where at least two servers in a heterogeneous server farm are equally most

energy efficient, we also prove that E* is guaranteed to outperform MEESF

in terms of the energy efficiency of the system. Having at least two servers

8

1.4 Contributions

that are equally energy efficient can be justified in practice by the fact that a

server farm is likely to comprise multiple servers of the same type purchased

at a time.

For the non-jockeying case, the main contributions of Chapter 4 are listed as

follows.

• We propose a job assignment policy, referred to as Most-energe-efficient-

server-first Accounting for Idle Power (MAIP). The MAIP requires the power

consumption of idle servers as an input parameter for decision making, and

provides a model of a realistic system with significant power consumption in

idle states. The MAIP is scalable, and requires only binary state information

of servers; this is appropriate for a real environment with frequently changing

server states.

• In our server farm, we apply the well-known Whittle’s index policy that

decomposes a complex RMABP problem into multiple sub-problems, each

assumed computationally feasible. In the general case, Whittle’s index does

not necessarily exist, and even if it does, a closed form solution is usually

unavailable.

• Remarkably we prove that when job sizes are exponentially distributed, the

Whittle’s index policy is equivalent to MAIP, and that it is asymptotically

optimal for our server farm composed of multiple groups of identical servers

as the numbers of servers in these groups tend to infinity.

• Extensive numerical results in Section 4.5 illustrate the performance and

the sensitivity to the shape of job-size distributions of these policies. We

numerically demonstrate the effectiveness of MAIP by comparing it to a

baseline policy. From the numerical results, the performance of MAIP in terms

9

Introduction

of energy efficiency is nearly insensitive to different job-size distributions

under the processor sharing discipline.

1.5 Publications

[1] J. Fu, J. Guo, E. W. M. Wong and M. Zukerman, “Energy-efficient heuristics

for insensitive job assignment in processor-sharing server farms," IEEE Journal on

Selected Areas in Communications, vol. 33, no. 12, pp. 2878-2891, Dec. 2015.

[Online Available] http://ieeexplore.ieee.org/xpl/articleDetails.jsp?arnumber=7279057

[2] J. Fu, J. Guo, E. W. M. Wong and M. Zukerman, “Energy-efficient heuristics for

job assignment in processor-sharing server farms," in Proc. IEEE INFOCOM, Hong

Kong SAR, China, pp. 882–890, Apr. 2015.

[Online Available] http://ieeexplore.ieee.org/xpl/login.jsp?tp=&arnumber=7218459

[3] Z. Rosberg, Y. Peng, J. Fu, J. Guo, E. W. M. Wong and M. Zukerman, “Insen-

sitive Job Assignment with Throughput and Energy Criteria for Processor-Sharing

Server Farms," IEEE/ACM Transactions on Networking, vol. 22, no. 4, pp. 1257–

1270, Aug. 2014.

[Online Available] http://ieeexplore.ieee.org/xpls/abs_all.jsp?arnumber=6583274

10

http://ieeexplore.ieee.org/xpl/articleDetails.jsp?arnumber=7279057

http://ieeexplore.ieee.org/xpl/login.jsp?tp=&arnumber=7218459

http://ieeexplore.ieee.org/xpls/abs_all.jsp?arnumber=6583274

Chapter 2

Maximization of the Ratio of

Long-Run Expected Reward to

Long-Run Expeced Cost

The main objective of our problem in this thesis is to maximize the energy efficiency

of a server farm. The energy efficiency is defined as the ratio of the long-run

average throughput to the long-run average power consumption. Here, we regard the

throughput and power consumption as the reward and cost of our system, then this

ratio becomes the ratio of long-run expected reward to long-run average cost. This

objective is more general than the traditional optimization problem focusing on the

expected reward (cost) per unit time.

In this chapter, we introduce and analyze the relationships between optimizing

cumulative reward, average reward and average reward divided by average cost. In

this chapter, we assume a Poisson arrival process and exponentially distributed job

sizes.

11

Maximization of the Ratio of Long-Run Expected Reward to Long-RunExpeced Cost

2.1 Server Farm Model

We study a heterogeneous server farm modeled as a multi-queue system. There are

K servers and the set of these servers is denoted by K = {1,2, . . . ,K}. Let N j be

the set of all states of server j 2K , where |N j|<+•. Decisions are made upon

arrival/departure events to allocate jobs to the servers (queues) in the server farm

(queueing system).

We also define X(t) = (X1(t),X2(t), . . . ,XK(t)) as the random variable repre-

senting the state at time t � 0 in the stochastic process of the multi-queue sys-

tem. The probability distribution of X(t) is denoted by P(t). Define vectors

n = (n1,n2, . . . ,nK) representing possible values of the random variable X(t), with

n j 2N j, j 2K . The set of all such n is denoted by N .

We consider a set of stationary policies, denoted F, which is a subset of the set

of all policies, where a policy is said to be stationary if it is non-randomized and

the action it takes at time t only depends on the state of the process at time t. The

decisions (actions) upon arrivals/departures rely on the values of the random variable

X(t) at the moments right before these arrivals/departures occur. A policy f gives

these decisions (actions) along the time line.

Let {Xf (t), t > 0}, denote the stochastic process under policy f with initial state

Xf (0) = x(0). For each j 2K , there exists a set of mappings R j = {R j : N j!R+}

with R j(n j) the reward rate for server j2K in state n j 2R j. Let R=(R j 2R j : j2

K ) denote a vector of these mappings, we define a function

g

f (R) = limt!+•

1t

Z t

0E(

Âj2K

R j(Xf

j (u))

)

du. (2.1)

If µ j(n j) and e j(n j) are the service rate and the energy consumption rate of server j in

state n j, respectively, then µ j,e j 2R j. Let µ = (µ j : j 2K ) and e = (e j : j 2K ).

In this thesis, our objective function is to maximize the ratio of the long-run average

12

2.2 Markov/Semi-Markov Decision Process

reward to the long-run average cost, when the reward is defined as the job throughput

and the cost is defined as the power consumption of the entire system, namely,

supf2F

g

f (µ)

g

f (e). (2.2)

2.2 Markov/Semi-Markov Decision Process

To solve the optimization problem with the objective function (2.2), in this section,

we introduce the well-known Semi-Markov Decision Process and then apply it as a

key tool to model our problem.

In this section, we focus on the maximization of the long-run expected non-

discounted reward, and, in later sections in this chapter, we will show the connections

between the long-run expected reward, the long-run average reward (Section 2.3) and

the long-run expected reward devided by the long-run expected cost (Section 2.4).

In our model defined in Section 2.1, there are K +1 key events, each of which

causes a change of state. One represents the arrival of jobs, labeled event 0, and the

other K events are departures from server 1, 2, . . ., K, labeled as event i, i = 1, 2,

. . ., K, respectively. Either an arrival or a departure will trigger a policy decision

(action) to rearrange the jobs into the system positions followed by an increment or a

decrement (by one) of the number of jobs in the system. In the definition of SMDP,

the increment/decrement of the number of jobs is also defined as part of the decision

actions. The rearrangement will be discussed in the case where jockeying is allowed.

For non-jockeying case, we simplify the rearrangement as the selection of a server to

allocate a new arriving job or decrement by one of the number of jobs of a server

associated with the occurred departure.

We define time points t0 ⌘ 0, t1, t2, . . . as the occurrence times for events where

ti is the time of the ith event, i = 1, 2, . . .. As defined above, any occurrence of an

event will trigger a decision action. For t > 0, let af (t) represent the action taken

13


at time t immediately following the event at t under policy f . The reward rate for

server j 2K during the period between two time points, say ti�1 and ti, depends on

the state Xf

j (ti) during this time period, and is represented by R j(Xf

j (ti)), R j 2R j.

For a stationary policy f 2 F, let V f (n,R) be the expected value of the accu-

mulative reward of the process of the system that starts from state n 2N until

it firstly goes into an absorbing state n0 2N with a vector of reward functions

(mappings) R = (R j 2R j : j 2K ) (i.e., R j(n j) is the reward rate of server j in

state n j). In particular, V f (n0,R)⌘ 0 for any f 2F and R j 2R j, j 2K . Without

loss of generality, we can extend the process associated with V f (n,R) to a long-run

process by setting both the transition probability, from state n0 to any other state

n 2N , and the reward rate in state n0 to be zero. For clarity of presentation, we

still refer to this extended long-run process as the process associated with V f (n,R).

If this extended process gives rise to an irreducible Markov chain with finite state

space, then, V f (n,R)<+• for f 2F [49, Corollary 6.20].

For n 2N and R = (R j 2R j : j 2K ), let

V (n,R) = supf2F

V f (n,R), (2.3)

and define the optimal policy f

⇤ 2F by

V f

⇤(n,R) =V (n,R), (2.4)

where f

⇤ exists since |F| < +•. Moreover, let t

f (n) denote the expected length

of a sojourn that the system stays in state n under policy f , and let Pf

n,n0 denote the

transition probability from state n to n0, n,n0 2N , under policy f . According to

the Hamilton-Jacobi-Bellman equation, we have

V (n,R) = supaf (n)

(

t

f (n) Âj2K

R j(n j)dt + Ân02N , n0 6=n

Pf

n,n0V (n0,R)

)

. (2.5)

14

2.3 Long-Run Average Reward

2.3 Long-Run Average Reward

In this section, we introduce how, in our problem, the optimization of the expected

accumulative reward V f (n,R) can be transferred to the optimization of the long-run

average reward by means of the g-revised reward.

Here, we consider a process that initializes at state n until its first return to the

same state, called process Pf

1 (n). Then, the expected cumulative reward of this

process is denoted by Uf (n,R). Define T f (n) as the expected time of the first return

to state n with initial state n. According to [49, Corollary 6.20], if process Pf

1 (n)

gives rise to an irreducible Markov chain with finite state space, then T f (n) is finite.

According to [49, Theorem 7.5], if T f (n) is finite, then the long-run average reward

under stationary policy f is equivalent to Uf (n,R)T f (n) .

We extend process Pf

1 (n) to a long-run version by repeating this process such

that those replicas are probabilistic replicas of Pf

1 (n). Then, the long-run average

reward of the extended Pf

1 (n) process is the same as Pf

1 (n), which, under certain

conditions, is equivalent to the ratio Uf (n,R)T f (n) . Namely, If Pf

1 (n) gives rise to an

irreducible Markov chain for any f 2 F, then, optimizing the long-run average

reward is same as optimizing the average reward of process Pf

1 (n).

We use the g-revised reward to maximize the expected average reward. The

g-revised reward (cost) is the reward obtained by subtracting a constant g from the

reward rate. There exists a value of g such that the policy optimizing the expected

accumulative g-revised reward will also optimize the expected average reward with

the original reward function.

Let T f

v (n) be the expected time of the process associated with V f (n,R). We

define the g-revised version of V f (n,R) as

V f ,g(n,R) =V f (n,R)�gT f

v (n).

15


Also, we set V g(n,R) = supf

V f ,g(n,R). According to [49, Theorem 7.6, Theorem

7.7], we obtain following corollary.

Corollary 2.1. If Uf (n,R) and Âj2K

R j(n j)tf (n) are finite, R = (R j 2R j : j 2K ),

for all f 2F and n 2N , then there exists a scaler g satisfying

V g(n,R) = supaf (n)

(

Âj2K

R j(n j)tf (n)+ Â

n02NPf

n,n0Vg(n0,R)�gt

f (n)

)

, (2.6)

and

g =Uf

⇤(n0,R)

T f

⇤(n0,R)= sup

f2F

Uf (n0,R)

T f (n0,R), (2.7)

where n0 is the absorbing state of V f (n,R),V f ,g(n,R) as defined before.

The optimal policy f

⇤ 2F prescribes an action that maximizes the right hand

side of (2.6) for any n 2N .

In particular, for our server farm problem, we set the absorbing state n0 to be the

empty state 0, which leads to only one available action: all servers are idle. Then, the

stationary policy f

⇤ for U(0,R) can be identified by solving (2.6). We will describe

the details of the algorithms for an optimal policy in later chapters where we focus

on more detailed situations.

2.4 Maximization of Average Reward divided by Av-

erage Cost

In previous section, we have explained the connection between the maximization of

long-run average reward and the maximization of V f (n,R) by means of the g-revised

reward introduced in Section 2.3.

We are now going to extend this g-revised reward to a more general situation

based on the result claimed in [21]; we refer to this result as e-revised reward. The

e-revised reward generalize g-revised reward in a way that the long-run expected

16

2.4 Maximization of Average Reward divided by Average Cost

reward per unit time, can be extended to the ratio of the long-run average reward to

the long-run average cost.

Define a vector C = (Cj 2R j : j 2K ) as the vector of cost functions. In the

vein of the definition of the g-revised reward function in Section 2.3, for a vector of

cost functions C, we define the e-revised version of V f (n,R) as

V f ,e(n,R,C) =V f (n,R)� eV f (n,C),

and

V e(n,R,C) = supf2F

V f ,e(n,R,C).

Similarly to Corollary 2.1, we obtain the following proposition for this e-revised

reward function.

Proposition 2.1. If Uf (n,R), Uf (n,C) and Âj2K

�

R j(n j)� eCj(n j)�

t

f (n) are fi-

nite, R = (R j 2R j : j 2K ),C = (Cj 2R j : j 2K ), for all f 2 F and n 2N ,

then, there exists a scaler e satisfying

V e(n,R,C) = supaf (n)

(

Âj2K

�

R j(n j)� eCj(n j)�

t

f (n)+ Ân02N

Pf

n,n0Ve(n0,R,C))

)

,

(2.8)

and

e =Uf

⇤(n0,R)

Uf

⇤(n0,C)= sup

f2F

Uf (n0,R)

Uf (n0,C), (2.9)

where n0 is the absorbing state of V f (n,R),V f ,e(n,R,C) as defined before.

The optimal policy f

⇤ 2F is a policy prescribes an action that maximizes the

right hand side of (2.8) for any n 2N .

Proof. This proposition is derived from Corollary 2.1 and [21, Theorem 1].

Define

Uf ,e,g(n,R,C) =Uf (n,R)� eUf (n,C)�gt

f (n),

17


and

Ue,g(n,R,C) = supf2F

Uf ,e,g(n,R,C).

Clearly, (2.9) is equivalent to Ue,0(n,R,C) = 0. Observe that Ue,0(n,R,C) is con-

tinuous in e 2 R. Since Uf (n,R) and Uf (n,C) are finite, then Ue,0(n,R,C)!+•

as e!�• and Ue,0(n,R,C)!�• as e!+•.

We then discuss the monotonicity of Ue,0(n,R,C) on e 2 R. For any e 2 R, we

have

Ue,0(n,R,C) =Uf

⇤e ,e,0(n,R,C)�Uf ,e,0(n,R,C).

Then, for any e1 < e, we obtain

Ue1,0(n,R,C)=Uf

⇤e1,e1,0(n,R,C)�Uf

⇤e ,e1,0(n,R,C)>Uf

⇤e ,e,0(n,R,C)=Ue,0(n,R,C).

(2.10)

Namely, Ue,0(n,R,C) is monotonically decreasing in e 2 R. In another word, there

exists a uniquely value for e satisfying (2.9).

For clarity of presentation, let e⇤ denote that value of e satisfying (2.9). Without

loss of generality, we set g = 0 and then adjust the variable e 2 R to be e⇤, i.e.,

Ue⇤,0(n,R,C) = 0. We refer to such an optimal policy as f

⇤, i.e., Uf

⇤,e⇤,0(n,R,C) =

Ue⇤,0(n,R,C) = 0. Then, (2.7) also holds true when the reward rates for states

n 2N are set to be

Âj2K

�

R j(n j)� e⇤Cj(n j)�

. (2.11)

That is, the optimal policy f

⇤ that prescribes the action maximzing the right hand

side of (2.6) for the same reward rate, also maximizes the long-run average reward,

which is zero, for the same reward rate.

According to [21, Theorem 1], such f

⇤ associated with e⇤ with the zero long-run

average reward, where the reward rate in state n is given by (2.11), also maximizes

the ratio Uf (n,R)/Uf (n,C).

18

2.5 Conclusions

As a consequence, the optimal solution that maximizes the ratio of the long-run

average reward to the long-run average cost can be obtained by conventional policy

iteration (or value iteration) method.

Although the computing complexity of a general algorithm derived on policy

iteration is exponentially increasing in the scale of the server farm (the number

of servers), we will prove in Chapter 3 that the algorithm for an optimal solution

proposed in [21] is of polynomial computing complexity.

2.5 Conclusions

We have established a connection between the expected e-revised cumulative reward

and the long-run average reward per unit cost. Then, the dynamic programing based

on the Hamilton-Jacobi-Bellman equation can be used to maximize the ratio of the

long-run expected reward to the long-run expected cost.

19

Chapter 3

Job-Assignment Heuristics with

Jockeying

In this chapter, we focus on job assignment where jockeying is allowed. If jockeying

is permitted, incomplete jobs can be reassigned to other vacant positions in the

system. The reassignment of jobs provides more degrees of freedom and matches a

real server farm where servers are allocated in a single physical data center, and a

shared DRAM-storage [30] or flash-storage [31] is employed. In general, stationary

policies with jockeying can significantly improve the performance of the system. In

addition, they are computationally scalable so that the resource optimization under

jockeying is more tractable.

For such a resource optimization problem, a straightforward heuristic approach

is achieved by greedily selecting the most energy-efficient server at each decision

epoch. Under certain conditions, we will prove that the straightforward policy, which

we refer to as Most Energy-Efficient Server First (MEESF) in this chapter, maximizes

the ratio of instantaneous job departure rate to instantaneous energy consumption

rate. However, we shall see in general that MEESF does not necessarily optimize the

ratio of long-run average job departure rate (i.e., job throughput) to long-run average

energy consumption rate (i.e., power consumption). This observation incentives us

21

Job-Assignment Heuristics with Jockeying

to seek a more robust heuristic job-assignment policy which improves the energy

efficiency and yet yields a higher job throughput than available under MEESF.

As mentioned in Section 1.4, the main contribution of this chapter includes:

• We prove that the computational complexity of the algorithm proposed in [21]

for an optimal solution is O(K logKB), where K is the number of servers and

B is the total number of positions to allocate jobs in the entire system. For

a system with homogeneous buffer size of each server, the algorithm is of

linear complexity in the buffer size and of quadratic-logarithmic complexity in

the number of servers. For the cubic power functions, the complexity of the

algorithm is quadratically increasing in the number of servers.

• We prove that the optimal insensitive policy in the jockeying case also optimize

the energy efficiency for all jockeying policies with exponentially distributed

job sizes.

• Despite the polynomial complexity mentioned above, the algorithm is not

sufficiently scalable for systems with tens of thousands of servers. According

to our numerical experiments, it takes around five days running time to obtain

an optimal solution for 1,000 servers with 100 buffers for each server. This

motivates our studies on nearly optimal and scalable methodologies.

• We propose a robust job-assignment policy for a finite-buffer PS server farm.

We demonstrate the effectiveness of E* by comparing it to the baseline MEESF

policy. Unlike MEESF that greedily chooses the most energy-efficient servers

for job assignment, E* aggregates an optimal number of bK � 2 most energy-

efficient servers to form a virtual server. In our design, E* always gives

preference to this virtual server and utilizes its service capacity in such a way

as to guarantee a higher job throughput than what is achievable with MEESF,

22

and yet improve the energy efficiency of the system. The decision variable bK

provides a degree of freedom for E* to fine-tune performance.

• We discuss the insight gained from our design of E*. This will lead us to

proposing a simple rule of thumb for determining the optimal bK value that

maximizes the energy efficiency of the system under E*. The resulting policy,

referred to as rate matching (RM), chooses the value of bK so that the aggregate

service rate of the virtual server matches the job arrival rate. We provide

extensive numerical results to demonstrate the effectiveness of RM relative to

E*.

• We devise E* as an insensitive job assignment policy, in that, the stationary

(steady-state) distribution of the number of jobs in the system depends on the

job size distribution only through its mean. This insensitivity property is useful

for enssuring the robustness and predictability of the performance of the server

farm under a wide range of job size distributions.

• We perform a rigorous analysis of E*. In particular, we prove that E* has

always a higher job throughput than MEESF. Under the realistic scenario

where at least two servers in a heterogeneous server farm are equally most

energy efficient, we also prove that E* is guaranteed to outperform MEESF

in terms of the energy efficiency of the system. Having at least two servers

that are equally energy efficient can be justified in practice by the fact that a

server farm is likely to comprise multiple servers of the same type purchased

at a time.

The remainder of the chapter is organized as follows. In Section 3.3, we introduce

the concept and construction of symmetric queue which possesses the insensitivity

of the queueing system where jockeying is allowed. In Section 3.4, an algorithm

for the optimal insensitive jockeying policy is introduced; we complement the

23


details in implementing the algorithm; and a theoretical proof is provided for the

polynomial computational complexity of the algorithm in the scale of the server farm.

In Section 3.5, we give a thorough analysis, including theoretical and numerical

results, for our proposed job-assignment heuristics which aim to maximize the energy

efficiency, under the insensitive structure.

3.1 Related Work

Queueing models associated with job assignment among multiple servers with and

without jockeying have been studied since the work of Haight [50] in 1958. Most

of the existing work has focused on policies that aim to improve the system’s

performance under the FCFS discipline such as Join-the-Shortest-Queue (JSQ). The

key aim of such policies is to balance system’s workload among all servers (queues),

and to optimize the throughput and the response time. The JSQ policy in the non-

jockeying case has been studied in [51–56] for FCFS and in [57–59] for processor

sharing.

For the non-jockeying case, Kingman [51] studied the non-jockeying case and

proved that assigning new arrival to the shortest queue would maximize the dis-

counted number of customers processed at any time t < +•. Then, Winston [52]

asserted that Join-the-Shortest-Queue with a finite number of servers maximizes

throughput during a fixed time t <+• in a FCFS homogeneous multi-queue system.

Later in 1987, Knessl et al. [55] analyzed a two-queue system with general service

times and infinite capacity under the Join-the-Shortest-Queue policy and provided

approximations for this system’s statistical characteristics. Later, JSQ became a

basic mechanism for computer networks and a popular model of multi-processor

architecture systems. Weber [53] considered multi-identical servers where if iden-

tical customers came in arbitrary arrival process and their job-size distribution is

characterized by a non-decreasing hazard function, JSQ will maximize the number

24

3.1 Related Work

of jobs served in a given time interval. For heterogeneous servers, Hordijk and Koole

[56] characterized, under certain condition, the optimal solution in the form that

“an arriving customer should be assigned to the queue with a faster server when

that server has a shorter queue" and such type of policies are referred to Shortest

Queue Faster Server Policy (SQFSP). However, fully characterizing optimal solu-

tion for a general heterogeneous problem is commonly considered difficult. Then,

approximation methodologies have been studied.

Bonomi [57] proved the optimality of JSQ for the processor sharing case under

a general arrival process, Markov departure processes and homogeneous servers.

A counter-example for the non-optimality of JSQ with non-exponential job-size

distribution was given by Whitt [58]. Gupta [59] provided an approximate analysis

for the performance of JSQ in a processor sharing model with a general job-size

distribution; also he intuitively explained and proved the optimality of JSQ in terms

of average delay for a system comprising servers with two different service rates.

Moreover, Altman et al. investigate optimal strategies for optimal centralized

load balancing in PS server farm and indicate that optimal policy for job assignment

in PS discipline is independent of service requirement of each request which is

dependent in FCFS [25].

Server farm applications of the JSQ with jockeying policies under the FCFS

discipline have been studied in [60–62]. Here, the jockeying action is triggered

when the difference between the queue sizes of the shortest queue and the longest

queue achieves a threshold value. Different threshold values clearly lead to different

policies. These publication focus on the expression of the equilibrium distribution

of the queue lengths; it has been shown that the service quality can be significantly

improved by jockeying.

For other job-assignments in processor sharing server farms without jockeying,

the energy efficiency of a multi-queue heterogeneous system with infinite buffer and

setup delay has been studied in [19, 63], where exact expressions of the value function

25


for the Markov Decision Process (MDP) are given. Hyytiä et al. [63] showed that

the M/G/1-LCFS model is only sensitive to the mean of the set-up delay, while, the

M/G/1-PS model is not insensitive with respect of the set-up delay. Li et al. in [64]

proposed a heuristic algorithm which aims to maximize the weighted probability

of the combined execution time (delay) and the energy consumption metric with

constraints on both the delay deadline and the energy budget in a heterogeneous

computing system.

For other job-assignments in processor sharing server farms without jockeying,

the energy efficiency of a multi-queue heterogeneous system with infinite buffer and

setup delay has been studied in [19, 63], where exact expressions of the value function

for the Markov Decision Process (MDP) are given. Hyytiä et al. [63] showed that

the M/G/1-LCFS model is only sensitive to the mean of the set-up delay, while, the

M/G/1-PS model is not insensitive with respect of the set-up delay. Li et al. in [64]

proposed a heuristic algorithm which aims to maximize the weighted probability

of the combined execution time (delay) and the energy consumption metric with

constraints on both the delay deadline and the energy budget in a heterogeneous

computing system.

To the best of our knowledge, processor sharing multi-queue system with jockey-

ing has been only studied in [21, 22], where the optimization problem to maximize

the energy efficiency of systems, as mentioned in Section 1.1, is modeled as a

Semi-Markov Decision Process (SMDP) [65].

In the general context of Markov Decision Process (MDP) or SMDP, significant

work has been done to optimize the expected average reward (cost). In [66], Lippman

optimized the finite horizon discounted reward in three models, namely, the M/M/c

finite capacity queue, M/M/1 service rate control problem and M/M/c finite capacity

queue with policy-dependent arrival rates, by using the concept of "uniformization",

a device by which a virtual exponential clock runs independently from both policy

and the state of the stochastic process. The concept of g-revised reward [67] provides

26

3.2 Model

a bridge between the optimization problems of the expected total reward and the

expected average reward. In addition, an optimal solution that maximizes the

expected average reward can be achieved by using a procedure proposed in [67]. In

particular, Stidham and Weber [68] considered a service rate control problem of a

single queue with left-skip-free transition structure for both exponential and non-

exponential service times, where the decisions depend on the number of customers

(jobs) in the queue and no discounting is considered. They provided a method to

prove the monotonicity of the optimal service rates, i.e., service rates are increasing

in queue length; some of their optimal policies are shown to be insensitive to the

shape of service time distribution. George and Harrison [69] studied the service rate

control problem of a single queue evolving as a birth-and-death Markov process, and

proved the existence of monotonic optimal service rates under weaker assumption

than those of Stidham and Weber [68].

The long-run average reward per unit cost (e.g. time consumption, energy

consumption and etc.) in [21] the long-run average service quality per unit time

which had been studied previously. With similar techniques as in [66–69], Rosberg et

al. [21] proposed an algorithm to find the stationary job assignment that maximizes

the energy efficiency by exploring servers’ heterogeneity. As is algorithm not

sufficiently scalable for a real local server farm, the authors of [21] proposed the

scalable job-assignment Slowest Server First (SSF) policy which aims to approximate

this optimality. SSF demonstrated numerically to be near-optimal under a cubic-

relationship between service rate and power consumption.

3.2 Model

For the reader’s convenience, Table 3.1 provides a list of major symbols that we

use in this chapter. Our model of a server farm consists of K independent servers,

each having a finite buffer for queuing jobs. For j = 1,2, . . . ,K, we denote by B j the

27


Table 3.1 Summary of major symbols

Symbol Definition

K Number of servers in the systemB j Buffer size of server jeBi Aggregate buffer size of the first i servers in the systemµ j Service rate of server je j Energy consumption rate of server j

µ j/e j Energy efficiency of server jl Job arrival ratebK Number of energy-efficient servers forming a virtual servere

µ

bK Aggregate service rate of the virtual servere

e

bK Aggregate energy consumption rate of the virtual servere

µ

bK/eebK Energy efficiency of the virtual serverbK⇤ Optimal value of bK chosen by the E* policybKRM Empirical value of bK chosen by the RM policyL f Job throughput of the system under policy f

E f Power consumption of the system under policy f

L f/E f Energy efficiency of the system under policy f

buffer size of server j where B j > 1. For notational convenience, we denote by eBi

the aggregate buffer size of the first i servers in the system, given by

eBk =k

Âl=1

Bl, k = 0,1, . . . ,K (3.1)

where eB0 = 0 by definition.

We denote by µ j the service rate of server j, defined as the units of jobs that can

be processed per time unit, and by e j the energy consumption rate of server j. Note

that, in the literature, the energy consumption rate of a server is usually related to the

server speed by a convex function of the form

e(µ) µ µ

b , b > 0 (3.2)

28

3.3 Insensitive Conditions with Jockeying

with b = 3 being the most commonly used value [70–72]. However, some researchers

have suggested that e(µ) is not necessarily convex [73, 74]. The job assignment

policies that we propose in this chapter do not require such an assumption.

We refer to the ratio µ j/e j as the energy efficiency of server j. Accordingly,

server i is defined to be more energy-efficient than server j if and only if µi/ei >

µ j/e j. Since our focus in this chapter is on developing energy-efficient job assign-

ment policies, for convenience of description, we label the servers according to their

energy efficiency; i.e. if i < j, then µi/ei � µ j/e j.

Remark 3.1. With this convention, for i < j, we have µie j� eiµ j � 0 since µi/ei �

µ j/e j.

We consider that jobs arrive at the system according to a Poisson process with

mean rate l . An arriving job is assigned to one of the servers with at least one vacant

slot in its buffer, subject to the control of an assignment policy. If all buffers are full,

the arriving job is lost.

We assume that job sizes (in units) are independent and identically distributed

random variables; with an cumulative distribution function (CDF) F(x), x � 0.

Without loss of generality, we normalize the average size of jobs to one. Each server

j serves its jobs at a total rate of µ j using the PS service discipline.


Rosberg et al. [21] proposed a class of insensitive stationary policies with jockeying

based on the concept of symmetric queue introduced in [75]. As in [21, 22], we

apply the symmetric queue model with state-dependent service rates defined in [75]

to our multi-queue system where jockeying is allowed. in Section 3.3.1, we begin

by providing an explanation based on [21, 22] of this adaptation to make this thesis

self-contained.

29


3.3.1 Adaptation of symmetric queue

A symmetric queue is a logical queue defined in [75] as a queue that has a symmetry

between the service rate allocated to each position in the queue and the probability

that a newly arrived job will join the queue in the corresponding position. In

particular, if there are n = |n| (n2N ) jobs in the queue, the jobs are labeled by their

positions, say 1,2, . . . ,n, and the total service rate is µ(n), then a proportion g(l,n)

of the total service effort µ(n) is directed to the job at position l, l = 1,2, . . . ,n.

• When the job at position l completes its service and leaves the system, the jobs

at positions l +1, l +2, . . . ,n move to positions l, l +1, . . . ,n�1, respectively.

• When a job arrives, it moves into position l with probability g(l,n+1). The

jobs originally located at positions l, l + 1, . . . ,n move to positions l + 1, l +

2, . . . ,n+1, respectively.

This logical queue, referred to as a symmetric queue, is general enough to rule a group

of job-assignment policies by adapting the values of the service efforts assigned to

its positions. The equilibrium distribution of the state of such a symmetric queue

(the number of jobs in the queue n = 0,1, . . . , eBK) was shown to be of product-form,

hence insensitive to general service requirement distributions [75, 76]. Kelly [75]

showed that a stationary symmetric queue is insensitive to the shape of the service

time distribution, given that it can be represented as a mixture of Erlang distributions.

This result was extended by Barbour [77] to arbitrarily distributed service times.

Taylor [76] showed that the canonical insensitive queueing model, the Erlang loss

system, can be described as a symmetric queue.

Let the buffer position in the multi-queue system be defined using a 2-tuple

notation ( j,m) for position m at server j where j 2K and m = 1,2, . . . ,B j. For

a policy f 2 F, we define a one-to-one mapping Qf from the buffer positions in

the multi-queue system (with domain set Qm = {( j,m) : j 2K , m = 1,2, . . . ,B j})

30


to the buffer positions in a logically-combined queue (whose range set is Qs = {l :

l = 1,2, . . . , eBK}). Note that there could be more than one such mapping Qf that

matches the underlying policy, since the service discipline at each server is processor

sharing (PS) so that all the relevant mappings associated with the positions of a given

server are equivalent. For clarity of presentation, an example for the mapping Qf ,

including its relavence to jockeying actions, is provided in Appendix A.1.

Because of the insensitivity property of the logically-combined queue, by the

symmetric queue construction of [75] shown below, all these mappings give the same

stationary distribution of the underlying stochastic process, {Xf (t) = |Xf (t)|, t > 0},

where |Xf (t)|= Âj2K

Xf

j (t). We define N = {0,1, . . . , eBK} for clarity of presentation.

In a logically-combined queue, a stationary policy f 2 F affects the equilibrium

state distribution of the underlying process {Xf (t), t > 0} by controlling the total

service rate µ(n), rewritten µ

f (n), for n 2N . We define a subset of F, referred

to as Fs, so that, for any policy in Fs, there exists a logically-combined queue as

defined above.

For any mapping Qf for a policy f 2 Fs, the multi-queue system can be im-

plemented as a symmetric queue on the Qs domain. To show this, we begin by

deriving the state-dependent service rates of Qs. With the logically-combined queue,

each server j serves only the jobs located at its associated positions using the PS

discipline. For policy f at state n, the total service rate µ

f (n) is given by

µ

f (n) def= Â

j2T f (n)µ j,

where T f (n) is the set of busy servers.

Under the PS discipline, the proportion of µ

f (n) allocated to the job at position

l def= Qf ( j,m) on the Qs domain is equivalent to that allocated to the job at its

31


Fig. 3.1 State transition diagram of the logically-combined queue under any insensi-tive jockeying policy f 2Fs.

corresponding position m of server j on the Qm domain, and is given by

g

f (l,n) =µ j

n jµf (n). (3.3)

To complete the construction of the logically-combined queue as a symmetric

queue, it remains to enforce a symmetry in the same manner as [75] between the

service rate allocated to each position in the logically-combined queue and the

probability that a newly arrived job will join the queue in the corresponding position.

That is, as in [21, 22], based on the one-to-one position mapping Qf from Qm to Qs,

the movements of jobs in the multi-queue system correspond to the movements of

jobs in the logically-combined queue.

Due to the insensitivity property of the symmetric queue, the state transition

process of the logically-combined queue in this context can be modeled as a birth-

death process (shown in Fig. 3.1) with birth rate l and death rates µ

f (n), n 2N .

Accordingly, the stationary distribution p

f = (pf

n , n2N ) of the process {Xf (t), t >

0} under any stationary policy f with jockeying, can be obtained by solving the

balance equations

lp

f

n = µ

f (n+1)pf

n+1, n = 0,1, . . . , eBK�1. (3.4)

32

3.4 Optimality for Insensitive Job Assignments

Then, the long-run job throughput of the system under policy f is equivalent to

the long-run average job departure rate, and can be obtained as

g

f (µ) =eBK

Ân=0

µ

f (n)pf (n), (3.5)

or

g

f (µ) = l

h

1�p

f (eBK)i

, (3.6)

for the stable system.

The power consumption of the system under policy f is equivalent to the long-run

average energy consumption rate, and can be obtained as

g

f (e) =eBK

Ân=0

e

f (n)pf (n) (3.7)

where e

f (n) is the total energy consumption rate in state n and is given by

e

f (n) def= Â

j2T f (n)e j. (3.8)

By definition, g

f (µ)/g

f (e) is the energy efficiency of the system under policy

f 2Fs.


Rosberg et al. [21] proposed an algorithm for an optimal policy that maximizes

the energy efficiency of a server farm defined in Section 3.2 among all insensitive

stationary job-assignment policies based on the concept of symmetric queue.

In this section, we briefly introduce the algorithm proposed in [21], and then

refine it in the way that the computing complexity of the algorithm is guaranteed to

be polynomial in the scale of the server farm.

33


We provide the pseudo-code of the algorithm in Appendix A.2, and the imple-

mentation, including source code and instruction, is given in [78]. Here, we firstly

explain the idea and theoretical basis for constructing this algorithm in Section 3.4.1

and complete the implementable details of the algorithm which are not described in

[21]. We then prove the computing complexity of the algorithm is polynomial in the

number of servers and size of buffers of the server farm in Section 3.4.2. Although

the polynomial algorithm provides a perfect benchmark in research experiments, the

computing time is not scalable for applications in a real server farm with thousands

of servers, as we will demonstrate by numerical results in Section 3.4.2.

3.4.1 Algorithm

By Proposition 2.1 in Chapter 2, maximizing the energy efficiency is equivalent in

maximizing the e-revised reward given by equation (2.8), where the optimal value e⇤

is conditioned by (2.9).

Based on the mapping from the server farm to a logically combined queue Qf , a

vector of state n 2N in the server farm can be mapped to a state n = |n|, n 2N , in

the logically combined queue. Thus, in the insensitive model defined in Section 3.3,

set V e(n) =V e(n,µ,e) where n = |n|. For clarity of presentation, in the insensitive

job-assignment problem with jockeying, we rewrite (2.8)

V e(n)�V e(n+1) = supT f (n)

1l

⇣

µ

f (n)(V e(n�1)�V e(n))+µ

f (n)� ee

f (n)⌘

,

(3.9)

for n = 2,3, . . . , eBK�1, where V e(1)�V e(0) = 0.

With

Y e(n) =V e(n�1)�V e(n), n = 1,2, . . . , eBK,

34


we can rewrite (3.9) as

Y e(n+1) = supT f (n)

1l

⇣

µ

f (n)Y e(n)+µ

f (n)� ee

f (n)⌘

. (3.10)

The resulting sequence of sets of busy servers T I�J�OPT(n), n = 1,2, . . . , eBK , forms

the optimal insensitive job-assignment stationary policy I-J-OPT.

As in [21], the bisection method can be used to find e⇤. According to (2.10) in

the proof for Proposition 2.1, the accumulative e-revised reward Ue,0(n,µ,e), for

each µ,e 2R, is monotonically decreasing in e. We start from a lower and an upper

bound of e⇤, say e1 and e2, which are usually set to be 0 and the energy efficiency

under any f 2Fs, respectively, and then calculate the corresponding Ue1+e2

2 ,0(0,µ,e)

by Bellman equation repeatedly until |e1� e2| < d , where d > 0 is a given small

positive real. In the program available online [78], d = 10�30 (For readers who are

interested in the implementation details, we strongly recommend the files including

user instruction and source code available in [78]).

Proposition 3.1. For the system defined in Section 3.2, if the job-sizes are expo-

nentially distributed, then a policy that maximizes the energy efficiency among all

insensitive policies is also optimal among all stationary policies.

Proof. The proof is given in Appendix A.3.

3.4.2 Polynomial Computational Complexity

To find an optimal policy remains solving the right hand side of (2.8). We propose

details for implementing dynamic programming for problem (2.8).

For clarity of presentation, we define the set of all available sets of busy servers

for state n 2N as T (n) = {T |T ⇢K }, let

yT ,e(n) =1l

Âj2T

⇥

(Y e(n)+1)µ j� ee j⇤

, (3.11)

35


for a set of busy servers T 2 T (n), Then, (3.10) can be rewritten as

Y e(n+1) = supT 2T (n)

yT ,e(n). (3.12)

For any policy f 2 Fs, n 2N , T ,T + { j} 2K and j /2 T , the increment of

yT ,e(n) for adding j into the busy set T is given by

yT [{ j},e(n)� yT ,e =1l

⇥

(Y e(n)+1)µ j� ee j)⇤

, (3.13)

of which the right hand side is independent from T . We hope to pick up all the j

with a higher value of the right hand side of (3.13) to comprise the set of busy servers,

T , which leads to a higher value of the resulting yT ,e(n). We obtain an order of

servers, say j1, j2, . . . , jK , which is descending in the value of the right hand side of

(3.13). Then, we define a set of integers, Le(n) = {l : { j1, j2, . . . , jl} 2 T (n)}, such

that there exists an l⇤ defined as

l⇤ = argmaxl2Le(n)y{ j1, j2,..., jl},e(n). (3.14)

To beak ties, we select the smallest l⇤ without loss of generality. An optimal soluton

is given by

T I�J�OPT (n) = { j1, j2, . . . , jl⇤}, (3.15)

Following this approach, the computational complexity of the algorithm for

I-J-OPT is O(D(K)eBK log(e2� e1)/d ), where the e2 and e1 denote the initial values

of the upper and lower bound of e⇤ for the bisection, d is a parameter for the accuracy

of the bisection, and O(D(K)) is the time complexity for ordering K servers based

on the the value of (3.13), which is O(K) for the cubic power functions assumed in

[21], and is O(K logK) for general power functions.

36


Table 3.2 The running times of the algorithm for I-J-OPT

K 10 60 110 210 310 410 460Running Time (s) <1 20 97 680 1935 4316 6177

Despite its polynomial complexity which provides a perfect benchmark in re-

search experiments, the algorithm for I-J-OPT is not scalable to realistic applications

involving server farm with thousands of servers. This will be illustrated as follows.

We begin with a set of small examples when B j = 100, j = 1,2, . . . ,K, and K ranges

between 10 and 460. The running times of the algorithm for I-J-OPT for these

examples are presented in Table 3.2. These results are based on an implementation

of the algorithm in C++ on an iMAC with 2.7 GHz Intel Core i5, 8.00GB installed

memory, and OS X 10.9.2 as the operating system. Then, a polynomial fitting of the

data in Table 3.2 is performed by OriginPro 8 SR3 v8.0932, and the running time as

a function of K is approximated as

0.428M2�7.245M+239.79(s).

This implies around 420,994.79 seconds (4.87 days) for K = 1,000. The algorithm

for I-J-OPT is not sufficiently scalable.

When B j = B is a constant for all j and the power function follows the cubic

assumption, i.e., e j = µ

3j and the time complexity for ordering, O(D(K)), is now

O(K), the approximate time complexity in the experiments above is consistent with

our previous result.

37


3.5 Jockeying Job-Assignment Heuristics

3.5.1 MEESF

MEESF is a straightforward heuristic approach that works by greedily choosing the

most energy-efficient server for job assignment. The SSF algorithm proposed in [21]

is a special case of MEESF by assuming for each server j its energy consumption

rate satisfies e j = µ

3j . It can be shown when all servers in the system have the same

buffer size, i.e., B1 = B2 = . . .= BK , that SSF maximizes the ratio of instantaneous

job departure rate to instantaneous energy consumption rate. In general, however,

this is not the case. We shall see later in Section 3.5.4 that MEESF does not

necessarily maximize the ratio of long-run average job departure rate (i.e., job

throughput) to long-run average energy consumption rate (i.e., power consumption).

This observation motivates us to design the more robust E⇤ policy.

3.5.2 E⇤

We design the E⇤ policy so that the bK� 2 most energy-efficient servers are aggregated

to form a virtual server.

Proposition 3.2. For the system defined in Section 3.2, consider a virtual server

obtained by aggregating the bK most energy-efficient servers in the system. If its

energy efficiency is defined by the ratiobKÂj=1

µ j/ÂbKj=1 e j, then, the virtual server is

more energy efficient than any server j � bK +1, i.e.,

ÂbKj=1 µ j

ÂbKj=1 e j

>µk

ek, k = bK +1, bK +2, . . . ,K (3.16)

if there exists at least one pair of servers u and v, where 1 u < v bK +1, such

that µu/eu > µv/ev.

Proof. The proof is straightforward and omitted here.

38


In particular, the decision variable bK provides a degree of freedom for E⇤ to

fine-tune its performance: The objective of E⇤ is to determine within the range

{2,3, . . . ,K} an optimal bK value, referred to as bK⇤, which maximizes the energy

efficiency.

3.5.3 Rate-Matching

Our proposed rule of thumb, RM, to determine the value bK⇤ of E⇤ simply picks up

an integer bKRM for which the aggregated service rate of the virtual server largest but

still lower than the average job arrival rate. More specifically, bKRM is chosen to be

the largest bK satisfyingbKÂj=1

µ j l .

We provide an intuition for this simple rule of thumb with the help of the

numerical results presented in Fig. 3.2. In this particular example, we have ten

servers. The server speeds and the energy consumption rates are randomly generated.

The job arrival rate is set to be the sum of the service rates of the six most energy-

efficient servers. The results in Fig. 3.2 are presented in the form of relative difference

between E* and MEESF in terms of each corresponding performance measure,

namely, energy efficiency in Fig. 3.2a, job throughput in Fig. 3.2b, and power

consumption in Fig. 3.2c.

On the basis of these results in Fig. 3.2, we argue that a different value of bK

selected for E⇤ other than bKRM is likely to reduce the energy efficiency. This happens

for several reasons.

• On one hand, if we choose a value of bK larger than bKRM, the excessive service

capacity made available with the virtual server can only increase the job

throughput marginally, but can substantially increase the power consumption

since it now uses more of those less energy-efficient servers. As a result, the

energy efficiency (defined in our context as the ratio of job throughput to power

39


consumption) is likely to decrease with an increasing value of bK in the range

{bKRM, bKRM +1, . . . ,K].

• On the other hand, if we choose a value of bK smaller than bKRM, the aggregated

service rate of the virtual server is not high enough to support the input

traffic in the long run. As a result, incoming jobs will be queued up and we

are forced to use more of those less energy-efficient servers for serving the

backlog. This will again increase power consumption. Since it can be shown

that the job throughput decreases with decreasing bK, accordingly the energy

efficiency is also likely to decrease with a decreasing value of bK in the range

{bKRM, bKRM�1, . . . ,2}.

3.5.4 Insensitive MEESF Policy

The insensitive MEESF policy specifies the set of servers T MEESF(n) designated

for serving existing jobs in the system at state n to be

T MEESF(n) = {1, . . . , i}, 1 n eBK (3.17)

where i K is the smallest integer satisfying Âij=1 B j � n. The server sets specified

in (3.17) indeed define the MEESF policy, thereby the available servers that are most

energy efficient are designated for service in state n.

The position mapping QMEESF of MEESF is defined iteratively as follows. The

B1 positions of server 1 (the most energy-efficient server) are mapped to the first

B1 positions of the logically-combined queue that are associated with server 1. The

B2 positions of server 2 (the second most energy-efficient server) are mapped to

the following B2 positions of the logically-combined queue that are associated with

server 2. The procedure continues until the BK positions of server K have been

mapped.

40


2 4 6 8 10−40

−20

0

20

K

Rel

ativ

e di

ffere

nce

(%)

(a) Energy efficiency.

2 4 6 8 100

0.6

1.2

1.8 x 10−3

K

Rel

ativ

e di

ffere

nce

(%)

(b) Job throughput.

2 4 6 8 10−20

0

20

40

60

K

Rel

ativ

e di

ffere

nce

(%)

(c) Power consumption.

Fig. 3.2 Illustration of optimizing bK for the E* policy.

Under the condition where all servers in the system have the same buffer size, i.e.,

B1 = B2 = . . .= BK , it can be shown that SSF maximizes the ratio of instantaneous

job departure rate to instantaneous energy consumption rate.

Proposition 3.3. For the system defined in Section 2.1, among all insensitive jock-

eying policies f 2 Fs, if B1 = B2 = . . . = BK and the power function follows the

cubic assumption, i.e. e j = µ

3j for all j = 1,2, . . . ,K, then the MEESF maximizes the

ratio of instantaneous job departure rate to instantaneous energy consumption rate,

41


namely,Â

j2T MEESF(n)µ j

Âj2T MEESF(n)

e j�

Âj2T f (n)

µ j

Âj2T f (n)

e j(3.18)

for any f 2Fs and n 2N .

Proof. Let L(n) represent the set of busy servers at state n, and assume without loss

of generality that the servers in the system have been ordered in descending order

with respect to their energy efficiencies. For a given number of jobs in the system n,

we use Sf (n) to represent the set of all reachable state vector n with |n|=ÂKj=1 n j = n

under policy f . In particular, according to the definition of MEESF, in our system

with jockeying, SMEESF(n) contains only one element. For clarity of presentation,

we denote the only state vector in SMEESF(n) by nMEESF(n).

We define the set L(n) = { j /2 L(n), j = 1,2, . . . ,K}. Since B1 = B2 = . . . =

BK = b, the minimal number of busy servers for a given number of jobs in the

system n is dn/be, which is equivalent to the number of elements in L(nMEESF(n)).

Hence, for any policy f and any state vector n0 2 Sf (n), the number of elements in

L(nMEESF(n))\L(n0) is greater than that in L(nMEESF(n))\ L(n0). Assume with-

out loss of generality, L(nMEESF(n))\L(n0) = {i1, i2, . . . , im} and L(nMEESF(n))\

L(n0) = { j1, j2 . . . , jm0}, m � m0. Also, according to the definition of MEESF, for

any j 2 L(nMEESF(n)) and j0 2 L(nMEESF(n)), we have µ j µ j0 .

Under the cubic assumption, for any policy f and any state vector n0 2 Sf (n),

for any i 2 L(nMEESF(n))\L(n0) and j 2 L(nMEESF(n))\ L(n0) satisfying µi 6= µ j,

we obtainµi�µ j

ei� e j=

1µ

2i +µ

2j +µiµ j

1µ

2i=

µi

ei, (3.19)

and

µi

ei

Âj2L(nMEESF(n))

µ j

Âj2L(nMEESF(n))

e j. (3.20)

42


Inequality (3.20) can be shown to hold by induction on the number of elements in

L(nMEESF).

Based on (3.19), (3.20) and the cubic assumption, if

Âj2L(nMEESF(n))\L(n0)

e j 6= Âj2L(nMEESF(n))\L(n0)

e j, (3.21)

then,

Âj2L(nMEESF(n))

µ j

Âj2L(nMEESF(n))

e j�

m0

Âk=1

(µik�µ jk)+mÂ

k=m0+1µik

m0Â

k=1(eik� e jk)+

mÂ

k=m0+1eik

=


µ j� Âj2L(nMEESF(n))\L(n0)

µ j


e j� Âj2L(nMEESF(n))\L(n0)

e j� 0. (3.22)

Therefore, if (3.21) holds true,

Âj2L(nMEESF(n))

µ j

Âj2L(nMEESF(n))

e j

�Â

j2L(nMEESF(n))µ j + Â

j2L(nMEESF(n))\L(n0)µ j� Â

j2L(nMEESF(n))\L(n0)µ j

Âj2L(nMEESF(n))

e j + Âj2L(nMEESF(n))\L(n0)

e j� Âj2L(nMEESF(n))\L(n0)

e j

=

Âj2L(n0)

µ j

Âj2L(n0)

e j. (3.23)

If (3.21) does not hold true, then (3.18) straightforwardly holds true. Hence, for any

reachable state vector n, (3.18) holds true. This proves the proposition.

43


Table 3.3 The Counter Example for the Instantaneous Optimality of MEESF

j µ j e j µ j/e j B j

1 1 2 0.5 22 1 4 0.25 23 0.25 1.1 5/22 ( < 0.25) 4

The power of three used here in the power function is not a very restrictive

assumption and is used only for simplicity of exposition to illustrate the instantaneous

energy efficiency attribute of the MEESF policy.

In general, however, this is not the case. We give a counter-example for the

instantaneous optimality of MEESF for general buffer sizes and general power

function. The counter-example for (3.18), in the general case with heterogeneous

buffer size and without the cubic assumption, is given as follows.

Consider a system with K = 3, and use the other parameters in Table 3.3. When

there are four jobs in the system, MEESF will use the first two servers and the

ratio of the instantaneous service rate to the instantaneous energy consumption rate

(the instantaneous energy efficiency) of the system is 13 ⇡ 0.3333. While, if we use

servers 1 and 3 to serve these jobs, the ratio will become 2562 ⇡ 0.4032 > 1/3.

Moreover, if we consider the state with six jobs in the system, then all three

servers will be busy under MEESF where the instantaneous energy efficiency is

2.257.1 ⇡ 0.3169. However, if we use servers 1 and 3, which have sufficient capacity

for the six jobs since B1 + B3 = 6, then the instantaneous energy efficiency is

2563 ⇡ 0.4023 > 2.25/7.1.

MEESF does not necessarily maximize the ratio of long-run average job departure

rate (i.e., job throughput) to long-run average energy consumption rate (i.e., power

consumption). This observation motivates us to design the more robust E* policy.

In [21], a special case of MEESF, when the cubic power function is assumed,

referred to as the Slowest-Server-First policy (SSF), has been numerically demon-

44


0 100 200 300 400 5000

0.3

0.6

0.9

1.2

Arrival rate

Rel

ativ

e di

ffere

nce

(%)

E*MEESF

(a)

0 100 200 300 400 5000

0.3

0.6

0.9

1.2

Arrival rate

Rel

ativ

e di

ffere

nce

(%)

(b)

Fig. 3.3 Energy efficiency of E⇤ and MEESF versus arrival rate.

strated to be close to an optimal solution that maximizes the energy efficiency of the

system.

Here, in Fig. 3.3, we present results for both the MEESF and E⇤ policies that

show their relative difference of energy efficiency versus the average arrival rate of

the system. In Fig. 3.3a, the y-axis represents the relative difference of both policies

to the optimal solution obtained by the algorithm in [21] under the cubic assumption.

Correspondingly, Fig. 3.3b gives the relative difference of energy efficiency of E⇤ to

MEESF for the same running of the experiment. For the sake of clarity, for policies

f1 and f2, we define

G(f1,f2) =

Lf1

Ef1� L

f2E

f2L

f2E

f2

. (3.24)

Then, the relative difference of energy efficiency of E⇤ to MEESF is G(E⇤,MEESF),

and the relative difference of energy efficiency of policy f to the optimal solution is

G(f ,OPT), where OPT represents the optimal solution.

In Fig. 3.3, the parameters are set as K = 50, µ j = 0.1 j, e j = µ

3j , and B j =

10, j = 1,2, . . . ,K. In Fig. 3.3a (Fig. 3.3b), it can be observed that both MEESF

and E⇤ policies are close to the optimal solution with relative difference smaller

45


0 10 20 30 40 500

0.2

0.4

0.6

Buffer size

Rel

ativ

e di

ffere

nce

(%)

E*MEESF

(a)

0 10 20 30 40 500

0.2

0.4

0.6

Buffer size

Rel

ativ

e di

ffere

nce

(%)

(b)

Fig. 3.4 Energy efficiency of E⇤ and MEESF versus buffer size.

than 1.2% and 0.3%, respectively, and that E⇤ always outperforms MEESF. These

two observations are consistent with the results in [21] and the argument given in

this chapter. In other words, MEESF approximates an optimal solution under the

cubic assumption, and in that case E* still outperforms MEESF in terms of energy

efficiency.

In Fig. 3.4, for the same parameters except r is fixed to 0.8, we present the

relationship between the relative energy efficiencies of E⇤ and MEESF versus the

buffer size of each server, where the buffer size is the same for all servers. Fig. 3.4a

gives the relative difference of E⇤ and MEESF to the optimal solution and Fig. 3.4b is

the relative difference of energy efficiency of E⇤ to MEESF. The trend of the curves

in Fig. 3.4 are similar to those in Fig. 3.3, namely, MEESF is close to the optimal

solution and E⇤ outperforms MEESF for all points in the figures.

In Fig. 3.5, we also present the cumulative distribution of the relative dif-

ference of energy efficiency of E⇤ to MEESF. With K = 50, we set e j = µ

3j

and B j, j = 1,2, . . . ,K. B1,B2, . . . ,BK are uniformly randomly generated from

{10,11, . . . ,15}, while the service rates µ1,µ2, . . . ,µK are uniformly randomly gen-

erated from [0.1,10]. Here, MEESF is equivalent to the SSF proposed in [21].

46


0 0.5 1 1.5 20

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

11

Relative difference (%)

Cum

ulat

ive

dist

ribut

ion

(a) r = 0.4

0 0.5 1 1.5 20

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1


Cum

ulat

ive

dist

ribut

ion

(b) r = 0.6

0 0.5 1 1.5 20

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

11


Cum

ulat

ive

dist

ribut

ion

(c) r = 0.8

Fig. 3.5 Relative difference of energy efficiency of E⇤ to MEESF.

In addition, in Fig. 3.5, we observe that the superiority of E⇤ over MEESF is

limited under these cubic-assumption cases which is consistent with the argument

in [21]. Also, in Fig. 3.5, E⇤ achieves higher energy efficiency than MEESF for all

cases with randomly generated parameters under the cubic assumption. In addition,

for the three traffic intensities, i.e. the ratio of the average arrival rate to the maximal

total service rate of the system, the cumulative distribution is stable. In other words,

the relative performance of E⇤ to MEESF is not very sensitive to the normalized

system load when the power function is assumed to follow the cubic assumption.

47


0 10 20 30 40 50 60 70 80 90 1000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

11


Cum

ulat

ive

dist

ribut

ion

Case 1Case 2Case 3

(a) r = 0.4

0 10 20 30 40 50 60 70 80 90 1000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1


Cum

ulat

ive

dist

ribut

ion

Case 1Case 2Case 3

(b) r = 0.6

0 10 20 30 40 50 60 70 80 90 1000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

11


Cum

ulat

ive

dist

ribut

ion

Case 1Case 2Case 3

(c) r = 0.8

Fig. 3.6 Relative difference of energy efficiency of E⇤ to MEESF.

3.5.5 Insensitive E⇤ Policy

The cubic assumption in [21] is far away from reality. In the following, we consider a

more general scenario by dropping the constraint of cubic power function. We argue

that, although MEESF approximates an optimal solution in the cubic-assumption

case, the performance of MEESF may be significantly improved in a general case

with independently randomly generated service rates and energy consumption rates

in terms of energy efficiency.

In Fig. 3.7, the cumulative distributions of the relative difference of energy

efficiency are presented. In Fig. 3.7, there are 10 server groups, where the servers in

48


−50 −45 −40 −35 −30 −25 −20 −15 −10 −5 00

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

11


Cum

ulat

ive

dist

ribut

ion

Case 1Case 2Case 3

(a) r = 0.4

−50 −45 −40 −35 −30 −25 −20 −15 −10 −5 00

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

11


Cum

ulat

ive

dist

ribut

ion

Case 1Case 2Case 3

(b) r = 0.6

−50 −45 −40 −35 −30 −25 −20 −15 −10 −5 00

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

11


Cum

ulat

ive

dist

ribut

ion

Case 1Case 2Case 3

(c) r = 0.8

Fig. 3.7 Relative difference of power consumption of E⇤ to MEESF.

the same server group have the same service rate, power consumption and PS buffer

size that are denoted by µi, ei and Bi, i = 1,2, . . . ,10, respectively. We uniformly

randomly generate µi, i = 1,2, . . . ,10, and the ratios ri, i = 2,3, . . . ,10, from [0.1,10]

and [0.1,1], respectively, where the service rates are ordered descendingly. With

µ1/e1 = 100, the energy efficiency of server group i is set to be µi/ei = riµi�1/ei�1,

r1.2i µi�1/ei�1, and r1.4

i µi�1/ei�1, i = 2,3, . . . ,10, for Case 1, Case 2, and Case 3 in

Fig. 3.7, respectively. These three cases represent three levels of server heterogeneity.

The buffer sizes Bi, i = 1,2, . . . ,10, are also uniformly randomly generated from

{10,11, . . . ,15}. These settings can be justified in that a newer server is likely to

49


have higher service rate and higher energy efficiency. Also, a server farm vendor

would likely purchase many servers with the same brand and same style at any given

time, and those servers would behave identically.

Compare to Fig. 3.5 with cubic power function, the relative differences in Fig. 3.7

are much more significant, namely, E⇤ substantially improves the energy efficiency

of MEESF without assuming a cubic power function. From Fig. 3.7, the degradation

of MEESF in terms of energy efficiency can be more than 100%. In Fig. 3.7c, in

around 27% of the experiments for Case 3, E⇤ improves MEESF by more than 10%

in terms of energy efficiency.

In Fig. 3.7, from Case 1 to Case 3, as the server heterogeneity becomes higher,

the superiority of E⇤ over MEESF also increases. As we recall our proof, if all the

servers are equivalent to each other, then MEESF will be equivalent to E⇤ in terms

of energy efficiency. We therefore argue that the increasing heterogeneity among

servers leads to increasing superiority of E⇤ over MEESF. This argument about the

continuous trend of energy efficiency versus server heterogeneity is consistent with

the spirit of this proof.

According to Fig. 3.7, the proportion of experiments where MEESF achieves

higher energy efficiency or lower power consumption than those of E⇤ can be

ignored. Then we also claim that E⇤ usually outperforms MEESF and can sig-

nificantly improve MEESF’s performance with respect to energy efficiency and

power consumption with randomly generated parameters. In addition, by varying

the normalized system load, the superiority of E⇤ over MEESF is increasing as the

normalized system load r increases.

The insensitive E* policy is derived by specifying the set of servers T E⇤(n)

designated for serving the existing jobs in the system at state n as

T E⇤(n) =

8

>

<

>

:

{1, . . . ,min(n, bK)}, 1 n eBbK�1

{1, . . . , i}, eBbK n eBK

(3.25)

50


where i K is the smallest integer satisfying Âij=1 B j � n. The server sets specified

in (3.25) indeed define the E* policy, in which preference is always given to the

virtual server for maximally utilizing its service capacity at any state n.

The position mapping QE⇤ of E* is defined iteratively as follows. In the first

iteration, the first buffer positions of servers 1,2, . . . , bK are mapped to the first bK

positions of the logically-combined queue in the order of the server labels 1,2, . . . , bK,

inheriting their original server speeds and energy consumption rates. In every

subsequent iteration until all the positions of the first bK servers have been mapped,

the next remaining position levels from the remaining buffers, say m bK positions,

are mapped to the next m positions of the logically-combined queue in the order

of the server labels. The BbK+1 positions of server bK + 1 are then mapped to the

next BbK+1 positions of the logically-combined queue that are associated with server

bK +1. The BbK+2 positions of server bK +2 are mapped to the next B

bK+2 positions of

the logically-combined queue that are associated with server bK +2. The iterations

terminate when all the positions of all server buffers have been mapped.

Here, we provide a rigorous analysis of the E* policy. First, we show that E*

is guaranteed to outperform MEESF in terms of job throughput. Then, we derive

conditions under which E* is guaranteed to outperform MEESF in terms of energy

efficiency.

Remark 3.2. By the nature of the two policies, we have µ

E⇤(n) � µ

MEESF(n) for

1 n eBK.

Proposition 3.4. For the stochastic job assignment problem studied in this chapter,

we have

L E⇤ �L MEESF. (3.26)

Proof. Using (3.6), we obtain the job throughput of the system under E* and that

under MEESF as

L E⇤ = l

h

1�p

E⇤(eBK)i

(3.27)

51


and

L MEESF = l

h

1�p

MEESF(eBK)i

, (3.28)

respectively. From (3.4), we derive for E* that

p

E⇤(n) =eBK

’i=n+1

µ

E⇤(i)l

p

E⇤(eBK), 0 n eBK�1. (3.29)

By normalization, we have

eBK�1

Ân=0

eBK

’i=n+1

µ

E⇤(i)l

p

E⇤(eBK)+p

E⇤(eBK) = 1, (3.30)

hence

p

E⇤(eBK) =1

ÂeBK�1n=0 ’

eBKi=n+1

µ

E⇤(i)l

+1. (3.31)

Likewise, for MEESF, we get

p

MEESF(eBK) =1

ÂeBK�1n=0 ’

eBKi=n+1

µ

MEESF(i)l

+1. (3.32)

It follows from Remark 3.2 that

eBK

’i=n+1

µ

E⇤(i)l

�eBK

’i=n+1

µ

MEESF(i)l

(3.33)

for 0 n eBK�1. Therefore, we have

p

E⇤(eBK) p

MEESF(eBK) (3.34)

and follows (3.26).

52


Now, we consider energy efficiency. For convenience, let

P(n) =p

E⇤(n)p

E⇤(eBK), 0 n eBK. (3.35)

It is plain that

P(n) =

8

>

>

>

<

>

>

>

:

eBK

’i=n+1

µ

E⇤(i)l

, 0 n eBK�1

1, n = eBK.

(3.36)

Similarly, let

P0(n) =p

MEESF(n)p

MEESF(eBK), 0 n eBK, (3.37)

so that

P0(n) =

8

>

>

>

<

>

>

>

:

eBK

’i=n+1

µ

MEESF(i)l

, 0 n eBK�1

1, n = eBK.

(3.38)

Lemma 3.1. For P(n) and P0(n) defined by (3.36) and (3.38), respectively, we have

eBK

Ân=eBx�1+1

P(n)eBK

Ân0=eBy�1+1

P0(n0)�eBK

Ân=eBy�1+1

P(n)eBK

Ân0=eBx�1+1

P0(n0)� 0 (3.39)

for any two integers x and y such that 1 x < y K.

Proof. Note that

eBK

Ân=eBx�1+1

P(n) =eBy�1

Ân=eBx�1+1

P(n)+eBK

Ân=eBy�1+1

P(n) (3.40)

andeBK

Ân0=eBx�1+1

P0(n0) =eBy�1

Ân0=eBx�1+1

P0(n0)+eBK

Ân0=eBy�1+1

P0(n0). (3.41)

53


Thus, proving (3.39) is equivalent to proving

eBy�1

Ân=eBx�1+1

P(n)eBK

Ân0=eBy�1+1

P0(n0)�eBK

Ân=eBy�1+1

P(n)eBy�1

Ân0=eBx�1+1

P0(n0)� 0. (3.42)

It suffices to show that, for any two integers l and m where eBx�1 +1 l eBy�1

and eBy�1 +1 m eBK , we have P(l)P0(m)�P(m)P0(l)� 0, or equivalently,

P(l)P(m)

=m

’i=l+1

µ

E⇤(i)l

� P0(l)P0(m)

=m

’i=l+1

µ

MEESF(i)l

. (3.43)

The inequality of (3.43) clearly holds since by Remark 3.2 we have µ

E⇤(i) �

µ

MEESF(i) for all i in the defined range.

Proposition 3.5. A sufficient condition for

L E⇤

E E⇤ �L MEESF

E MEESF (3.44)

to hold is thatµ j

e j=

µ1

e1, j = 2,3, . . . , bK. (3.45)

Proof. From (3.7), we derive for E* that

E E⇤ =eBK

Ân=1

e

E⇤(n)pE⇤(n)

=bK

Ân=1

p

E⇤(n)n

Âj=1

e j +

eBbK

Ân=bK+1

p

E⇤(n)bK

Âj=1

e j +K

Âi=bK+1

eBi

Ân=eBi�1+1

p

E⇤(n)i

Âj=1

e j. (3.46)

54


Interchanging the summations in (3.46), we obtain

E E⇤ =bK

Âj=1

e j

bK

Ân= j

p

E⇤(n)+bK

Âj=1

e j

eBbK

Ân=bK+1

p

E⇤(n)+

0

@

bK

Âj=1

e j

K

Âi=bK+1

+K

Âj=bK+1

e j

K

Âi= j

1

A

eBi

Ân=eBi�1+1

p

E⇤(n)

=bK

Âj=1

e j

eBK

Ân= j

p

E⇤(n)+K

Âj=bK+1

e j

eBK

Ân=eB j�1+1

p

E⇤(n).

(3.47)

Since B j � 1 for all j = 1,2 . . . ,K, we have eB j�1 +1� j for 1 j bK, and we can

rewrite the elements of E E⇤ in (3.47) as

E E⇤ =bK

Âj=1

e j

eB j�1

Ân= j

p

E⇤(n)+bK

Âj=1

e j

eBK

Ân=eB j�1+1

p

E⇤(n)+K

Âj=bK+1

e j

eBK

Ân=eB j�1+1

p

E⇤(n)

=bK

Âj=1

e j

eB j�1

Ân= j

p

E⇤(n)+K

Âj=1

e j

eBK

Ân=eB j�1+1

p

E⇤(n).

(3.48)

Similar to the way we derive the expression of E E⇤ in (3.48), we obtain L E⇤ as

L E⇤ =eBK

Ân=1

µ

E⇤(n)pE⇤(n) =bK

Âj=1

µ j

eB j�1

Ân= j

p

E⇤(n)+K

Âj=1

µ j

eBK

Ân=eB j�1+1

p

E⇤(n), (3.49)

so that

L E⇤

E E⇤ =L E⇤/p

E⇤(eBK)

E E⇤/p

E⇤(eBK)=

ÂbKj=1 µ j Â

eB j�1n= j P(n)+ÂK

j=1 µ j ÂeBKn=eB j�1+1

P(n)

ÂbKj=1 e j Â

eB j�1n= j P(n)+ÂK

j=1 e j ÂeBKn=eB j�1+1

P(n). (3.50)

On the other hand, from (3.7), for the MEESF, we derive that

E MEESF =eBK

Ân0=1

e

MEESF(n0)pMEESF(n0) =K

Âi0=1

eBi0

Ân0=eBi0�1+1

p

MEESF(n0)i0

Âj0=1

e j0 . (3.51)

55


Interchanging the summations in (3.51), this time we obtain

E MEESF =K

Âj0=1

e j0K

Âi0= j0

eBi0

Ân0=eBi0�1+1

p

MEESF(n0) =K

Âj0=1

e j0

eBK

Ân0=eB j0�1+1

p

MEESF(n0). (3.52)

Similar to the way we derive the expression of E MEESF in (3.52), we obtain L MEESF

as

L MEESF =eBK

Ân0=1

µ

MEESF(n0)pMEESF(n0) =K

Âj0=1

µ j0

eBK

Ân0=eB j0�1+1

p

MEESF(n0), (3.53)

and then we obtain

L MEESF

E MEESF =L MEESF/p

MEESF(eBK)

E MEESF/p

MEESF(eBK)=

ÂKj0=1 µ j0Â

eBKn0=eB j0�1+1

P0(n0)

ÂKj0=1 e j0Â

eBKn0=eB j0�1+1

P0(n0). (3.54)

Given L E⇤/E E⇤ in the form of (3.50) and L MEESF/E MEESF in the form of

(3.54), for the inequality of (3.44) to hold, it requires

bK

Âj=1

K

Âj0=1

µ je j0

eB j�1

Ân= j

P(n)eBK

Ân0=eB j0�1+1

P0(n0)+K

Âj=1

K

Âj0=1

µ je j0

eBK

Ân=eB j�1+1

P(n)eBK

Ân0=eB j0�1+1

P0(n0)

�bK

Âj=1

K

Âj0=1

e jµ j0

eB j�1

Ân= j

P(n)eBK

Ân0=eB j0�1+1

P0(n0)+K

Âj=1

K

Âj0=1

e jµ j0

eBK

Ân=eB j�1+1

P(n)eBK

Ân0=eB j0�1+1

P0(n0).

(3.55)

First, we show that

K

Âj=1

K

Âj0=1

µ je j0

eBK

Ân=eB j�1+1

P(n)eBK

Ân0=eB j0�1+1

P0(n0)�K

Âj=1

K

Âj0=1

e jµ j0

eBK

Ân=eB j�1+1

P(n)eBK

Ân0=eB j0�1+1

P0(n0).

(3.56)

In particular, we observe in (3.56) that:

56


• For j = j0, we have

µ je j0

eBK

Ân=eB j�1+1

P(n)eBK

Ân0=eB j0�1+1

P0(n0) = e jµ j0

eBK

Ân=eB j�1+1

P(n)eBK

Ân0=eB j0�1+1

P0(n0).

(3.57)

• For any two integers x and y where 1 x < y K, we have

µxey

eBK

Ân=eBx�1+1

P(n)eBK

Ân0=eBy�1+1

P0(n0)+µyex

eBK

Ân=eBy�1+1

P(n)eBK

Ân0=eBx�1+1

P0(n0)

� exµy

eBK

Ân=eBx�1+1

P(n)eBK

Ân0=eBy�1+1

P0(n0)� eyµx

eBK

Ân=eBy�1+1

P(n)eBK

Ân0=eBx�1+1

P0(n0)

= (µxey� exµy)

"

eBK

Ân=eBx�1+1

P(n)eBK

Ân0=eBy�1+1

P0(n0)�eBK

Ân=eBy�1+1

P(n)eBK

Ân0=eBx�1+1

P0(n0)

#

� 0,(3.58)

where the final inequality follows from Remark 3.1 and Lemma 3.1.

Now, for the inequality of (3.55) to hold, it remains to find a sufficient condition

yielding

bK

Âj=1

K

Âj0=1

µ je j0

eB j�1

Ân= j

P(n)eBK

Ân0=eB j0�1+1

P0(n0)�bK

Âj=1

K

Âj0=1

e jµ j0

eB j�1

Ân= j

P(n)eBK

Ân0=eB j0�1+1

P0(n0).

(3.59)

In (3.59), we observe that:

• For j = j0, µ je j0 = e jµ j0 .

• For j = 1,2, . . . , bK and j0 = j+1, j+2, . . . ,K, µ je j0 � e jµ j0 � 0.

• For j = 2,3, . . . , bK and j0 = 1,2, . . . , j�1, µ je j0 � e jµ j0 0.

57


Therefore, for the inequality of (3.59) to hold, it is sufficient to have (3.45), which

enforces

µ je j0 � e jµ j0 = 0, j = 2,3, . . . , bK, j0 = 1,2, . . . , j�1. (3.60)

This completes the proof.

From Proposition 3.5, we can obtain the following two corollaries.

Corollary 3.1. If (3.44) is satisfied with parameters K = K⇤, bK = bK⇤, l = l

⇤,

B j = B⇤j , µ j/e j = µ

⇤j /e

⇤j , j = 1, . . . ,K⇤, then (3.44) still holds true for all K � K⇤

with bK = bK⇤, l = l

⇤, B j = B⇤j , µ j/e j = µ

⇤j /e

⇤j , j = 1, . . . ,K⇤.

Corollary 3.2. If µ j/e j = c, j = 1,2, . . . ,K, then we have

L E⇤

E E⇤ =L MEESF

E MEESF (3.61)

for each bK = 1, . . . ,K.

Corollary 3.3. If e j/µ j = e1/µ1, j = 1,2, . . . ,k where k = 1,2, . . . ,K, then we have

L E⇤

E E⇤ �L MEESF

E MEESF (3.62)

for each bK = 1,2, . . . ,k.

Corollary 3.2 suggests that, if all servers in the system are equally energy efficient,

the energy efficiency of E* is equivalent to that of MEESF. Nevertheless, even in

the case of a homogeneous server farm, E* has a higher job throughput than that

of MEESF. On the other hand, Corollary 3.3 suggests that, if at least two servers

in a heterogeneous server farm are both most energy-efficient, E* is guaranteed

to outperform MEESF in terms of energy efficiency. We argue that the latter is a

realistic scenario since in practice a server farm is likely to comprise multiple servers

of the same type purchased at a time.

58


Moreover, for scenarios where µ1/e1 > µ2/e2, we have the following theoretical

results. We derive an extended policy (denoted by E⇤k

) that is the same as E⇤, except

that a new arrival will not be assigned to the second server until the queue size of the

first server is no less than k for some k = 1,2, . . . ,B1. That is, under E⇤k

, the second

server is idle if the queue size of the first server is no larger than k . Hence, E⇤ is a

special case of E⇤k

for k = 1.

Proposition 3.6. If K = 4, B j � 2, j = 1,2,3,4, (p

2�1)/2µ3 � µ2 � µ1, e3/µ3 �p

2e2/µ2 and l � µ1+µ2+µ3, then we have L E⇤/E E⇤ �L MEESF/E MEESF where

bK = 2 and k = 1,2, . . . ,B1.

Proof. See Appendix A.9.

Based on Proposition 3.6 and Corollary 3.1, we have the following corollary.

Corollary 3.4. If K � 4, B j � 2, j = 1,2, . . . ,K, (p

2�1)/2µ3 � µ2 � µ1, e3/µ3 �p

2e2/µ2 and l � µ1+µ2+µ3, then we have L E⇤/E E⇤ �L MEESF/E MEESF where

bK = 2.

Next, we will analyze the feasible region of bK, denoted by FbK , which is defined

to be

FbK =

(

bK :L E⇤

E E⇤ �L MEESF

E MEESF , bK = 2,3, . . . ,K

)

. (3.63)

We do not consider bK = 1 since E⇤ with bK = 1 is equivalent to MEESF as discussed

before.

For any policy f 2Fs, we define

pf

n =p

f (n)L f

µ

f (n), n 2N (3.64)

and

qf

n =pf

n

pf

eBbK

, n 2N . (3.65)

59


We have ÂeBKn=0 pf

n = 1, based on the definition of pf

n , n = 0,1, . . . , BK .

To formulate an optimization problem, we rewrite our objective function E f/L f

as a weighted sum of decision variables.

E f

L f

=eBK

Ân=0

Ff (n) · pf

n , (3.66)

where Ff (n) = e

f (n)/µ

f (n). We refer to pf

n as the virtual probability of state n in

steady state. We use the word virtual to indicate that these quantities are not the

real steady-state probabilities for the original system defined in Section 3.2. Then,

the virtual probability of the system’s total service rate µ is denoted by pf (µ) and

defined as

pf (µ) = Ân:µf (n)=µ

pf

n . (3.67)

As the set of service rates is an attribute of the system which is independent of the

policy used, this set is common to all the E⇤ policies, including MEESF, considered

here. We denote this set of service rates by R = {0,r1,r2, . . . ,rK} where

r j =j

Âi=1

µi.

For E⇤ and MEESF, the total power consumption of the system is uniquely de-

termined by its total service rate, i.e., the corresponding virtual probability of the

system’s total power consumption is exactly the same as the system’s total service

rate. We rewrite (3.66) as

E f

L f

= Âµ2R

Ff (µ)pf (µ) (3.68)

where Ff (µ) is the ratio of the total power consumption to total service rate of the

system when the total service rate is µ; in our case, f 2 {MEESF,E⇤}. Note that

f here is not strongly restricted to MEESF and E⇤, but to the set of policies where

60


Fig. 3.8 Virtual probability of service rate.

the total power consumption can be uniquely determined by a given value of the

total service rate. Then, Fig. 3.8 can also represent the virtual probabilities of the

power consumption. Our objective is then equivalent to minimizing the average

value of the ratio of the total power consumption to the total service rate based on

the virtual probabilities. Neither (3.66) nor (3.68) is amenable to a solution by linear

programming because the corresponding constraints, which are related to decision

variables µ

f (n), are non-linear.

In Fig. 3.8, we present the virtual probabilities versus system total service rates

under MEESF and E⇤, where K = 10, B j = 10, j = 1,2, . . . ,10, bK = 5, l = Â5j=1 µ j

and µ j and e j are uniformly randomly generated in [0.1,10] and [0.1,100], respec-

tively. In this figure, we split R into the following three mutually exclusive subsets.

RA = {0,r1,r2, . . . ,rbK�1},

RB = {rbK},

RC = {rbK+1,rbK+2, . . . ,rK},

(3.69)

where RA[RB[RC = R. We define A, B and C as the time period during which

the total service rate of the system is in RA,RB and RC, respectively. For clarity of

61


presentation, we also define the state set for A, B and C as

N f

q

= {n 2N |µf (n) 2 Rq

}, q 2 {A,B,C}

µn,f is the total service rate of the system as defined before. The average ratio of

the total power consumption to the total service rate in each part based on virtual

probability is given by

Ff

q

=

Ân2N f

q

Ff (n)pf

n

Ân2N f

q

pf

n, q 2 {A,B,C} (3.70)

where Ff (n) is the ratio of the total power consumption to the total service rate in

state n under policy f .

Lemma 3.2. For the system defined in Section 3.2, if l � rbK�1 and B j+1 � B j � 1

for all j = 1,2, . . . , bK�2, then,

FMEESFA �FE⇤

A .


Lemma 3.3. For bK 2 {1,2, . . . ,K}, we have

FMEESFB = FE⇤

B (3.71)

and

FMEESFC = FE⇤

C . (3.72)


62


As defined in (3.63), the condition of the feasible region is L E⇤/E E⇤ �L MEESF/E MEESF�

0, and, by (3.68) and (3.70), this condition is equivalent to

Âq2{A,B,C}

"

FMEESFq

Âµ2R

q

pMEESF(µ)�FE⇤q

Âµ2R

q

pE⇤(µ)

#

� 0. (3.73)

We observe in Fig. 3.8 that the virtual probability of service rate r5 for E⇤ is higher

than that of MEESF, but for service rates greater than r5, the virtual probability

is higher under MEESF. In Fig. 3.8, we shadow the zones that represent the dif-

ferences of the virtual probabilities for the two policies. For these shadow zones,

dense-slash represents pMEESF(µ)< pE⇤(µ) and sparse-slash is for the cases where

pMEESF(µ) � pE⇤(µ). Based on the definition (3.64), the size of all the shadow

zones for the case of pMEESF(µ)< pE⇤(µ) will be equivalent to those for the com-

plementary cases, i.e.,

Âµ:pMEESF(µ)<pE⇤(µ)

h

pE⇤(µ)� pMEESF(µ)i

= Âµ:pMEESF(µ)�pE⇤(µ)

h

pMEESF(µ)� pE⇤(µ)i

.

(3.74)

From Fig. 3.8, we see that the virtual probabilities satisfy pMEESF(µ)< pE⇤(µ) for

RB, and pMEESF(µ)> pE⇤(µ) for RC, and that the virtual probabilities are near-zero

for RA.

Lemma 3.4. For bK 2 {1,2, . . . ,K}, we have

pMEESF(µ)� pE⇤(µ),µ 2 RC. (3.75)


Proposition 3.7. If l �p

5+12 r

bK�1,

Ân2N MEESF

A

pMEESFn < Â

n2N E⇤A

pE⇤n , (3.76)

63


then, {2,3, . . . ,K�1}✓FbK.


When pMEESF(r j)� pE⇤(r j) decreases rapidly as r j decreases from rbK�1 to 0,

then the condition (3.76) will hold true. The descent rate is strongly related to the

value of l/rbK , which should be made as high as possible. Although l can be infinite,

bK cannot be equal to K. If bK = K, the condition (3.76) can not hold true, and the

shadowed zones, as illustrated in Fig. 3.8 will diminish in C which leads to higher

energy efficiency for MEESF.

To find the feasible region FbK as defined in (3.63), define a set

fFbK =

(

2,3, . . . ,min

"

K�2,max

j :j+1

Âi=1

µi l

!#)

(3.77)

and give Proposition 3.8 as follows.

Proposition 3.8. For the system defined in Section 3.2, if K � 4 and B j � 3, j =

1,2, . . . ,K, then,

fFbK \ fF 0

bK ✓FbK.

where fF 0bK✓ {2,3, . . . ,K} and m 2 fF 0

bKif and only if the following conditions hold

true.

Condition 3.1. B1 B2 . . . Bm�1.

Condition 3.2.

(p

3+1)rm�1 rm p

2�12

rm+1

andm+1Âj=1

e j

m+1Âj=1

µ j

� 94

mÂj=1

e j

mÂj=1

µ j

.


64


0 10 20 30 40 50 60 70 80 90 1000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

11


Cum

ulat

ive

dist

ribut

ion

Case 1Case 2Case 3

(a) r = 0.4

0 10 20 30 40 50 60 70 80 90 1000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

11


Cum

ulat

ive

dist

ribut

ion

Case 1Case 2Case 3

(b) r = 0.6

0 10 20 30 40 50 60 70 80 90 1000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

11


Cum

ulat

ive

dist

ribut

ion

Case 1Case 2Case 3

(c) r = 0.8

Fig. 3.9 Relative difference of energy efficiency of E⇤ to RM.

Proposition 3.8 does not rely on Proposition 3.7 which is used to describe our

intuition for fFbK .

3.5.6 Approximate E⇤

In this section, we numerically demonstrate that the RM policy approximates the

policy E⇤ with optimal bK. In Fig. 3.9, we present the cumulative distribution of the

relative difference of energy efficiency between E⇤ and RM with the same settings

as those used in Fig. 3.7. The relative difference of energy efficiency of E⇤ to RM is

G(E⇤,RM). Results for RM in Figs. 3.9a and 3.9b are close to those for MEESF in

65


Figs. 3.7a and 3.7b. We nevertheless observe a clear improvement from MEESF in

Fig. 3.7c to RM in 3.9c; the relative difference of MEESF in Fig. 3.7c is higher than

5% for 40% of the experiments in Case 3 while, for RM in Fig. 3.9c, it is only in

17% of our experiments that the enery efficiency degrades from E⇤ to MEESF for

over 5%.

In Fig. 3.9, we observe that the values of the relative difference are less than

5% for over 78% of the experiments in all cases, which numerically demonstrate

the high accuracy of RM for approximating E⇤. Note that the curves with r = 0.4,

r = 0.6 and r = 0.8 in Figs. 3.9a-3.9c, respectively, are similar to each other. That

is, the accuracy of RM for approximating E⇤ is relatively stable with three different

traffic intensities, which is practical to a real environment with uncertain arrival rate.

In Fig. 3.10, we present the histogram of the difference between the values of bK

for E⇤ and RM. The settings are the ones already used in Fig. 3.9.

By observing Fig. 3.10, the difference between valus of bK for E⇤ and RM vary

within the range of {�2, . . . ,1}. That is, for a system with 50 servers, we only need

to explore at most four integers for an optimal bK. In addition, in Fig. 3.10, as the

normalized system load grows from 0.6 to 0.8, the range of the distance between

RM an E⇤ remains the same, and points to the stability of our system in a real

environment with varying l .

3.6 conclusion

We have proposed a new approach that gives rise to an insensitive job-assignment

policy for the popular server farm model comprising a parallel system of finite-buffer

PS queues with heterogeneous server speeds and energy consumption rates. Unlike

the straightforward MEESF approach that greedily chooses the most energy-efficient

servers for job assignment, an important feature of the more robust E* policy is to

aggregate an optimal number of most energy-efficient servers as a virtual server.

66

3.6 conclusion

−3 −2 −1 0 1 2 30

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Difference

Rel

ativ

e fre

quen

cy

Case 1Case 2Case 3

(a) r = 0.4

−3 −2 −1 0 1 2 30

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Difference

Rel

ativ

e fre

quen

cy

Case 1Case 2Case 3

(b) r = 0.6

−3 −2 −1 0 1 2 30

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Difference

Rel

ativ

e fre

quen

cy

Case 1Case 2Case 3

(c) r = 0.8

Fig. 3.10 Difference of the value of bK of E⇤ to RM.

The policy E* is designed to give preference to this virtual server and utilize its

service capacity in such a way that both the job throughput and the energy efficiency

of the system can be improved. We have provided a rigorous analysis of the E*

policy, and have shown that E* has always a higher job throughput than MEESF;

and that there exist realistic and sufficient conditions under which E* is guaranteed

to outperform MEESF in terms of the energy efficiency. We have further proposed

a rule of thumb to form the virtual server by simply matching its aggregate service

rate to the job arrival rate. Extensive experiments based on random settings have

confirmed the effectiveness of the resulting RM policy. In addition, the algorithm

67


proposed in [21] was shown to have polynomial computational complexity. The

optimal solution obtained by this algorithm was also shown to bw optimal for

general jockeying policies when job sizes are exponentially distributed. Despite the

polynomial complexity, the optimal policy was shown not to be sufficiently scalable

for a practical system with tens of thousands of servers.

68

Chapter 4

Job Assignment Heuristics without

Jockeying

The data-center industry, with over 500 thousand data centers worldwide [5], has

been growing in parallel with the dramatic increase in global Internet traffic. An

estimated 91 billion kWh of electricity was consumed by U.S. data centers in 2013,

and this consumption rate continues to grow, resulting in $13 billion annually for

electricity bills and potentially nearly 100 million metric tons of carbon pollution per

year by 2020 [7]. Servers account for the major portion of energy consumption of

data centers [8]. Our aim here is to describe optimal scheduling/dispatching strategies

for incoming job requests in a server farm so as to improve energy efficiency.

This has been a topic of interest for some time, with approaches such as speed

scaling to reduce energy consumption by controlling server speed(s) [10, 11, 16, 79,

80]. Right-sizing of server farms has been done by powering servers on/off accord-

ing to traffic load [17, 18, 45, 81], and by switching servers between active/sleep

mode according to number of waiting jobs [82]. In [20, 83] resource allocation

methodologies are used to distribute the power budget for energy conservation.

Rapid improvements in computer hardware have resulted in frequent upgrades

of parts of the server farms. This, in turn, has led to server farms with a range of

69

Job Assignment Heuristics without Jockeying

different computer resources (heterogeneous servers) being deployed [23]. Such

heterogeneity significantly complicates optimization, since each server needs to

be considered individually. Despite the complexity, we are able to improve here

the energy efficiency of heterogeneous server farm via appropriate scalable job

assignment policies that are applicable to server farms with tens of thousands of

servers. For the purposes of this chapter, a server farm is postulated to have a fixed

number of servers with no possibility of powering off during the time period under

consideration; this, in practice, could apply to periods during which no powering off

takes place. In this way, the job assignment policies described here can be combined

with the right-sizing techniques mentioned above, as appropriate. Note that frequent

powering off/on increases wear and tear and the need for costly replacement and

maintenance [84].

Here, we consider a system in which idle servers may have non-negligible energy

consumption rate [32]. In [16, 21, 22], job assignment policies have been discussed

without consideration of idle power (energy consumption rate of idle servers), and

in [17, 45, 85], such policies have been considered for a server pool with identical

servers. In the server farms considered, servers are not assumed to be identical. To

the best of our knowledge, there is no published work that considers idle power in

designing job assignment methodologies for a given heterogeneous server pool.

Other job assignment policies in the literature (e.g., [57, 59, 62]) have considered

scenarios with infinite buffer size and have sought to minimize delay. We consider a

server farm with parallel finite-buffer queues, and with heterogeneous server speeds,

energy consumption rates, and buffer sizes. As in [21], the energy efficiency of a

server farm is defined as the ratio of the long-run expected job departure rate divided

by the expected energy consumption rate. It forms the objective function of our

optimization strategy. This objective function represents the amount of useful work

(e.g., data rate, throughput, processes per second) per watt.

70

The processor sharing (PS) discipline is imposed on each queue, so that all jobs

on the same queue share the processing capacity and are served at the same rate.

The PS discipline avoids unfair delays for those jobs that are preceded by extremely

large jobs, making it an appropriate model for web server farms [24, 25], where

job-size distributions are highly variable [27, 86]. The finite buffer size queuing

model with PS discipline can be applied in situations where a minimum service

rate is required for processing a job in the system [87]. In communication systems,

broader applications of PS queues have been studied; e.g., [28, 29].

A key feature of our approach is to model the problem as an instance of the

Restless Multi-Armed Bandit Problem (RMABP) [33] in which, at each epoch, a

policy chooses a server to be tagged for a new job assignment (other servers are

said to be untagged). The general RMABP has been proved SPACE-hard [34], and

this has led to studies in scalable and near-optimal approximations, such as index

policies. An index policy selects a set of tagged servers at any epoch according to

their state-dependent indices.

We consider a large-scale realistically-dimensioned server farm that cannot reject

a job if it has buffer space available. Such a situation would occur, for instance,

where a server farm owner is unable to replace all older servers simultaneously, and

so legacy inefficient servers are needed to meet a service level agreement. Buffering

spill-over creates dependencies between servers, and requires us to postulate uncon-

trollable states [35]. In other words, the constraint on the number of tagged servers

in conventional RMABP has to be replaced by a constraint on the number of tagged

servers in controllable states. As far as we are aware, there are no theoretical results

on the asymptotic optimality of an index policy for a multi-server system with finite

buffer size, where loss of jobs happens if and only if all buffers are full. A further

discussion on existing related work is provided in Section 4.1.

As mentioned in Section 1.4, the main contribution of this chapter is listed as

follows.

71


• We propose a new job assignment method to attempt maximization of energy

efficiency. The new proposed policy is referred to as the Most energy-efficient

available server first Accounting for Idle Power (MAIP). MAIP prioritizes

the most energy-efficient servers that are available (that is, servers with at

least one vacant slot in their buffers) and requires the energy consumption

rate of idle servers as an input parameter for decision making. This policy

provides a model of a real system with significant energy consumption rate

in idle states. MAIP is scalable and requires only binary state information

of servers, making it suitable for an environment with frequently changing

server states. Our server farm model is centralized and is applicable to a

local system with frequently changing information, which for our case is the

binary information of server states. We note that Google has built a centralized

control mechanism for network routing and management that monitors all link

states and is scalable for Google’s building-scale data center [88].

• We prove, remarkably, that when job sizes are exponentially distributed, the

Whittle’s index policy is equivalent to MAIP, and that it is asymptotically

optimal for our server farm comprised of multiple groups of identical servers

as the numbers of servers in these groups tend to infinity. It is reasonable to

assume that if the total number of servers in a server farm is very large, then

the number of servers bought in a single batch, or over a short period of time

during which the technology is not improving, will also be large. In any case,

there is cost benefit in buying in bulk, so that the number of servers purchased

at once is likely to be large. More importantly, the typical lifespan of a server

is in the range of 3 years [89, 90]. Accordingly, a modern server farm is likely

to be categorized into several server groups, each of which contains a large

number of servers of the same or similar type and attributes that were bought

at the same time, or over a short time period. The well-known Whittle’s index

72

4.1 Related Work

relaxation enables decomposition of a complex RMABP problem into multiple

sub-problems, assumed computationally feasible [33]. Note that, in the general

case, Whittle’s index does not necessarily exist, and even if it does, a closed

form solution is often unavailable. As mentioned before and in Section 4.1, the

buffer constraint in our case enforces the need for uncontrollable states and, in

turn, prevents direct application of previous asymptotic optimality results on

RMABP to our problem.

• We demonstrate numerically the effectiveness of MAIP by comparing it with

a baseline policy that prioritizes the most energy-efficient available serves but

ignores idle power. Although, as mentioned above, powering off servers is not

considered here, the performance of the baseline policy can be significantly

improved by taking the idle power into consideration (MAIP) in terms of

energy efficiency. MAIP is demonstrated numerically to be almost insensitive

to the shape of job-size distributions. We also demonstrate the applicability of

MAIP for a large server farm with significant cost of job reassignment.

The remainder of this chapter is organized as follows. In Section 4.1, we proived

the related work for job assignment policies. In Section 4.2, we define our server

farm model. In Section 4.3, we propose the MAIP policy, and in Section 4.4, we

give the proof for the asymptotic optimality of MAIP. We present numerical results

in Section 4.5 and conclusions are given in Section 4.6.

4.1 Related Work

Queueing models associated with job assignment among multiple servers with and

without jockeying (reassignment actions of incomplete jobs) have been studied since

1958 [50]. Most existing work has focused on job assignment policies that aim to

73


improve the system performance under a first-come-first-served (FCFS) discipline

such as Join-the-Shortest-Queue (JSQ).

For the non-jockeying case, JSQ under PS has been analyzed in [24, 57–59].

Bonomi [57] proved optimality of JSQ for the processor sharing case under a general

arrival process, a Markov departure process, and homogeneous servers while, for a

non-exponential job-size distribution, a counter-example to optimality of JSQ has

been given by Whitt [58]. Gupta [59] provided an analysis for the approximation

of the state distribution of a system under JSQ with homogeneous PS servers and

general job-size distributions. Gupta also showed the optimality of JSQ in terms of

average delay for a system comprising servers with two different service rates.

Server farm applications of JSQ with jockeying policies for FCFS have been

studied in [60–62]. In these papers, when the difference between the longest and

shortest queue sizes achieves a threshold, a jockeying action is triggered. Different

values of the threshold clearly result in different JSQ policies. These publications

focus on the calculation of the equilibrium distribution of the lengths of queues.

Energy efficiency for a multi-queue heterogeneous system with infinite buffers

and set-up delay has been studied in [19], where the authors assume zero energy

consumption rate when a server is idle. Hyytiä et al. [19] have shown that the

M/G/1-LCFS is insensitive to the shape of the distribution of set-up delay while

this insensitivity is lost in the M/G/1-PS. In [91], energy-efficient job assignment

in a system with heterogeneous servers and homogeneous job has been considered,

where jobs were queued in an infinite public buffer without waiting room on each

server and no cost were consumed for an idle server. The authors analyzed the

coinciding of individual optimality (minimizing the cost of one job) and social

optimality (minimize the sum of the cost of all jobs) that were, both proved in this

paper, threshold style in certain situations.

Energy-aware PS multi-queue systems with jockeying have been studied in

[21, 22], where the optimization problem is characterized as a semi-Markov decision

74

4.1 Related Work

process (SMDP) [65]. In that paper, maximization of the ratio of job throughput to

power consumption (the ratio of the long-run average job departure rate to the long-

run average energy consumption rate) is introduced as a measure of performance.

The use of long-run average reward per unit cost (e.g., time consumption, energy

consumption, etc.) as an objective function in [21] generalizes long-run average

service quality per unit time, studied previously.

In this chapter, we consider a large-scale realistically-dimensioned server farm

model with heterogeneous servers. The jockeying case discussed in [21, 22] is more

appropriate for a localized server farm, in which the cost of jockeying actions is

negligible. For a server farm with significant jockeying costs, a simple scalable

job assignment policy without jockeying is more attractive. A similar dynamic

programming methodology in which the computational complexity increases linearly

in the number of states can be applied to our case. Unfortunately, in the non-

jockeying server farm the number of states increases exponentially in the number of

servers, so that the optimal solution is limited to very small cases with a few servers.

As in Section 4, to show the asymptotic optimality MAIP in this chapter, we

consider an interesting and useful framework: bandit problem. Gittins and Jones

published the well-known index theorem in 1974 [92, 93]. In 1979, Gittins [94] gave

the optimal solution for traditional multi-armed bandit problem (MABP) in the form

of the so-called Gittins index policy. More details concerns Gittins indices can be

found in [92, Chapter 2.12] (and references therein). While studies for the traditional

MABP consider only that one machine (projector/bandit/process) is played at a time

and that only played machine changes states, Whittle [33] published a more general

model, restless multi-armed bandit problem (RMABP) and proposed the so-called

Whittle’s index as an approximation for optimality. In 1990, Weber and Weiss [95]

proved, under certain conditions, the asymptotical optimality of Whittle index, which

has been conjectured by Whittle. Papadimitriou and Tsitsiklis [34] proved that the

optimization of RMABP is PSPACE-hard. The Whittle index policy is derived from

75


a relaxation of the original linear optimization problem. Bertsimas and Niño-Mora

[96] analyzed the performance region for the RMABP problem with different levels

of relaxations, and obtained a hierarchy of linear programming (LP) relaxations.

They then proved that the performance region of a lower order relaxation is involved

in that of a higher order relaxation, where the highest order relaxation is equivalent

to the original problem.

In 2001, Niño-Mora claimed a partial conservation law (PCL) for the optimality

of RMABP [97], which is an extended version for the general conservation law (GCL)

published in 1996 [98]. Later, in [35], he defined a group of problems that satisfies

PCL-indexibility and proposed a new index policy that improves Whittle’s. The new

index policy is optimal for problems under PCL-indexibility. PCL-indexibility leads

to, and is stronger, than the indexibility introduced by Whittle. More details in the

optimality of problems with indexibility can be found in [99].

Recently, in [100], the proof of the asymptotic optimality of Whittle’s index in

[95] has been extended to cases with serveral classes of bandits, with arrivals of

new bandits and with multiple actions per bandit. Also, Verloop proposed an index

policy which is not restricted to indexable models and is numerically demonstrated

to be near optimal in non-asymptotic region. In [101], Larrañaga et al. analyzed

a system with multiple users (modeled as multiple bandits), aiming at minimizing

the average cost comprising of convex holding costs and user impatience. The user

impatience here represents a cost triggered by the voluntary leaving of customers

that have been waiting too long in the system. Larrañaga et al. [101] analyzed a

system with multiple users (modeled as multiple bandits), aiming at minimizing the

average cost: a combination of convex (also non-decreasing) holding costs and user

impatience. This work contains uncontrollable states as a bandit cannot be played

when the number of corresponding users is zero, but the non-decreasing holding

cost constraint, which simplifies the asymptotic optimality argument, cannot be

guaranteed in our problem.

76

4.1 Related Work

Stolyar [102] studied a switching model with multiple parallel servers (queues)

based on discrete time Markov chain, and proposed a MaxWeight discipline under

heavy traffic and an assumption of Resource Pooling which ensures the stability of

the multi-queue system. The MaxWeight discipline is a special case of so-called

cµ-rule, and has been proved in [102] to asymptotically minimizes the cumulative

holding cost during a finite time interval. A more generalized algorithm has been

proposed in [103] for a similar queueing system. In [104], Mandelbaum and Stolyar

considered a similar model in continuous time. They proved that a simple generalized

cµ-rule asymptotically minimizes instantaneous and cumulative holding costs in a

queueing with multiple-parallel flexible servers and multi-class jobs when the system

is in heavy traffic and a stablility condition is satisfied. The holding cost rate in

[104] is assumed to be increasing and covex which is more general than the holding

cost rate discussed in [102]. However, the model discussed in [102, 103] is more

general in that it is applicable to queueing system with arbitrary dependence between

servers but this general model is not amenable for a large-scale system. Stolyar

[105] proposed a MinDrift rule to combine dynamic programming and Lagrangian

relaxation and to approximate an optimal solution, for the output queued problem

under a complete resource pooling (CPR) condition, that minimizes the customer

workload, which can be specified as average job number. In particular, for the

customer workload, earlier results in [104] have shown the asymptotical optimality

of the generalized cµ rule (Gcµ-rule) under the CPR condition for the input queued

problem. Work in [102–105] focuses on infinite buffer size and heavy-traffic regime,

where server is only idle in an ignorable time. MaxWeight, Gcµ-rule and Mindrift

are typical index policies.

Nazarathy and Weiss [106] proposed a method for the control problem of a

multi-server queueing system over a finite time horizon. This method decomposes

the original problem into local sub-problems where each of the decisions relies only

on the local information of a sub-problem. They obtained the optimal solution of the

77


control problem in the fluid region by means of separated continuous linear program

(SCLP). Their heuristic method for the original control problem is shown to converge

to the optimal solution in the fluid limit, in other words, the method asymptotically

minimizes the total cost over a finite time horizon.

Ayesta et al. [107] have studied a preemptive queue with infinite buffer and

multiple users as a model for the flow-level behavior of end-users in a narrowband

HDR wireless channel (CDMA 1xEV-DO). They discussed conditions for the sta-

bility and the asymptotic optimality of policies under which users (channels) are

selected. They showed the importance of the tie-breaking rule on policies, such as

priority-index policy, best-rate (BR) policy and best-rate priority (BRP) policies.

They discuss conditions for the stability and asymptotic optimality of policies for

selecting users (channels). They claim that any BR policy is stable (satisfying the

maximum stability condition, which have been considered as certain conditions in

[108, 109]), but priority-index policies are essentially not stable. Those priority-

index policies, which are also referred to as index policies, have been studied for

decades. For the whittle-index-based policies, Taboada et al. in [110] studied the

time-varying channel problem with general job-size distribution and proposed a new

index rule, referred to as Attained Service Potential Improvement, in the case with

decreasing hazard rate flow (job) size distribution. They showed numerically that the

new rule outperforms cµ-rule, MaxRate scheduler, and Proportional Fair, which are

well-known opportunistic disciplines and are based on the spirit of Whittle index.

In [111], Atar and Shifrin analyzed a G/G/1 queue with a finite buffer and multiple

classes of jobs; all jobs share the finite buffer capacity of this queue. They improved

a methodology, which is based on cµ-rule. This methodology admits/rejects jobs

for the case with infinite buffer size where decision maker only need to know the

lowest priority of those arrival classes. However, in finite buffer case, the order of the

priorities of those job classes must be known. Asymptotic optimality of their method

was also proved under certain conditions. Glazebrook et al. in [112, 113] defined

78

4.2 Model

full indexability as an extended version of the indexability defined by Whittle, where

more flexible resource allocation was considered. Classical indexability requires

a binary set of actions, i.e. {0,1}, while, the full indexability allows for action

variables to be chosen from a more general set [0,1].

For the whittle-index-based policies, Taboada et al. in [110] studied the time-

varying channel problem with infinite buffer size and general job-size distribution

and proposed a new index rule, referred to as Attained Service Potential Improvement

under decreasing hazard rate distributions for flow (job) size. Numerically, they

showed that the new rule outperforms cµ-rule, MaxRate scheduler, and Proportional

Fair, which are well-known opportunistic disciplines based on Whittle index.

This large, but by no means complete, collection of related work, contains no

asymptotic optimality result directly applicable to our problem of a multi-server

queue system with a buffer constraint on each queue, requiring the presence of

uncontrollable states as mentioned before. As far as we are aware, there is no result

on the asymptotic optimality of an index policy in this context.

4.2 Model

We consider a heterogeneous server farm modeled as a multi-queue system where

reassignment of incomplete jobs is not allowed. For the reader’s convenience, Table

4.1 provides a list of symbols that are frequently used in this chapter.

The server farm has a total of K � 2 servers, forming the set K = {1,2, . . . ,K}.

These servers are characterized by their service rates, energy consumption rates, and

buffer sizes. For j 2K , we denote by µ j the service rate of server j and by B j its

buffer size. The energy consumption rate of server j is e j when it is busy and e

0j

when it is idle, respectively, where e j > e

0j � 0. We refer to the ratio µ j/(e j� e

0j )

as the effective energy efficiency of server j. Note that this definition of energy

79


Table 4.1 Summary of Frequently Used Symbols

Symbol Definition

K Set of servers in the systemK Number of servers in the systemB j Buffer size of server jµ j Service rate of server je j Energy consumption rate of server j when it is busye

0j Energy consumption rate of server j when it is idle

µ j/(e j� e

0j ) Effective energy efficiency of server j

l Job arrival rateL f Job throughput of the system under policy f

E f Power consumption of the system under policy f

L f/E f Energy efficiency of the system under policy f

efficiency for each server in the system that takes into account the effect of idle

power is key to the design of the MAIP policy proposed in this chapter.

Job arrivals follow a Poisson process with rate l , indicating the average number

of arrivals per time unit. An arriving job is assigned to one of the servers with at

least one vacant slot in its buffer, subject to the control of an assignment policy f . If

all buffers are full, the arriving job is lost.

We assume that job sizes (in units) are independent and identically distributed,

and normalize without loss of generality the average size of jobs to one. Each server

j serves its jobs at a total rate of µ j using the PS service discipline.

Our considerations are limited to realistic cases, and assume that the ratio of the

arrival rate to the total service rate, i.e., r

def= l/ÂK

j=1 µ j, is sufficiently large to be

economically justifiable but not too large to violate the required quality of service.

We refer to r as the normalized offered traffic.

The job throughput of the system under policy f , which is equivalent to the

long-run average job departure rate, is denoted by L f . The power consumption

of the system under policy f , which is equivalent to the long-run average energy

80

4.3 Job Assignment Policy: MAIP

consumption rate, is denoted by E f . By definition, L f/E f is the energy efficiency

of the system under policy f .

4.3 Job Assignment Policy: MAIP

Here we provide details of the MAIP policy. Note that, with a non-jockeying policy,

the server farm scheduler makes assignment decisions only at arrival events and

assigns a new job to one of the available servers in the system. MAIP is designed in

such a way that takes into account the effect of idle power. The key idea of MAIP

can be conveniently explained by using a simple example.

Consider a system with two servers only, where µ1 = µ2 = 1, e1 = 2, e

01 = 1,

e2 = 2.5, and e

02 = 2. It is clear that in this example e1 < e2 and e

01 < e

02 . If a job

arrives when both servers are idle, the scheduler has two choices:

1. Assigning the job to server 1 makes server 1 busy. The energy consumption

rate of the whole system becomes e1 + e

02 = 4.

2. Assigning the job to server 2 makes server 2 busy. The energy consumption

rate of the whole system becomes instead e2 + e

01 = 3.5.

Since (e1 + e

02 )> (e2 + e

01 ), which is equivalently (e1� e

01 )> (e2� e

02 ), and since

both servers have the same service rate, choosing server 2 for serving the job in

this particular example turns out to be better in terms of the energy efficiency of the

system, despite the fact that server 2 consumes more power when busy than server 1

does.

Intuitively, in the situation where power consumption of idle servers in a system

is not necessarily negligible, the energy used by the system can be categorized

into two parts: productive and unproductive. The productive part contributes to

job throughput, whereas the unproductive part is a waste of energy. For a server j,

when it is idle, the service rate is 0 accompanied by an energy consumption rate of

81


e

0j ; when it is busy, the service rate becomes µ j and the energy consumption rate

increases to e j. We regard the additional service rate µ j�0 as a reward at the cost of

an additional energy consumption rate e j� e

0j . In other words, if jobs are assigned

to server j, the productive power used to support the service rate µ j is effectively

e j� e

0j . Accordingly, productive power is our main concern in the design of MAIP.

Since MAIP aims for energy-efficient job assignment, for convenience of descrip-

tion, we label the servers according to their effective energy efficiency. In particular,

in the context of MAIP, server i is defined to be more energy-efficient than server j

if and only if µi/(ei� e

0i )> µ j/(e j� e

0j ). That is, for any pair of servers i and j, if

i < j, we have µi/(ei� e

0i )� µ j/(e j� e

0j ). Then, MAIP works by always selecting

a server with the highest effective energy efficiency among all servers that contain at

least one vacant slot in their buffers, where ties are broken arbitrarily. As a result of

this design, MAIP is a simple approach that requires only binary state information

(i.e., available or unavailable) from each server for its implementation.

4.4 Analysis

Here we give a precise definition of our optimization problem as described informally

in Section 4.4.1. In Section 4.4.2, the Whittle’s index policy for our problem is

described, and in Section 4.4.3, with exponentially distributed job sizes, we prove

the indexability of our server farm model and present a closed-form expression of

Whittle’s index, producing the equivalence of Whittle’s index policy and MAIP.

In Section 4.4.4, the proof of asymptotic optimality of Whittle’s index policy is

presented, leading to the asymptotic optimality of MAIP. Namely, the difference

between MAIP and an optimal solution, which maximizes the energy efficiency of

the server farm, asymptotically tends to zero as K!+•.

82

4.4 Analysis

4.4.1 Stochastic Process

For j = 1,2, . . . ,K, let N j denote the set of all states of server j, where the state,

n j is the number of jobs in the queue of or being served by server j. Thus, N j =

{0,1, . . . ,B j} where B j � 2 is the buffer size for server j.

For server j, states 0,1, . . . ,B j�1 are called controllable states, while the state

B j is termed uncontrollable state. The set of controllable states, in which server

j is available to be tagged, is denoted by N {0,1}j = {0,1, . . . ,B j � 1} while, for

the uncontrollable state in the set N {0}j = {B j}, server j is forced to be untagged

because it cannot accept jobs.

For t � 0, we define X(t) = (X1(t),X2(t), . . . ,XK(t)) to be a vector of random

variables (or a random vector) representing the state at time t of the stochastic process

of the multi-queue system. Define the vectors n = (n1,n2, . . . ,nK) as values of X(t)

with n j 2N j, j 2J , representing the states of the system. The set of all such states

n is denoted by N , and the sets of uncontrollable and controllable states in N are

given by

N {0} =n

n 2N | n j 2N {0}j , 8 j 2J

o

,

N {0,1} =n

n 2N | n /2N {0}o

.(4.1)

Decisions made upon job arrivals rely on the values of X(t) just before the

corresponding arrivals occur. We use af

j (t), j 2J as an indicator of activity at time

t under policy f , so that af

j (t) = 1 if server j is tagged, and af

j (t) = 0, otherwise.

Then Â j2J af

j (t) 1 for all t > 0. All job assignment policies considered are

stationary, so that we can also use af

j (n), n 2N , to represent the action we take

when the system is in state n. A policy f comprises those af

j (t) for all t > 0 and,

according to the definition of stationary policy, f can also be represented by the

sequence of af (n) = (af

1 (n),af

2 (n), . . . ,af

K(n)), for all n 2N .

Let {Xf (t), t > 0}, represent the stochastic process under policy f with initial

state Xf (0) = x(0). For each j = 1,2, . . . ,K, define a mapping R j : N j! R, where

83


R j(n j) is the reward rate of server j in state n j. Let R j be the set of all such mappings.

Then, for a given vector of mappings R = (R1,R2, . . . ,RK), we define the long-run

average reward under policy f to be

g

f (R) = limt!+•

1tE(

Z t

0Â

j2JR j(X

f

j (u))du

)

. (4.2)

We refer to R as the reward rate function. Along similar lines, for each j = 1,2, . . . ,K,

we consider µ j(n j) and e j(n j), the service rate and power consumption of server j

in state n j, respectively, as rewards; that is, µ j,e j 2R j. As mentioned in Section

4.2, we define µ j(n j) = µ j and e j(n j) = e j for n j > 0 with µ j(0) = 0 and e j(0) =

e

0j where µ j > 0, e j > e

0j � 0, j 2J . For the vectors µ = (µ1,µ2, . . . ,µK) and

e = (e1,e2, . . . ,eK), the long-run average job service rate of the entire system is then

g

f (µ) and the long-run average energy consumption rate of the system is g

f (e).

For simplicity, we refer to the long-run average job service rate and the long-run

average energy consumption rate as the job throughput and power consumption,

respectively. As in [21, 22], and as informally discussed earlier, the energy efficiency

of the system is the ratio of the job throughput to the power consumption. The

problem of maximizing energy efficiency is then encapsulated in

maxf2F

g

f (µ)

g

f (e). (4.3)

Based on the definitions above, we rigorously define the MAIP as follows:

if n 2N {0,1}, aMAIPj (n) =

8

>

<

>

:

1, j = minargmaxj:n j2N

{0,1}j

{Q j},

0, otherwise,

if n 2N {0}, aMAIPj (n) = 0, f or all j = 1,2, . . . ,K.

(4.4)

84

4.4 Analysis

4.4.2 Whittle’s Index

Gittins and Jones published the well-known index theorem for SFABP in 1974

[92, 93], in 1979, Gittins [94] produced the optimal solution for the general multi-

armed bandit problem (MABP) in the form of the so-called Gittins’ index policy.

Additional details concerning the history of the Gittins index can be found in [92,

Chapter 2.12] along with proofs. Relaxing the constraint that only one machine

(project/bandit/process) is played at a time, and that only the played machine changes

state, Whittle [33] published a more general model, the restless multi-armed bandit

(RMAB) and proposed an index, the so-called Whittle’s index, as an approximation

for optimality.

The general definition of Whittle’s index is given here; a closed-form expression

will be provided in Section 4.4.3 for the case when job sizes are exponentially

distributed.

According to [21, Theorem 1], there exists a value e⇤ > 0 given by

e⇤ = maxf2F

⇢

g

f (µ)

g

f (e)

�

, (4.5)

so that our optimization problem (4.3) can be written as

supf2F

8

>

<

>

:

g

f (R) : Âj2J :Xf

j (t)2N{0,1}

j

af

j (t) = 1, t � 0

9

>

=

>

;

(4.6)

whereR = (R1,R2, . . . ,RK),

R j 2R j,

R j(n j) = µ j(n j)� e⇤e j(n j),

j 2J .

85


Following the Whittle’s index approach, we relax the problem (4.6) as

supf

limt!+•

1tE(

Z t

0Â

j2JR j(X

f

j (u))du

)

,

s.t. E

8

>

<

>

:

Âj2J :Xf

j (t)2N{0,1}

j

af

j (t)

9

>

=

>

;

= 1.

(4.7)

Seldom in the literature is it well explained that this means that the af

j (t) become

random variables, so that sometimes more than one server will be tagged simultane-

ously. This is not consistent with the framework of our original problem and is also

unrealistic.

The linear constraint in (4.7) is covered by the introduction of a Lagrangian

multiplier n 2R, namely,

infn

supf

limt!+•

1tE(

Z t

0

"

Âj2J

R j(Xf

j (u))

�n Âj2J :Xf

j (t)2N{0,1}

j

af

j (u)

3

7

5

du

9

>

=

>

;

+n . (4.8)

For a given n , we can now decompose (4.8) into K sub-problems:

supf

limt!+•

1tE⇢

Z t

0

h

R j(Xf

j (u))�naf

j (u)i

du

�

, (4.9)

where af

j (u) = 0 when Xf

j (u) 2N {0}j , for 0 < u < t, j 2J .

In [33], Whittle defined a n-subsidy policy for a project (server) as an optimal

solution for (4.9), which provides the set of states where the given project will be

passive (untagged), and introduced the following definition.

86

4.4 Analysis

Definition 4.1. With n 2R, let D(n) be the set of states of a project which is passive

under a n-subsidy policy. The project is indexable if D(n) increases monotonically

from /0 to the set of all possible states for the project as n increases from �• to +•.

In particular, if a project (server) j is indexable and there is a n

⇤ satisfying

n j /2 D(n) for n n

⇤ and n j 2 D(n) otherwise, then this value n

⇤ is the value of

Whittle’s index for project (server) j at state n j. The Whittle’s index values may be

different for different servers at different states. Then, Whittle’s index policy for the

multi-queue system will select a controllable server (a server in controllable states)

with highest Whittle’s index to be tagged (with others untagged) at each decision

making epoch.

4.4.3 Indexability

We give the closed form of the optimal solution for Problem (4.9), namely, the Whit-

tle’s index policy, for the case with exponentially distributed job sizes. Our approach

uses the theory of semi-Markov decision processes and the Hamilton-Jacobi-Bellman

equation. This formulation requires the exponential job size assumption.

For this section, we define R j(n j) = µ j(n j)� e⇤e j(n j), j 2J , where e⇤ is

defined as in (4.5). For policy f j, let V f j,nj (n j,R j) denote the expected value of the

cumulative reward of a process for server j 2J with reward rate R j(n j)�naf jj (n j)

that starts in state n j 2N j and ends when it first goes into an absorbing state n0j 2N j.

In particular, V f jj (n0

j) = 0 for any f j. Here, f j is a stationary policy for server j,

which determines whether it is tagged or not according to its current state Xf jj (t).

Because state 0 is reachable from all other states, we can assume without loss of

generality that n0j = 0 for all j 2J .

Now, let PHj represent a process of Xf j

j (t) for server j 2J that starts from

state 0 until it reaches state 0 again, where f j is constrained to those policies

satisfying af jj (0) = 1. The set of all such policies is denoted by FH

j . It follows from

87


[49, Corollary 6.20 and Theorem 7.5] that the average reward of process PHj is

equivalent to the long-run average reward of the system.

Now an application of the g-revised reward [49, Theorem 7.6, Theorem 7.7],

yields the following corollary.

Corollary 4.1. For a server j as defined in Section 4.2 and a given n <+•, with

R j(n j) = µ j(n j)� e⇤e j(n j) < +•, there exists a real g, with Rgj(n j) = R j(n j)� g

such that if policy f

⇤j 2 FH

j maximizes V f j,nj (n j,R

gj), then f

⇤j also maximizes the

long-run average reward of server j with reward rate R j(n j)�af

⇤j

j (n j)n , n j 2N j,

among all policies in FHj . In particular, this value of g, denoted by g⇤, is equivalent

to above maximized long-run average reward.

In other words, if we compare the maximized average reward of process PHj

under policy f

⇤j and policy f

0j with a

f

0j

j (0) = 0 (and all the actions for non-zero

states are the same as f

⇤j ), then the one with higher average reward is the optimal

policy for (4.9). Note that, in our server farm model, if af

0j

j (0) = 0, the actions for

non-zero states are meaningless since the corresponding server (queue) will never

leave state 0.

We start by finding this f

⇤j as in Corollary . Let V n

j (n j,Rgj) = sup

f jV f j,n

j (n j,Rgj).

The maximization of V f j,nj (n j,R

gj) can be written using the Hamilton-Jacobi-Bellman

equation as

V n

j (n j,Rgj) = max

f j

(

⇣

R j(n j)�g�naf jj (n j)

⌘

t

f jj (n j)+ Â

n2N j

Pf jj (n j,n)V n

j (n,Rgj)

)

,

(4.10)

where t

f jj (n j) is the expected sojourn time in state n j under f j, and Pf j

j (n j,n),

n j,n 2N j, is the transition probability from state n j to state n for the next epoch.

For the multi-queue system, policy f j comprises a sequence of af jj (n j) for n j 2N j.

We rewrite (4.10) as

88

4.4 Analysis

V n

j (n j,Rgj) = max

(

⇣

Rgj(n j)�n

⌘

t

1j (n j)+ Â

n2N j

P1j (n j,n)V n

j (n,Rgj),

Rgj(n j)t

0j (n j)+ Â

n2N j

P0j (n j,n)V n

j (n,Rgj)

)

, (4.11)

where t

1j (n j) and t

0j (n j), are the expected sojourn time in state n j for af j

j (n j)= 1, and

af jj (n j) = 0, respectively, and P1(n j,n) and P0(n j,n), n j,n 2N j, are the transition

probability for af jj (n j) = 1 and af j

j (n j) = 0, respectively.

For (4.11), there is a specific n , referred to as n

⇤j (n j,R

gj), satisfying

n

⇤j (n j,R

gj)t

1j (n j) = Â

n2N j

P1j (n j,n)V n

j (n,Rgj)� Â

n2N j

P0j (n j,n)V n

j (n,Rgj)

+Rgj(n j)(t

1j (n j)� t

0j (n j)). (4.12)

For an indexable server j, we define a policy as follows:

if n < n

⇤j (n j,R

gj), j will be tagged

if n > n

⇤j (n j,R

gj), j will be untagged, and

if n = n

⇤j (n j,R

gj), j can be either tagged or untagged.

(4.13)

The quantities n

⇤(n j,Rgj), n j 2N j, j 2J , constitute Whittle’s index [33] in this

context, and (4.13) defines the optimal solution for Problem (4.9). According to

(4.12), although the value of n

⇤j (n j,R

gj) may appear to rely on n , we prove later on

that here the value of n

⇤j (n j,R

gj) can be expressed in closed form and is independent

of n , and that the server farm is indexable according to the definition in [33].

Proposition 4.1. For the system defined in Section 4.2, for each j 2J , we have

n

⇤j (n j,R

gj) =

l (µ j� e⇤e j�g)µ j

, n j = 1,2, . . . ,B j�1. (4.14)

Proof. The proof is given in Appendix B.1.

89


Moreover, the optimal policy f

⇤j that maximizes V f j,n

j (n j,Rgj) also maximizes

the average reward of process PHj with the value of g specified in Corollary 4.4.3

among all policies in FHj . For the optimal n-subsidy policy, it remains to compare

f

⇤j with a

f

⇤j

j (0) = 1 and f

0j with a

f

0j

j (0) = 0.

Proposition 4.2. For the system defined in Section 4.2, for each j 2J , we have

n

⇤j (0,R

gj) =

l

µ j

�

µ j� e⇤e j + e⇤e0j�

. (4.15)


Now Proposition 4.3 is a straightforward consequence of Propositions 4.1 and

4.2.

Proposition 4.3. For the system defined in Section 4.2, if job-sizes are exponentially

distributed then the Whittle’s index of server j at state n j is

n

⇤j (n j,R

gj) = l (1� e⇤

e j� e

0j

µ j), n j = 0,1, . . . ,B j�1. (4.16)

Thus, the system is indexable.


It is clear that the Whittle’s index policy, which prioritizes server(s) with the

highest index value at each decision epoch, is equivalent to the MAIP policy defined

in (4.4), when job sizes are exponentially distributed. The form of Whittle’s index

in the case for general job-size distributions remains unclear. In Section 4.5, we

numerically demonstrate the sensitivity of MAIP to highly varying job sizes.

4.4.4 Asymptotic optimality

We will prove the asymptotic optimality of MAIP as the number of servers becomes

large when job sizes are exponentially distributes and the number of servers is scaled

90

4.4 Analysis

under appropriate and reasonable conditions for large server farms (as discussed in

Section 1.1).

We will apply the proof methodology of Weber and Weiss [95] for the asymptotic

optimality of index policies to our problem though, as we have already stated in

Section 4.1, this proof cannot be directly applied to our problem because of the

presence of uncontrollable states. We define an additional (virtual) server, designated

as server K+1, to handle the blocking case when all actual servers are full; this server

has only one state (server K + 1 never changes state) with zero reward rate. This

virtual server is only used in the proof of asymptotic optimality in this section. For

this server, |N {0,1}K+1 |= 1 and N {0} = /0. In addition, we define K + =K [{K+1}

as the set of servers including this extra zero-reward server. The set of controllable

states of these K +1 servers is defined as Ñ {0,1} =S

j2K + N {0,1}j , and the set of

uncontrollable states is Ñ {0} =S

j2K + N {0}j .

In this section, those servers with identical buffer size, service rate, and energy

consumption rate are grouped as a server group, and we label these server groups as

server groups 1,2, . . . , K. For servers i, j of the same server group, N {0,1}i =N {0,1}

j

and N {0}i = N {0}

j . For clarity of presentation, we define Ñ {0,1}i and Ñ {0}

i , i =

1,2, . . . , K as, respectively, the sets of controllable and uncontrollable states of servers

in server group i. We regard states for different server groups as different states; that

is, Ñ {0,1}j \ Ñ {0,1}

i = /0 and Ñ {0}j \ Ñ {0}

i = /0 for different server groups i and j,

i, j = 1,2, . . . , K. Let Zf

i (t) be the random variable representing the proportion of

servers in state i 2 Ñ {0,1}[ Ñ {0} at time t under policy f . As previously, we label

states i 2 Ñ {0,1}[ Ñ {0} as 1,2, . . . , I, where I = | Ñ {0,1}[ Ñ {0}|, and use Zf (t)

to denote the random vector (Zf

1 (t),Zf

2 (t), . . . ,Zf

I (t)). Correspondingly, actions

af

j (n j), n j 2N j, j 2K +, correspond to actions af (i), i 2 Ñ {0,1}[ Ñ {0}.

Let z,z0 2 RI be instantiations of Zf (t), t > 0, f 2F. Transitions of the random

vector Zf (t) from z to z0 can be written as z0 = z+ ei,i0 , where ei,i0 is a vector

of which the ith element is + 1K+1 , the i0th element is � 1

K+1 and otherwise is zero,

91


i, i0 2 Ñ {0,1}[ Ñ {0}. In particular, for the server farm defined in Section 4.2, servers

in server group j only appear in state i 2 Ñ {0,1}j [ Ñ {0}

j ; that is, the transition from

z to z0 = z+ei,i0 , i 2 Ñ {0,1}j [ Ñ {0}

j , i0 2 Ñ {0,1}j0 [ Ñ {0}

j0 , j, j0 = 1,2, . . . , K, j 6= j0

never occur. We address such impossible transitions by setting the corresponding

transition probabilities to zero. The states i 2 Ñ {0,1} are ordered according to

descending index values, where all states i 2 Ñ {0} follow the controllable states

in the ordering, with af (i) = 0 for i 2 Ñ {0}. Then, we set the state i 2N {0,1}K+1 of

the zero-reward server, which is also a controllable state, to come after all the other

controllable states but to precede the uncontrollable states. Because of the existence

of the zero-reward server K + 1, the number of servers in controllable states can

always meet the constraint (4.7). Note here that we artificially move the state of

server K+1 and the uncontrollable states to places in the ordering that do not accord

with their indices, which are zero. We will show later that such movements do

not affect the long-run average performance of Whittle’s index policy, which exists

and is equivalent to MAIP in our context. The position of a state in the ordering

i = 1,2, . . . , I is also defined as its label.

Let g

OR(f) be the long-run average reward of the original problem (4.6) (that is,

without relaxation) under policy f , and g

LR(f) the long-run average reward of the re-

laxed problem (4.7) under policy f . In addition, let g

OR = maxf

�

g

OR(f)

, the maxi-

mal long-run average reward of the original problem, and g

LR = maxf

�

g

LR(f)

, the

maximal long-run average reward of the relaxed problem. From the definition of our

system, g

LR(f)/K,gOR(f)/K max j2K ,n j2N j R j(n j) < +•, where R j(n j) is the

reward rate of server j in state n j as defined before. Let index denote the Whittle’s

index policy, then, we obtain g

OR(index)/K g

OR/K g

LR/K. Following the idea

of [95], we prove, under Whittle’s index policy, that g

OR(index)/K� g

LR/K ! 0

when K is scaled in a certain way.

To demonstrate asymptotic optimality, we now describe the stationary policies,

including Whittle’s index policy, in another way. Let uf

i (z) 2 [0,1], z 2 RI , i =

92

4.4 Analysis

1,2, . . . , I, be the probability for a server in state i 2 Ñ {0,1}[ Ñ {0} to be tagged

(af (i) = 1) when Zf (t) = z. Then, 1�uf

i (z) is the probability for a server in state i

to be untagged (af (i) = 0).

Define N +i , i 2 Ñ {0,1}[ Ñ {0} to be the set of states that precede state i in the

ordering. Then, for Whittle’s index policy, we obtain

uindexi (z) = 1

zimin

8

<

:

zi,max

8

<

:

0,1

K +1� Â

i02N +i

zi0

9

=

;

9

=

;

. (4.17)

Our multi-queue system is stable, since any stationary policy will lead to an

irreducible Markov chain for the associated process and the number of states is finite.

Then, for a policy f 2 F, the vector Xf (t) converges as t! • in distribution to a

random vector Xf . In the equilibrium region, let p

f

j be the steady state distribution

of Xf

j for server j, j 2K +, under policy f 2F, where p

f

j (i), i 2N j, is the steady

state probability of state i. For clarity of presentation, we extend vector p

f

j to a vector

of length I, written p

f

j , of which the ith element is p

f

j (i), if i 2N j, and otherwise,

0. The long-run expected value of Zf (t) is ÂK+1j=1 p

f

j /(K + 1). In the server farm

defined in Section 4.2, the long-run expected value of Zf (t) will be a member of the

set

Z =

(

z 2 RI| Âi2 Ñ {0,1}[ Ñ {0}

zi ⌘ 1,

8i 2 Ñ {0,1}[ Ñ {0}, zi � 0

)

. (4.18)

Write q1(z,zi,z0i) and q0(z,zi,z0i), z 2 Z , i 2 Ñ {0,1} [ Ñ {0}, as the average

transition rate of the ith element in vector z from zi to z0i, under tagged and untagged

actions, respectively. Then, the average transition rate of the ith element of z under

93


policy f is given by

qf (z,zi,z0i) = uf

i (z)q1(z,zi,z0i)+(1�uf

i (z))q0(z,zi,z0i).

We consider the following differential equation for a stochastic process, denoted by

zf (t) = (zf

1 (t),zf

2 (t), . . . ,zf

I (t)),

dzf

i (t)dt

= Âz0i

h

z0i(t)qf (zf (t),z0i,z

f

i (t))

�zf

i (t)qf (zf (t),zf

i (t),z0i)i

. (4.19)

Because of the global balance at an equilibrium point of limt!+•R t

0 zf (u)du/t, if

exists, denoted by zf , dzf (t)/dt|zf (t)=zf

= 0. Let OPT represent the optimal solution

of the relaxed problem (4.7) and recall that index represents the Whittle’s index

policy. Since uindexi (zindex) = uOPT

i (zindex), following the proof of [95, Theorem 2],

we obtain dzOPT (t)/dt|zOPT (t)=zindex = 0 and zindex = zOPT , if both zindex and zOPT

exist. The existence of zindex and zOPT will be discussed later.

For a small d > 0, we define Rd ,f as the average reward rates during the time

period that |Zf (t)� zf (t)| d under policy f with Zf (0) = zf (0), and Rm/K =

supf

limsupt!+• |R(Xf (t))/K|<+• is an upper bound of the absolute value of the

reward rate divided by K. Then,

g

LR

K� g

OR(index)K

limd!0

limt!+•

1t

Z t

0

Rm

KPn

|ZOPT (u)� zOPT (t)|> d

o

+Rm

KPn

|Zindex(u)� zindex(t)|> d

o

+Rd ,OPT

KPn

|ZOPT (u)� zOPT (t)| d

o

�Rd ,index

KPn

|Zindex(u)� zindex(t)| d

o

#

du. (4.20)

94

4.5 Numerical results

The server farm is decomposed into K server groups, with number of servers in

the ith group denoted by Ki, i = 1,2, . . . , K. Then, K = ÂKi=1 Ki. Following the proof

of [95, Proposition], for any Ki = K0i n, K0

i = 1,2, . . ., i = 1,2, . . . , K, n = 1,2, . . .,

d > 0 and f is set to be either index or OPT ,

limn!+•

limt!+•

1t

Z t

0P{|Zf (u)� zf (u)|> d}du = 0. (4.21)

We provide a justification of (4.21) in Appendix B.4, following [114, Chapter 7].

Then, as n! +•, the existence of an equilibrium point of limt!+•R t

0 Zf (u)du/t

leads to the existence of zf = limt!+•R t

0 zf (u)du/t (using the Lipschitz continuity

of the right side of Equation (4.19) as a function of zf (t)). We obtain

limn!+•

limd!0

Rd ,OPT

K� Rd ,index

K

!

= 0,

and

limn!+•

✓

g

LR

K� g

OR(index)K

◆

= 0. (4.22)

Finally, g

OR(index)/K� g

OR/K ! 0 as n! +•; that is, MAIP (Whittle’s index

policy) approaches the optimal solution in terms of energy efficiency as the number

of servers in each server group tends to infinity at the appropriate rate.


In this section, we provide extensive numerical results obtained by simulation to

evaluate the performance of the MAIP policy. All results are presented in the form of

an observed mean from multiple independent runs of the corresponding experiment.

The confidence intervals at the 95% level based on the Student’s t-distribution are

maintained within ±5% of the observed mean. For convenience of describing the

95


results, given two numerical quantities x > 0 and y > 0, we define the relative

difference of x to y as (x� y)/y.

In all experiments, we have a system of servers that are divided into three server

groups. Servers in each server group i, i = 1,2,3, have the same buffer size, service

rate and energy consumption rate, denoted by Bi, µi, ei and e

0i , respectively. We

consider this to be a realistic setting since in practice a server farm is likely to

comprise multiple servers of the same type purchased at a time. If not otherwise

specified, we assume that job sizes are exponentially distributed.

Recall that, as defined in Section 4.2, the job throughput is the average job

departure rate (jobs per second), the power consumption is the average energy

consumption rate (Watt), and the energy efficiency is the ratio of job throughput to

power consumption (jobs per Watt second). Also, we have normalized the average

job size to one (Byte) in Section 4.2.

4.5.1 Effect of Idle Power

Recall that MAIP is designed to take into account the effect of idle power. To

demonstrate the effect of idle power on job assignment, here we consider a baseline

policy, called Most energy-efficient available server first Neglecting Idle Power

(MNIP). As its name suggests, MNIP is a straightforward variant of MAIP that

neglects idle power and hence treats e

0j = 0 for all j 2K in the process of selecting

servers for job assignment. We compare MAIP with MNIP in terms of energy

efficiency, job throughput and power consumption under various system parameters.

For the set of experiments in Fig. 4.1, each server group has 15 servers, where we

set Bi = 10 and e

0i /ei = 0.4i�0.3 for i = 1,2,3, and randomly generate µi and ei as

µ1 = 6.86, e1 = 6.86, µ2 = 3.64, e2 = 3.72, µ3 = 2.87, e3 = 3.15. The normalized

offered traffic r is varied from 0.01 to 0.9.

96


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

5

10

15

20

25

30

35

40

45

50

ρ

Re

lativ

e d

iffe

ren

ce (

%)

(a)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−5

−4

−3

−2

−1

0

1

2

3

4

5

ρ

Re

lativ

e d

iffe

ren

ce (

%)

(b)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−35

−30

−25

−20

−15

−10

−5

0

ρ

Re

lativ

e d

iffe

ren

ce (

%)

(c)

Fig. 4.1 Performance comparison with respect to normalized system load r . (a)Relative difference of L MAIP/E MAIP to L MNIP/E MNIP. (b) Job throughput. (c)Relative difference of E MAIP to E MNIP.

The results in Fig. 4.1 are presented in the form of the relative difference of

MAIP to MNIP in terms of each corresponding performance measure. We observe

in Fig. 4.1b that, in all cases, both policies have almost the same performance in job

throughput. We also observe in Fig. 4.1a and Fig. 4.1c that, in the case where r ! 0

or r ! 1, the two policies are close to each other in terms of both energy efficiency

and power consumption. This is because in such trivial and extreme cases the system

is at all times either almost empty or almost fully occupied. However, in the realistic

97


0 100 200 300 400 500 600 7000.6

0.65

0.7

0.75

0.8

0.85

0.9

0.95

1

Number of servers

En

erg

y e

ffic

ien

cy

MAIPMNIP

(a)

0 100 200 300 400 500 600 7000

250

500

750

1000

1250

1500

1750

2000

Number of servers

Job

th

rou

gh

pu

t

MAIPMNIP

(b)

0 100 200 300 400 500 600 7000

250

500

750

1000

1250

1500

1750

2000

2250

2500

2750

3000

Number of servers

Po

we

r co

nsu

mp

tion

MAIPMNIP

(c)

Fig. 4.2 Performance comparison with respect to number of servers K. (a) Energyefficiency. (b) Job throughput. (c) Power consumption.

cases where r is not too large and not too small, MAIP significantly outperforms

MNIP with a gain of over 45% in energy efficiency at r = 0.4,0.5.

In Fig. 4.2, we use the same settings as in Fig. 4.1, except that we fix the

normalized offered traffic r at 0.6 and vary the number of servers K from 3 to 690.

Note that here we increase K by increasing the number of servers in each of the

three server groups. We observe in Fig. 4.2b that, under such a medium traffic load,

the service capacity is sufficiently large, so that with both MAIP and MNIP almost

all jobs can be admitted and hence the job throughput is almost identical to the

98


0 10 20 30 40 50 600

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

11


Cu

mm

ula

tive

dis

trib

utio

n

β = 0.5

β = 1

β=1.5

(a)

0 10 20 30 40 50 600

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

11


Cu

mm

ula

tive

dis

trib

utio

n

β=0.5

β=1

β=1.5

(b)

0 10 20 30 40 50 600

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

11


Cu

mm

ula

tive

dis

trib

utio

n

β=0.5

β=1

β=1.5

(c)

Fig. 4.3 Cumulative distribution of relative difference of L MAIP/E MAIP toL MNIP/E MNIP. (a) r = 0.4. (b) r = 0.6. (c) r = 0.8.

arrival rate for all values of K. As a result, the job throughput increases almost

linearly with respect to the number of servers K. We observe in Fig. 4.2c that, for

both policies, the power consumption also increases almost linearly with respect

to the number of servers K. However, it is clear that the power consumption of

MAIP increases at a significantly smaller rate than that of MNIP, which results in a

substantial improvement of the energy efficiency by nearly 36% in all cases as seen

in Fig. 4.2a.

99


For the set of experiments in Fig. 4.3, we again consider a system where each

server group has 15 servers and we set Bi = 10 for i = 1,2,3. We introduce a param-

eter b , where different values of b lead to different levels of server heterogeneity.

In particular, we consider three different values for b , i.e., b = 0.5,1,1.5. We

set e

0i /ei = (0.4i� 0.3)b for i = 1,2,3. The set of service rates µi are randomly

generated from the range [1,10] and are arranged in a non-increasing order, i.e.,

µ1 � µ2 � µ3. For the set of energy consumption rates ei, we first choose two

real numbers a1 and a2 randomly from [0.5,1]. Then, with µ1/e1 = 200, we set

µi/ei = ab

i�1µi�1/ei�1 for i = 2,3.

The results in Fig. 4.3 are obtained from 1000 experiments and are plotted in the

form of cumulative distribution of the relative difference of MAIP to MNIP in terms

of the energy efficiency. We observe in Fig. 4.3 that MAIP significantly outperforms

MNIP by up to 60%. It can also be observed from Fig. 4.3a and Fig. 4.3b that

MAIP outperforms MNIP by more than 10% in nearly 100% of the experiments

for the case b = 0.5. In addition, we observe in Fig. 4.3 that, as the level of server

heterogeneity (i.e., the value of b ) becomes higher, the performance improvement

of MAIP over MNIP in general becomes larger, although the gain is decreasing

when the normalized offered traffic r approaches 0.8, similar to what is observed in

Fig. 4.1a.

4.5.2 Effect of Jockeying Cost

Recall that MAIP is designed as a non-jockeying policy, which is more appropriate

than jockeying policies for job assignment in a large-scale server farm. As discussed

in Section 4.1, jockeying policies suit a small server farm where the cost associated

with jockeying is negligible. In large-scale systems, the cost associated with jockey-

ing can be significant and may have a snowball effect on the system performance.

Here, we demonstrate the benefits of MAIP in a server farm where jockeying costs

100


are high, by comparing it with a jockeying policy known as Most Energy Efficient

Server First (MEESF) proposed in [22].

The settings of servers in each of the three server groups are based on the

benchmark results of Dell PowerEdge rack servers R610 (August 2010), R620

(May 2012) and R630 (April 2015) [115]. Specifically, we normalize µ3 and e3 to

one and then set µ1/µ3 = 3.5, e1/e3 = 1.2, e

01/e1 = 0.2, µ2/µ3 = 1.4, e2/e3 = 1.1,

e

02/e2 = 0.2 and e

03/e3 = 0.3. We also set Bi = 10 for i = 1,2,3 and r = 0.6. The

number of servers K is varied from 3 to 270, where we increase K by increasing the

number of servers in each of the three server groups.

Let us assume that each jockeying action incurs a (constant) delay D. That is,

when a job is reassigned from server i to server j, it will be suspended for a period D

before resumed on server j. Clearly, when D > 0, this is equivalent to increasing the

size of the job and hence its service requirement. Accordingly, for a given system,

a non-zero cost per jockeying action indeed increases the traffic load. We consider

three different values for D. The case where D = 0 is for zero jockeying cost, the

case where D = 0.0005 indicates a relatively small cost per jockeying action, and the

case where D = 0.01 represents a large cost per jockeying action. The results are

presented in Fig. 4.4 for the energy efficiency and in Fig. 4.5 for the job throughput.

For the case where D = 0, we have a similar observation in Fig. 4.5a to that in

Fig. 4.2b. That is, under a medium traffic load, the service capacity is sufficiently

large, so that both MAIP and MEESF yield a job throughput that is almost identical

to the arrival rate for all values of K. We observe in Fig. 4.4a that, in this case,

MEESF consistently outperforms MAIP in terms of the energy efficiency, even

though with a very small margin.

For the case where D = 0.0005, we observe in Fig. 4.5b that, since the cost per

jockeying action is relatively small, the service capacity turns out to be sufficiently

large so that the job throughput of MEESF is not affected. We also observe in

Fig. 4.4b that, when the number of servers K is small, the energy efficiency of

101


0 30 60 90 120 150 180 210 240 2700.3

0.6

0.9

1.2

1.5

1.8

2.1

Number of servers

En

erg

y e

ffic

ien

cy

MAIPMEESF

(a)

0 30 60 90 120 150 180 210 240 2700.3

0.6

0.9

1.2

1.5

1.8

2.1

Number of servers

En

erg

y e

ffic

ien

cy

MAIPMEESF

(b)

0 30 60 90 120 150 180 210 240 2700.3

0.6

0.9

1.2

1.5

1.8

2.1

Number of servers

En

erg

y e

ffic

ien

cy

MAIPMEESF

(c)

Fig. 4.4 Performance comparison in terms of the energy efficiency with respect tothe number of servers K. (a) D = 0. (b) D = 0.0005. (c) D = 0.01.

MEESF is still better than that of MAIP. However, when the number of servers K

is large, the energy efficiency of MEESF is clearly degraded. This is because, as K

increases, the number of jockeying actions required for a job on average increases.

With a non-zero cost per jockeying action, it can substantially increase the power

consumption, since we are forced to use more of those less energy-efficient servers

to meet the increased traffic load.

The effect is more profound when D is increased to 0.01. In this case, as shown

in Fig. 4.4c and Fig. 4.5c, the cost associated with jockeying is so high that both the

102


0 30 60 90 120 150 180 210 240 2700

30

60

90

120

150

180

210

240

270

300

330

Number of servers

Job

th

rou

gh

pu

t

MAIPMEESF

(a)

0 30 60 90 120 150 180 210 240 2700

30

60

90

120

150

180

210

240

270

300

330

Number of servers

Job

th

rou

gh

pu

t

MAIPMEESF

(b)

0 30 60 90 120 150 180 210 240 2700

30

60

90

120

150

180

210

240

270

300

330

Number of servers

Job

th

rou

gh

pu

t

MAIPMEESF

(c)

Fig. 4.5 Performance comparison in terms of the job throughput with respect to thenumber of servers K. (a) D = 0. (b) D = 0.0005. (c) D = 0.01.

job throughput and the energy efficiency of MEESF are significantly degraded, due

to the substantially increased traffic load.

4.5.3 Sensitivity to the Shape of Job-Size Distributions

The workload characterizations of many computer science applications, such as Web

file sizes, IP flow durations, and the lifetimes of supercomputing jobs, are known

to exhibit heavy-tailed distributions [27, 86]. Here, we are interested to see if the

performance of MAIP is sensitive to the job-size distribution. To this end, in addition

103


−5 −4.5 −4 −3.5 −3 −2.5 −2 −1.5 −1 −0.5 00

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

11


Cu

mu

lativ

e d

istr

ibu

tion

DeterministicPareto−1Pareto−2

(a)

−5 −4.5 −4 −3.5 −3 −2.5 −2 −1.5 −1 −0.5 00

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

11


Cu

mu

lativ

e d

istr

ibu

tion


(b)

−5 −4.5 −4 −3.5 −3 −2.5 −2 −1.5 −1 −0.5 00

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

11


Cu

mu

lativ

e d

istr

ibu

tion


(c)

Fig. 4.6 Cumulative distribution of relative difference of L MAIP,F/E MAIP,F toL MAIP,Exponential/E MAIP,Exponential; (a) r = 0.4. (b) r = 0.6. (c) r = 0.8.

to the exponential distribution, we further consider three different distributions, i.e.,

deterministic, Pareto with the shape parameter set to 2.001 (Pareto-1 for short), and

Pareto with the shape parameter set to 1.98 (Pareto-2 for short). In all cases, we set

the mean to be one.

We use the same experiment settings as in Fig. 4.3 with b = 1. In each experiment,

we obtain the energy efficiency of MAIP, and compute the relative difference of

the one using each corresponding distribution to the one using the exponential

distribution. Fig. 4.6 plots the cumulative distribution of the relative difference results

obtained from the 1000 experiments for each particular value of the normalized

104

4.6 Conclusion

offered traffic r . We observe in Fig. 4.6 that all relative difference results are

between �5% and 0. Given that the confidence intervals of these simulation results

are maintained within ±5% of the observed mean with a 95% confidence level, the

energy efficiency of MAIP seems not to be too sensitive to the job-size distribution.

4.6 Conclusion

We have studied the stochastic job assignment problem in a server farm comprising

multiple processor sharing servers with different service rates, energy consumption

rates and buffer sizes. Our aim has been to maximize the energy efficiency of the

entire system, defined as the ratio of the long-run average job departure rate to the

long-run average power consumption, by effectively assigning jobs/requests to these

servers. To this end, we have introduced MAIP job assignment policy and have

proved its equivalence to the Whittle’s index policy when job sizes are exponentially

distributed. MAIP only requires information of binary states of servers, and can

be implemented by using a binary variable for each server. This policy does not

require any estimation or prediction of average arrival rate. MAIP has been proven

to approach optimality as the numbers of servers in server groups tend to infinity

when job sizes are exponentially distributed. This asymptotic property is appropriate

to a large-scale server farm that is likely to purchase and upgrade a large number of

servers with the same style and attributes at the same time. The proof for asymptotic

optimality has been completed by applying the ideas of Weber and Weiss [95] to

our multi-queue system with inevitable uncontrollable states. Extensive numerical

results have illustrated the significant superiority of MAIP over MNIP (the baseline

policy) in a general situation where energy efficiency of each server differs from

its effective energy efficiency. MAIP has been shown numerically to give similar

energy efficiency results in cases of exponential and Pareto job-size distributions,

which indicates that it is appropriate to a server farm with highly varying job sizes.

105


Through a numerical example, we have shown that MAIP is more appropriate than

MEESF for a server farm with non-zero jockeying cost in the example, which is also

useful for a real large-scale system with significant cost of job reassignment.

106

Chapter 5

Conclusions

In this thesis, we have analyzed energy-efficient job assignment policies in server

farms, aiming at maximization of the energy efficiency (the ratio of the long-run

average throughput to the long-run average power consumption). We have considered

a large-scale server farm consisting of heterogeneous servers, where an optimal job

assignment policy is computationally infeasible. We have sought and proposed

near-optimal policies with guaranteed performance in terms of energy efficiency.

Both theoretical and numerical analysis has been provided for these policies.

In Chapter 2, we have introduced the connection between the optimization

of expected e-revised cumulative reward and the optimization of the ratio of the

long-run average reward to the long-run average cost. Then, the corresponding

optimal solution can be approached by conventional value iteration or policy iteration

methodologies. In this way, an algorithm has been designed and implemented for

an optimal insensitive jockeying policy in terms of energy efficiency in Chapter 3.

This algorithm has been proved to have polynomial complexity, but, according to

the numerical results for a server farm with one thousand servers, it has been shown

that this algorithm takes around five days for an optimal solution. Thus, the optimal

insensitive jockeying policy is more appropriate to be used as a benchmark in lab

107

Conclusions

experiments, and scalable near-optimal solutions are still necessary in modern server

farms with tens of thousands of servers or other computing components.

We then have studied two classified job assignments: jockeying and non-jockeying

job assignment. In Chapter 3, for the jockeying case, we have proposed a new ap-

proach that gives rise to an insensitive job assignment policy for the popular server

farm model comprising a parallel system of finite-buffer PS queues with hetero-

geneous server speeds and energy consumption rates. The insensitivity has been

possessed by constructing a logically combined queue based on the concept of the

symmetric queue [75]. MEESF is a generalized version of the SSF policy proposed

in [21], and has been proved, under a certain condition, to provide the highest

instantaneous ratio of the total service rate to the total power consumption.

Unlike the straightforward MEESF approach that greedily chooses the most

energy-efficient servers for job assignment, one important feature of the more robust

E⇤ policy is to aggregate an optimal number of most energy-efficient servers as a

virtual server. The aggregate service rate in E⇤ is controlled by a parameter bK. E⇤

is designed to give preference to this virtual server and utilize its service capacity

in such a way that both the job throughput and the energy efficiency of the system

can be improved. We have provided a rigorous analysis of the E⇤ policy where it

has been shown that E⇤ has always a higher job throughput than that of MEESF

and there exist realistic and sufficient conditions under which E⇤ is guaranteed to

outperform MEESF in terms of the energy efficiency of the system. Also, to seek the

optimal value of bK in E⇤, we have given a proof that, under a certain condition, if the

value of bK is within a given set determined by the job arrival rate, then the E⇤ policy

with this bK value achieves higher or equal energy efficiency compared to MEESF.

We have further proposed a rule of thumb (RM) to form the virtual server by

simply matching its aggregate service rate to the job arrival rate. Experiments based

on random settings have confirmed the effectiveness of the resulting RM policy.

Noting that the fundamentally important model of parallel PS queues has broader

108

applications in communication systems, our proposed solution for insensitive energy-

efficient job assignment has potentially wider applicability to green communications

and networking.

In Chapter 4, for the non-jockeying case, we have studied the stochastic job

assignment problem in a server farm comprising multiple processor sharing servers

with different service rates, energy consumption rates and buffer sizes. Our aim has

been to maximize the energy efficiency of the entire system, defined as the ratio of the

long-run average job departure rate to the long-run average power consumption, by

effectively assigning jobs/requests to these servers. To this end, we have introduced

MAIP job assignment policy and have proved its equivalence to the Whittle’s index

policy when job sizes are exponentially distributed. MAIP only requires information

of binary states of servers, and can be implemented by using a binary variable for

each server. This policy does not require any estimation or prediction of average

arrival rate. MAIP has been proven to approach optimality as the numbers of servers

in server groups tend to infinity when job sizes are exponentially distributed. This

asymptotic property is appropriate to a large-scale server farm that is likely to

purchase and upgrade a large number of servers with the same style and attributes

at the same time. The proof for asymptotic optimality has been completed by

applying the ideas of Weber and Weiss [95] to our multi-queue system with inevitable

uncontrollable states. Extensive numerical results have illustrated the significant

superiority of MAIP over MNIP (the baseline policy) in a general situation where

energy efficiency of each server differs from its effective energy efficiency. MAIP

has been shown numerically to give similar energy efficiency results in cases of

exponential and Pareto job-size distributions, which indicates that it is appropriate to

a server farm with highly varying job sizes. Through a numerical example, we have

shown that MAIP is more appropriate than MEESF for a server farm with non-zero

jockeying cost in the example, which is also useful for a real large-scale system with

significant cost of job reassignment.

109

References

[1] M. Arregoces and M. Portolani, Data center fundamentals. Cisco Press,

2003.

[2] U. E. P. Agency, Washington, DC, USA, “EPA report on server and data

center energy efficiency,” 2007, tech. Rep.

[3] T. C. Group, “Smart 2020: Enabling the low carbon economy in the informa-

tion age,” 2008, tech. Rep.

[4] A. Beloglazov, R. Buyya, Y. C. Lee, A. Zomaya et al., “A taxonomy and survey

of energy-efficient data centers and cloud computing systems,” Advances in

Computers, vol. 82, no. 2, pp. 47–111, 2011.

[5] Emerson Network Power, “State of the data center 2011,” 2011.

[Online]. Available: http://www.emersonnetworkpower.com/en-US/Solutions/

infographics/Pages/2011DataCenterState.aspx

[6] Cisco, “The zettabyte era: Trends and analysis,” May 2015. [Online].

Available: http://www.cisco.com/c/en/us/solutions/collateral/service-provider/

visual-networking-index-vni/VNI_Hyperconnectivity_WP.pdf

[7] Natural Resources Defense Council, “America’s data centers consuming

massive and growing amounts of electricity,” Aug. 2014. [Online]. Available:

http://www.nrdc.org/media/2014/140826.asp

111

http://www.emersonnetworkpower.com/en-US/Solutions/infographics/Pages/2011DataCenterState.aspx

http://www.emersonnetworkpower.com/en-US/Solutions/infographics/Pages/2011DataCenterState.aspx

http://www.cisco.com/c/en/us/solutions/collateral/service-provider/visual-networking-index-vni/VNI_Hyperconnectivity_WP.pdf

http://www.cisco.com/c/en/us/solutions/collateral/service-provider/visual-networking-index-vni/VNI_Hyperconnectivity_WP.pdf

http://www.nrdc.org/media/2014/140826.asp

References

[8] D. Kliazovich, P. Bouvry, F. Granelli, and N. L. S. Fonseca, “Energy con-

sumption optimization in cloud data centers,” Cloud Services, Networking

and Management, 2015.

[9] I. Mitrani, “Managing performance and power consumption in a server farm,”

Ann. Oper. Res., vol. 202, no. 1, pp. 121–134, Jan. 2013.

[10] F. Yao, A. Demers, and S. Shenker, “A scheduling model for reduced CPU

energy,” in Proc. IEEE FOCS, Washington, D.C., USA, Jul. 1995, pp. 374–

382.

[11] A. Wierman, L. L. Andrew, and A. Tang, “Power-aware speed scaling in pro-

cessor sharing systems: Optimality and robustness,” Performance Evaluation,

vol. 69, no. 12, pp. 601–622, Dec. 2012.

[12] N. Bansal, K. Pruhs, and C. Stein, “Speed scaling for weighted flow time,”

SIAM J. Comput., vol. 39, no. 4, pp. 1294–1308, Oct. 2009.

[13] N. Bansal, H.-L. Chan, and K. Pruhs, “Speed scaling with an arbitrary power

function,” ACM Transactions on Algorithms (TALG), vol. 9, no. 2, p. 18, Mar.

2013.

[14] L. L. H. Andrew, M. Lin, and A. Wierman, “Optimality, fairness, and ro-

bustness in speed scaling designs,” in Proc. ACM SIGMETRICS, Columbia

University, New York, USA, Jun. 2010, pp. 37–48.

[15] M. Andrews, S. Antonakopoulos, and L. Zhang, “Energy-aware scheduling

algorithms for network stability,” in Proc. IEEE INFOCOM, Shanghai, China,

Apr. 2011, pp. 1359–1367.

[16] S. Albers, F. Müller, and S. Schmelzer, “Speed scaling on parallel processors,”

Algorithmica, vol. 68, no. 2, pp. 404–425, Feb. 2014.

112

References

[17] A. Gandhi and M. Harchol-Balter, “How data center size impacts the effec-

tiveness of dynamic power management,” in Proc. 2011 49th Annual Allerton

Conference on Communication, Control, and Computing (Allerton). Monti-

cello, IL, USA: IEEE, Sep. 2011, pp. 1164–1169.

[18] M. Lin, A. Wierman, L. L. H. Andrew, and E. Thereska, “Dynamic right-

sizing for power-proportional data centers,” IEEE/ACM Trans. Netw., vol. 21,

no. 5, pp. 1378–1391, Oct. 2013.

[19] E. Hyytia, R. Righter, and S. Aalto, “Task assignment in a heterogeneous

server farm with switching delays and general energy-aware cost structure,”

Performance Evaluation, vol. 75-76, pp. 17–35, 2014.

[20] A. Gandhi, M. Harchol-Balter, R. Das, and C. Lefurgy, “Optimal power

allocation in server farms,” in ACM SIGMETRICS 2009, Seattle, USA, Jun.

2009, pp. 157–168.

[21] Z. Rosberg, Y. Peng, J. Fu, J. Guo, E. W. M. Wong, and M. Zukerman,

“Insensitive job assignment with throughput and energy criteria for processor-

sharing server farms,” IEEE/ACM Trans. Netw., vol. 22, no. 4, pp. 1257–1270,

Aug. 2014.

[22] J. Fu, J. Guo, E. W. M. Wong, and M. Zukerman, “Energy-efficient heuristics

for insensitive job assignment in processor-sharing server farms,” IEEE J. Sel.

Areas Commun., vol. 33, no. 12, pp. 2878–2891, Dec. 2015.

[23] W. Q. M. Guo, A. Wadhawan, L. Huang, and J. T. Dudziak, “Server

farm management,” Jan. 2014, US Patent 8,626,897. [Online]. Available:

http://www.google.com/patents/US8626897

113

http://www.google.com/patents/US8626897

References

[24] V. Gupta, M. Harchol-Balter, K. Sigman, and W. Whitt, “Analysis of join-

the-shortest-queue routing for web server farms,” Perform. Eval., vol. 64, no.

9-12, pp. 1062–1081, Oct. 2007.

[25] E. Altman, U. Ayesta, and B. J. Prabhu, “Load balancing in processor sharing

systems,” Telecommun. Syst., vol. 47, no. 1-2, pp. 35–48, Jun. 2011.

[26] K. Sigman, “Processor sharing queues,” 2013. [Online]. Available:

http://www.columbia.edu/~ks20/4404-Sigman/4404-Notes-PS.pdf

[27] M. E. Crovella and A. Bestavros, “Self-similarity in world wide web traffic:

evidence and possible causes,” Networking, IEEE/ACM Transactions on,

vol. 5, no. 6, pp. 835–846, Dec. 1997.

[28] S. Gunawardena and W. Zhuang, “Service response time of elastic data traffic

in cognitive radio networks,” IEEE J. Sel. Areas Commun., vol. 31, no. 3, pp.

559–570, Mar. 2013.

[29] F. Liu, K. Zheng, W. Xiang, and H. Zhao, “Design and performance analysis

of an energy-efficient uplink carrier aggregation scheme,” IEEE J. Sel. Areas

Commun., vol. 32, no. 2, pp. 197–207, Jan. 2014.

[30] J. Ousterhout, P. Agrawal, D. Erickson, C. Kozyrakis, J. Leverich, D. Maz-

ières, S. Mitra, A. Narayanan, G. Parulkar, M. Rosenblum, S. M. Rumble,

E. Stratmann, and R. Stutsman, “The case for RAMClouds: Scalable high-

performance storage entirely in DRAM,” SIGOPS Operating Systems Review,

vol. 43, no. 4, pp. 92–105, Dec. 2009.

[31] A. M. Caulfield, L. M. Grupp, and S. Swanson, “Gordon: Using flash memory

to build fast, power-efficient clusters for data-intensive applications,” ACM

SIGPLAN Notices, vol. 44, no. 3, pp. 217–228, Mar. 2009.

114

http://www.columbia.edu/~ks20/4404-Sigman/4404-Notes-PS.pdf

References

[32] L. A. Barroso and U. Holzle, “The case for energy-proportional computing,”

Computer, vol. 40, no. 12, pp. 33–37, Dec. 2007.

[33] P. Whittle, “Restless bandits: Activity allocation in a changing world,” J. Appl.

Probab., vol. 25, pp. 287–298, 1988.

[34] C. H. Papadimitriou and J. N. Tsitsiklis, “The complexity of optimal queuing

network control,” Math. Oper. Res., vol. 24, no. 2, pp. 293–305, May 1999.

[35] J. Niño-Mora, “Dynamic allocation indices for restless projects and queue-

ing admission control: a polyhedral approach,” Mathematical programming,

vol. 93, no. 3, pp. 361–413, 2002.

[36] A. Gandhi, “Dynamic server provisioning for data center power management,”

Ph.D. dissertation, School of Computer Science, Carnegie Mellon University,

Jun. 2013.

[37] D. Wong and M. Annavaram, “Knightshift: scaling the energy proportionality

wall through server-level heterogeneity,” in Proc. MICRO. Vancouver,

Canada: IEEE, Dec. 2012, pp. 119–130.

[38] Z. Liu, M. Lin, A. Wierman, S. H. Low, and L. L. H. Andrew, “Greening

geographical load balancing,” in Proc. ACM SIGMETRICS. Portland, Oregan:

ACM, Jun. 2011, pp. 233–244.

[39] J. Li, Z. Li, K. Ren, and X. Liu, “Towards optimal electric demand manage-

ment for internet data centers,” IEEE Trans. Smart Grid, vol. 3, no. 1, pp.

183–192, Mar. 2012.

[40] Y. Yao, L. Huang, A. Sharma, L. Golubchik, and M. Neely, “Data centers

power reduction: A two time scale approach for delay tolerant workloads,” in

Proc. IEEE INFOCOM. Orlando, USA: IEEE, Mar. 2012, pp. 1431–1439.

115

References

[41] J. Cao, K. Li, and I. Stojmenovic, “Optimal power allocation and load distri-

bution for multiple heterogeneous multicore server processors across clouds

and data centers,” IEEE Trans. Comput., vol. 63, no. 1, pp. 45–58, Jan. 2014.

[42] Z. Liu, M. Lin, A. Wierman, S. H. Low, and L. L. H. Andrew, “Geographical

load balancing with renewables,” ACM SIGMETRICS PER, vol. 39, no. 3, pp.

62–66, Dec. 2011.

[43] P. Wang, Y. Qi, X. Liu, Y. Chen, and X. Zhong, “Power management in

heterogeneous multi-tier web clusters,” in Proc. ICPP. San Diego, USA:

IEEE, Sep. 2010, pp. 385–394.

[44] Q. Zhang, M. F. Zhani, S. Zhang, Q. Zhu, R. Boutaba, and J. L. Hellerstein,

“Dynamic energy-aware capacity provisioning for cloud computing environ-

ments,” in Proc. ICAC. San Jose, California, USA: ACM, Sep. 2012, pp.

145–154.

[45] T. Lu, M. Chen, and L. L. H. Andrew, “Simple and effective dynamic provi-

sioning for power-proportional data centers,” Parallel and Distributed Systems,

IEEE Transactions on, vol. 24, no. 6, pp. 1161–1171, Apr. 2013.

[46] K. M. Tarplee, R. Friese, A. A. Maciejewski, and H. J. Siegel, “Efficient and

scalable computation of the energy and makespan pareto front for heteroge-

neous computing systems,” in Proc. FedCSIS. Krakow, Poland: IEEE, Sep.

2013, pp. 401–408.

[47] T. Vondra and J. Sedivy, “Maximizing utilization in private iaas clouds with

heterogenous load through time series forecasting,” International Journal on

Advances in Systems and Measurements, vol. 6, no. 1 and 2, pp. 149–165,

2013.

116

References

[48] H. Khazaei, J. Misic, and V. B. Misic, “Performance analysis of cloud com-

puting centers using M/G/m/m+ r queuing systems,” IEEE Trans. Parallel

Distrib. Syst., vol. 23, no. 5, pp. 936–943, Apr. 2012.

[49] S. M. Ross, Applied probability models with optimization applications. Dover

Publications (New York), 1992.

[50] F. A. Haight, “Two queues in parallel,” Biom., vol. 45, no. 3-4, pp. 401–410,

1958.

[51] J. F. C. Kingman, “Two similar queues in parallel,” Ann. Math. Stat., vol. 32,

no. 4, pp. 1314–1323, Dec. 1961.

[52] W. Winston, “Optimality of the shortest line discipline,” J. Appl. Probab.,

vol. 14, no. 1, pp. 181–189, Mar. 1977.

[53] R. R. Weber, “On the optimal assignment of customers to parallel servers,” J.

Appl. Probab., vol. 15, no. 2, pp. 406–413, Jun. 1978.

[54] A. Ephremides, P. Varaiya, and J. Walrand, “A simple dynamic routing prob-

lem,” IEEE Trans. Autom. Control, vol. 25, no. 4, pp. 690–693, 1980.

[55] C. Knessl, B. Matkowsky, Z. Schuss, and C. Tier, “Two parallel M/G/1 queues

where arrivals join the system with the smaller buffer content,” IEEE Trans.

Comput., vol. 35, no. 11, pp. 1153–1158, Nov. 1987.

[56] A. Hordijk and G. Koole, “On the assignment of customers to parallel queues,”

Probability in the Engineering and Informational Sciences, vol. 6, no. 04, pp.

495–511, Oct. 1992.

[57] F. Bonomi, “On job assignment for a parallel system of processor sharing

queues,” IEEE Trans. Comput., vol. 39, no. 7, pp. 858–869, 1990.

117

References

[58] W. Whitt, “Deciding which queue to join: Some counterexamples,” Oper.

Res., vol. 34, no. 1, pp. 55–62, Jan./Feb. 1986.

[59] V. Gupta, “Stochastic models and analysis for resource management in server

farms,” Ph.D. dissertation, School of Computer Science, Carnegie Mellon

University, 2011.

[60] Y. Zhao and W. K. Grassmann, “Queueing analysis of a jockeying model,”

Oper. Res., vol. 43, no. 3, pp. 520–529, May/Jun. 1995.

[61] I. J. B. F. Adan, J. Wessels, and W. H. M. Zijm, “Matrix-geometric analysis of

the shortest queue problem with threshold jockeying,” Oper. Res. Lett., vol. 13,

no. 2, pp. 107–112, Mar. 1993.

[62] Y. Sakuma, “Asymptotic behavior for MAP/PH/c queue with shortest queue

discipline and jockeying,” Oper. Res. Lett., vol. 38, no. 1, pp. 7–10, Jan. 2010.

[63] E. Hyytiä, R. Righter, and S. Aalto, “Energy-aware job assignment in server

farms with setup delays under LCFS and PS,” in Proc. ITC 26, Sep. 2014, pp.

1–9.

[64] K. Li, X. Tang, and K. Li, “Energy-efficient stochastic task scheduling on het-

erogeneous computing systems,” IEEE Trans. Parallel Distrib. Syst., vol. 25,

no. 11, pp. 2867–2876, Nov. 2014.

[65] D. P. Bertsekas, Dynamic programming and optimal control. Athena Scien-

tific Belmont, MA, 1995.

[66] S. A. Lippman, “Applying a new device in the optimization of exponential

queuing systems,” Oper. Res., vol. 23, no. 4, pp. 687–710, Aug. 1975.

[67] J. Wijngaard and S. Stidham, “Forward recursion for Markov decision pro-

cesses with skip-free-to-the-right transitions, part I: theory and algorithm,”

Math. Oper. Res., vol. 11, no. 2, pp. 295–308, May 1986.

118

References

[68] S. Stidham and R. R. Weber, “Monotonic and insensitive optimal policies

for control of queues with undiscounted costs,” Oper. Res., vol. 37, no. 4, pp.

611–625, Jul./Aug. 1989.

[69] J. M. George and J. M. Harrison, “Dynamic control of a queue with adjustable

service rate,” Oper. Res., vol. 49, no. 5, pp. 720–731, Sep.-Oct. 2001.

[70] F. Yao, A. Demers, and S. Shenker, “A scheduling model for reduced CPU

energy,” in Proc. IEEE FOCS, Milwaukee, WI, USA, Oct. 1995, pp. 374–382.

[71] N. Bansal, K. Pruhs, and C. Stein, “Speed scaling for weighted flow time,” in

Proc. ACM-SIAM SODA, New Orleans, LA, USA, Jan. 2007, pp. 805–813.

[72] S. Albers and H. Fujiwara, “Energy-efficient algorithms for flow time mini-

mization,” ACM TALG, vol. 3, no. 4, p. 49, Nov. 2007.

[73] N. Bansal, H.-L. Chan, and K. Pruhs, “Speed scaling with an arbitrary power

function,” in Proc. ACM-SIAM SODA, New York, NY, USA, Jan. 2009, pp.

693–701.

[74] A. Wierman, L. L. H. Andrew, and A. Tang, “Power-aware speed scaling in

processor sharing systems,” in Proc. IEEE INFOCOM, Rio de Janeiro, Brazil,

Apr. 2009, pp. 2007–2015.

[75] F. Kelly, “Networks of queues,” Adv. Appl. Probab., vol. 8, no. 2, pp. 416–432,

Jun. 1976.

[76] P. G. Taylor, “Insensitivity in stochastic models,” in Queueing Networks, R. J.

Boucherie and N. M. van Dijk, Eds. New York, NY: Springer, 2011, ch. 3,

pp. 121–140.

[77] A. D. Barbour, “Networks of queues and the method of stages,” Adv. Appl.

Probab., vol. 8, no. 3, pp. 584–591, Sep. 1976.

119

References

[78] J. Fu, “Implementation for an optimal insensitive jockeying policy,” 2016.

[Online]. Available: http://www.ee.cityu.edu.hk/~zukerman/Codes/JingFu/

JSAC-2015-code.rar

[79] L. Wang, F. Zhang, J. A. Aroca, A. V. Vasilakos, K. Zheng, C. Hou, D. Li, and

Z. Liu, “GreenDCN: A general framework for achieving energy efficiency in

data center networks,” IEEE J. Sel. Areas Commun., vol. 32, no. 1, pp. 4–15,

Jan. 2014.

[80] Y. Tian, C. Lin, and M. Yao, “Modeling and analyzing power management

policies in server farms using stochastic Petri nets,” in Proc. e-Energy 2012.

Madrid, Spain: IEEE, May 2012, pp. 1–9.

[81] Y. Yao, L. Huang, A. B. Sharma, L. Golubchik, and M. J. Neely, “Power cost

reduction in distributed data centers: A two-time-scale approach for delay

tolerant workloads,” IEEE Trans. Parallel Distrib. Syst., vol. 25, no. 1, pp.

200–211, Jan. 2014.

[82] D. Niyato, S. Chaisiri, and L. B. Sung, “Optimal power management for

server farm to support green computing,” in Proc. IEEE/ACM CCGRID 2009.

Shanghai: IEEE Computer Society, May 2009, pp. 84–91.

[83] S. Li, S. Wang, T. Abdelzaher, M. Kihl, and A. Robertsson, “Temperature

aware power allocation: An optimization framework and case studies,” Sus-

tainable Computing: Informatics and Systems, vol. 2, no. 3, pp. 117–127, Sep.

2012.

[84] M. Pore, Z. Abbasi, S. K. S. Gupta, and G. Varsamopoulos, “Techniques to

achieve energy proportionality in data centers: A survey,” in Handbook on

Data Centers. Springer, Mar. 2015, pp. 109–162.

120

http://www.ee.cityu.edu.hk/~zukerman/Codes/JingFu/JSAC-2015-code.rar

http://www.ee.cityu.edu.hk/~zukerman/Codes/JingFu/JSAC-2015-code.rar

References

[85] E. Gelenbe and R. Lent, “Energy-qos trade-offs in mobile service selection,”

Future Internet, vol. 5, no. 2, pp. 128–139, Apr. 2013.

[86] M. Harchol-Balter, Performance Modeling and Design of Computer Systems:

Queueing Theory in Action. Cambridge University Press, 2013.

[87] T. Bonald and A. Proutiere, “Insensitivity in processor-sharing networks,”

Performance Evaluation, vol. 49, no. 1, pp. 193–209, Sep. 2002.

[88] A. Singh, J. Ong, A. Agarwal, G. Anderson, A. Armistead, R. Bannon,

S. Boving, G. Desai, B. Felderman, P. Germano, A. Kanagala, J. Provost,

J. Simmons, E. Tanda, J. Wanderer, U. Hölzle, S. Stuart, and A. Vahdat,

“Jupiter rising: A decade of clos topologies and centralized control in Google’s

datacenter network,” in Proc. ACM SIGCOMM, London, UK, Aug. 2015, pp.

183–197.

[89] L. A. Barroso, J. Dean, and U. Hölzle, “Web search for a planet: The Google

cluster architecture,” IEEE Micro, vol. 23, no. 2, pp. 22–28, Apr. 2003.

[90] C. Calero, Handbook of research on web information systems quality. IGI

Global, 2008.

[91] O. T. Akgun, D. G. Down, and R. Righter, “Energy-aware scheduling on

heterogeneous processors,” IEEE Trans. Autom. Control, vol. 59, no. 3, pp.

599–613, Feb. 2014.

[92] J. Gittins, K. Glazebrook, and R. R. Weber, Multi-armed bandit allocation

indices: 2nd edition. Wiley, Mar. 2011.

[93] J. C. Gittins and D. M. Jones, “A dynamic allocation index for the sequential

design of experiments,” in Progress in Statistics, J. Gani, Ed. Amsterdam,

NL: North-Holland, 1974, pp. 241–266.

121

References

[94] J. C. Gittins, “Bandit processes and dynamic allocation indices,” Journal of

the Royal Statistical Society. Series B (Methodological), pp. 148–177, 1979.

[95] R. R. Weber and G. Weiss, “On an index policy for restless bandits,” J. Appl.

Probab., no. 3, pp. 637–648, Sep. 1990.

[96] D. Bertsimas and J. Niño-Mora, “Restless bandits, linear programming re-

laxations, and a primal-dual index heuristic,” Oper. Res., vol. 48, no. 1, pp.

80–90, Feb. 2000.

[97] J. Niño-Mora, “Restless bandits, partial conservation laws and indexability,”

Advances in Applied Probability, vol. 33, no. 1, pp. 76–98, 2001.

[98] D. Bertsimas and J. Niño-Mora, “Conservation laws, extended polymatroids

and multiarmed bandit problems; a polyhedral approach to indexable systems,”

Mathematics of Operations Research, vol. 21, no. 2, pp. 257–306, May 1996.

[99] J. Niño-Mora, “Dynamic priority allocation via restless bandit marginal pro-

ductivity indices,” TOP, vol. 15, no. 2, pp. 161–198, Sep. 2007.

[100] I. M. Verloop, “Asymptotically optimal priority policies for indexable and

non-indexable restless bandits,” to appear in Ann. Appl. Probab., 2015.

[101] M. Larranaga, U. Ayesta, and I. M. Verloop, “Asymptotically optimal index

policies for an abandonment queue with convex holding cost,” Queueing

Systems, pp. 1–71, May 2015.

[102] A. L. Stolyar, “Maxweight scheduling in a generalized switch: State space

collapse and workload minimization in heavy traffic,” Ann. Appl. Probab.,

vol. 14, no. 1, pp. 1–53, Feb. 2004.

[103] A. L. Stolyar, “Maximizing queueing network utility subject to stability:

Greedy primal-dual algorithm,” Queueing Systems, vol. 50, no. 4, pp. 401–

457, Aug. 2005.

122

References

[104] A. Mandelbaum and A. L. Stolyar, “Scheduling flexible servers with convex

delay costs: Heavy-traffic optimality of the generalized cµ-rule,” Oper. Res.,

vol. 52, no. 6, pp. 836–855, Dec. 2004.

[105] A. L. Stolyar, “Optimal routing in output-queued flexible server systems,”

Probability in the Engineering and Informational Sciences, vol. 19, no. 02,

pp. 141–189, Apr. 2005.

[106] Y. Nazarathy and G. Weiss, “Near optimal control of queueing networks over

a finite time horizon,” Ann. Oper. Res., vol. 170, no. 1, pp. 233–249, Sep.

2009.

[107] U. Ayesta, M. Erausquin, M. Jonckheere, and I. M. Verloop, “Scheduling in a

random environment: stability and asymptotic optimality,” IEEE/ACM Trans.

Netw., vol. 21, no. 1, pp. 258–271, Feb. 2013.

[108] S. Borst, “User-level performance of channel-aware scheduling algorithms in

wireless data networks,” IEEE/ACM Trans. Netw., vol. 13, no. 3, pp. 636–647,

Jun. 2005.

[109] S. Aalto and P. Lassila, “Flow-level stability and performance of channel-

aware priority-based schedulers,” in Proc. EURO-NF Conference on Next

Generation Internet (NGI). Paris, France: IEEE, Jun. 2010, pp. 1–8.

[110] I. Taboada, F. Liberal, and P. Jacko, “An opportunistic and non-anticipating

size-aware scheduling proposal for mean holding cost minimization in time-

varying channels,” Performance Evaluation, vol. 79, pp. 90–103, Sep. 2014.

[111] R. Atar and M. Shifrin, “An asymptotic optimality result for the multiclass

queue with finite buffers in heavy traffic,” Stoch. Syst., vol. 4, no. 2, pp.

556–603, 2014.

123

References

[112] K. D. Glazebrook, D. J. Hodge, and C. Kirkbride, “General notions of index-

ability for queueing control and asset management,” The Annals of Applied

Probability, vol. 21, no. 3, pp. 876–907, Jun. 2011.

[113] K. D. Glazebrook, D. J. Hodge, C. Kirkbride, and R. J. Minty, “Stochastic

scheduling: A short history of index policies and new approaches to index

generation for dynamic resource allocation,” Journal of Scheduling, vol. 17,

no. 5, pp. 407–425, Oct. 2014.

[114] J. I. Freidlin, Markand Szücs and A. D. Wentzell, Random perturbations of

dynamical systems. Springer Science & Business Media, 2012, vol. 260.

[115] Standard Performance Evaluation Corporation, tested by Dell Inc. [Online].

Available: https://www.spec.org/power_ssj2008/results/

[116] F. Kelly, Reversibility and stochastic networks. Cambridge University Press,

1979.

124

https://www.spec.org/power_ssj2008/results/

Appendix A

Theoretical results for Jockeying

Job-Assignments

The system model and notation used in this appendix has been defined in Chapter 3.

A.1 An Example for Mapping Qf

In this example, we consider a server farm with K = 3 and B j = 4, j = 1,2,3. The

multi-queue system is shown in Fig. A.1. We define QE⇤ , bK = 2, as the mapping that

yeilds a logically combined queue shown in Fig. A.2, where the slots in the buffers

of the multi-queue system has been one-to-one mapped to the slots in the buffer of

the single queue.

We then demonstrate how to address an arrival/departure event in the logically

combined queue, when there are 9 jobs (being served) in the system, i.e. state

n = 9. If a new job arrives, as described in Section 3.3.1, according to a probability

g(l,9), l = 1,2, . . . , BK , we demonstrate in Fig. A.3 that a position (slot) is selected

to allocate the new job, such that, in the view of the logically combined queue, all

the jobs in and after this position should be moved backward by one slot. Jockeying

125

Theoretical results for Jockeying Job-Assignments

Fig. A.1 An example for multi-queue system.

Fig. A.2 An example for multi-queue system.

is required when we move backward these jobs. In Fig. A.4, we demonstrate the

correspondence between jockeying in the logically combined queue and that in the

multi-queue system.

In a similar way, if a job at a slot is completed, in the view of the logically

combined queue, all the jobs after this slot should be moved forward by one slot.

In Figs. A.5 and A.6, we present an example for departure event and the correspon-

dence between the two systems with respect to jockeying upon this departure event,

respectively.

A.2 Pseudo-Code of the Algorithm for an Insensitive

Optimal Policy

Here, we present the pseudo-code of the algorithm for an optimal policy, called

Algorithm 1. The theoretical analysis is given in Sections 3.4.1 and 3.4.2.

126

A.3 Proof of Proposition 3.1

(a) (b)

Fig. A.3 An example for arrival event. (a) Multi-queue system. (b) Logicallycombined queue.

(a) (b)

Fig. A.4 An example for jockeying upon arrival event. (a) Multi-queue system. (b)Logically combined queue.


Here, we provide a proof that I-J-OPT is optimal among all jockeying policies in the

case where the job-sizes are exponentially distributed. We begin with background

information on the symmetric queue model, which has also been given in Chapter 3.

When the service requirements of jobs are independently and identically distributed,

the symmetric queue [116, chapter 3.3], which is insensitive to the shape of job-size

distribution, is briefly constructed as follows. If there are n jobs being processed in the

queue, they should be located at the first n positions of the queue. When a departure

occurs at position l, the jobs in positions l +1, l + 2, . . . ,n move to positions l, l +

1, . . . ,n�1, respectively. A proportion g(l,n) of the total service effort is directed

to the job in position l when there are n jobs being processed in the queue. When

a new job arrives, it is assigned to the position l with probability g(l,n+1). Then,

jobs previously in positions l, l + 1, . . . ,n move to positions l + 1, l + 2, . . . ,n+ 1,

127


(a) (b)

Fig. A.5 An example for departure event. (a) Multi-queue system. (b) Logicallycombined queue.

(a) (b)

Fig. A.6 An example for jockeying upon departure event. (a) Multi-queue system.(b) Logically combined queue.

respectively. For our multi-queue server farm model, we can achieve insensitivity

by mapping the multi-queue system into a symmetric queue, as described in [21];

the available mappings are not unique. Because many configurations of multi-queue

server farm model can be mapped to the same symmetric queue configuration.

Let FJ be the set of all stationary jockeying policies. Let Fs be a subset of FJ ,

which are insensitive to the shape of the job-size distribution. Let Fsd be a subset

of Fs, where the policies in Fsd are deterministic policies. We define |n|, n 2N

(N is the set of all states defined in Chapter 3), as the total number of jobs in

the multi-queue system in state n, i.e. |n| = ÂKj=1 n j. Let p

f

m,L be the steady-state

probability that under policy f , there are m jobs in the multi-queue system and the

set of active servers is L. The probability of state n under policy f is p

f

n , as defined

128


Algorithm 1 Algorithm for an Insensitive Optimal Policy.Require: service rate µ = (µ1,µ2, . . . ,µK), power consumption e = (e1,e2, . . . ,eK),

buffer size B = (B1,B2, . . . ,BK), and arrival rate l .Ensure: An optimal policy that maximizes the energy efficiency among all insensi-

tive jockeying policies, namely, the sets of busy servers for each state, T OPT(n),n = 1,2, . . . , BK .

1: e1 02: e2 The value of E/T under any feasible policy3: e e1+e2

24: f0 any feasible policy for initialization5: W e OPTIMALCOSTPERCIRCLE(e,f0,f⇤)6: while |W e|> s/2 do7: if W e > s/2 then8: e1 e9: else

10: e2 e11: end if12: e e1+e2

213: Y e(0) 014: for n = 1 to BK , increment by 1 do15: j1, j2, . . . , jK represent servers decendingly ordered in terms of the value

of the right hand side of (3.13).16: l⇤ argmaxl2Le(n)y{ j1, j2,..., jl},e(n).17: T OPT(n) { j1, j2, . . . , jl⇤}.18: Y e(n) yT OPT(n),e(n).19: end for20: f

⇤ is the policy comprising of T OPT(n), n = 1,2, . . . , BK .21: W e g

f

⇤(µ)� eg

f

⇤(e).

22: end while

129


in Section 2.1. Acccordingly, we can write

p

f

m,L = Â|n|=m,

L(n)=L,n2N

p

f

n . (A.1)

Lemma A.1. Assume job-sizes to be exponentially distributed. For any policy

f 2FJ, we define a policy f

0 derived from f as follows. For a given state n 2N

under policy f , we reassign a job from server j1 to server j2, where n j1 > 2 and

1 6 n j2 6 b� 1. We perform this reassignment once and only once whenever the

process reaches n and the conditions for n j1 and n j2 hold. For other states n0 6= n,

n0 2N , the decisions made in n0 are the same as those in policy f . In this way,

policy f

0 is derived from f only by the reassignment at state n. Then, p

f

0

m,L = p

f

m,L.

Proof. Because of the memoryless property, the remaining work of each job is of the

same exponential distribution and will not change after the reassignment. The total

departure rate and the power consumption are also maintained after the reassignment,

since the set of active servers L does not change. After the reassignment in state

n, the state becomes state n0. We refer to the new sojourn time in state n0 as the

period which starts from the moment t1 that reassignment occurs to the moment t2

that the first arrival or departure occurs after t1. Because the arrival and departure

rate remain the same, the length of the new sojourn time in state n0 is statistically

equal to the length of the original period from t1 to the end of this sojourn in state n

without reassignment. When this sojourn ends at t2, the policy is again f , because

we consider here jockeying policies. The modification from n to n0 does not change

the steady state probability p

f

m,L, therefore, p

f

0

m,L = p

f

m,L.

Lemma A.2. There exists a deterministic policy f 2Fsd, which is optimal in Fs.

130


Proof. We consider a formulation for state n as follows.

V ef

⇤(n) = minp+(n,n0)p�(n,n0)

8

>

<

>

:

C (n)+ l

l+µ(n) · Ân02N

|n0|=|n|+1

p+(n,n0) ·V ef

⇤(n0)

+ µ(n)l+µ(n) Â

n02N|n0|=|n|�1

p�(n,n0) ·V ef

⇤(n0)

9

>

=

>

;

(A.2)

where p+(n,n0) and p�(n,n0) are the transition probabilities from state n to n0. They

satisfy

Ân02N

|n0|=|n|+1

p+(n,n0) = 1,

and

Ân02N

|n0|=|n|�1

p�(n,n0) = 1.

For such an policy f

⇤, if (A.2) holds for all n 2 N , then f

⇤ is opitmal among

policies in Fs. In a similar way, we define policy f

d 2Fsd as

V ef

d(n) = C (n)+ l

l+µ(n) · minn02N

|n0|=|n|+1

V ef

⇤(n0)+ µ(n)l+µ(n) min

n02N|n0|=|n|�1

V ef

⇤(n0). (A.3)

According to (A.2) and (A.3), for policy f

d , if (A.3) holds true for all n 2N , then

under f

d , (A.2) also holds true for all n 2N . That is, f

d is an optimal solution

among all policies in Fs. This proves the lemma.

Let S(L,m) = {n0,n1, . . . ,nk} be the set of all states in N , where L is the set

of active servers for states ni, i = 0, . . . ,k with |ni| = m, i = 0, . . . ,k. Those states

are always reachable under non-jockeying policies, but may not be reached under

jockeying policies. We refer to the set S(L,m) that contains reachable states under

policy f 2FJ as reachable S(L,m) under policy f . Henceforth, we only consider

jockeying policies.

131


We define a state n 2N that is reachable under a policy f 2 F0, F0 ⇢ FJ as

a possible state for F0. Recall that our multi-queue server farm model is mapped

to a single symmetric queue in the same way for all policies in Fs (as discussed

above) and that no hole is allowed in the single symmetric queue. Thus, for a given

set of active servers L and a given number of jobs in the system m, under any policy

f

s 2 Fs, the positions of jobs in the multi-queue system are also decided. Thus,

only one possible state for Fs exists in S(L,m) for given L and m. For a given

m 2 [0, BK], under any policy f

s 2 Fs, there always exists at least one reachable

S(L,m), since two jobs can not arrive or be completed simultaneously. In any

policy f 2 FJ , when the process steps into any state n 2 S(L,m), we reassign a

job from server j1 to server j2, where n j1 > 2 and 1 6 n j2 6 b�1. We repeat this

reassignment until the positions of jobs are the same as those in the possible state

for Fs in S(L,m). The indices j1 and j2 are variables which can be different at

each reassignment. We refer to f

0 2FJ as the new policy after the reassignments.

Based on Lemma A.1, each of these reassignments will not change the steady-state

probability p

f

m,L, and, we obtain p

f

m,L = p

f

0

m,L.

For a given set of active servers and number of jobs in the system, the positions

of jobs in the multi-queue system in a reachable state, under policy f

0, are the same

as those in a possible state for Fs. The reachable states under policy f

0 are therefore

the possible states for Fs.

With S(L0,m+1) = {n00,n01, . . . ,n0k0} and S(L00,m�1) = {n000,n001, . . . ,n00k00}, the

routing probabilities in f

0 are obtained by

q+(n,n00) =k0

Âj=0

kÂ

i=0

pniÂk

l=0 pnlp+(ni,n0j),

and

q�(n,n000) =k00

Âj=0

kÂ

i=0

pniÂk

l=0 pnlp�(ni,n00j ),

(A.4)

132

A.4 Proof of Lemma 3.2

where q+(n,n00) and q�(n,n000) are the routing probabilities of policy f

0, and n, n00and n000 are the reachable states in S(L,m), S(L0,m+ 1) and S(L00,m� 1) under

policy f

0, respectively. For any state n and unreachable state n under policy f

0,

q+(n, n) = 0 for |n|= |n|+1 and q�(n, n) = 0 for |n|= |n|�1.

Since reachable states in f

0 are the possible states for Fs, and for any given

number of jobs in the system m there exists at least one reachable S(L,m) under

policy f

0, there should be a policy f

s 2 Fs that has the same Markov chain as f

0.

Then, p

f

n = p

f

0n and p

f

m,L = p

f

0

m,L. That is, p

f

m,L = p

f

s

m,L. Since L decides the current

service rate and power consumption of the system, f and f

s also have the same

expected throughput and power consumption.

The proof for Proposition 3.1 is given next.

Proof. Use f

⇤ 2 Fsd to denote I-J-OPT. Assume that there exists a policy f0 2

FJ, f0 6= f

⇤, satisfyingE

f0

Tf0

<E

f

⇤

Tf

⇤. (A.5)

Let f1 be the insensitive jockeying policy associated to f0, where f1 and f0 have the

same expected power consumption and throughput. Thus, policy f1 2Fs, satisfies

Ef0

Tf0

=E

f1

Tf1

<E

f

⇤

Tf

⇤. (A.6)

Since f

⇤ is optimal in Fsd , it is also optimal in Fs (Lemma A.2), and this contradicts

(A.6). The theorem is now proved.


Some notation have been defined in Chapter 3.

133


Proof. We can rewrite FA as

FSSFA =

Ân2N SSF

A

FSSF(n)pSSFn

Ân2N SSF

A

pSSFn

=

bK�1Â

i=1

BiÂ

k=1Fi(

ril

)kbK

’j=i+1

(r jl

)B j

bK�1Â

i=1

BiÂ

k=1( ri

l

)kbK

’j=i+1

(r jl

)B j

, (A.7)

where Fi = Âij=1 e j/ri. Since for j = 1,2, . . . , bK�2, B j+1 � B j � 1, (A.7) is equiv-

alent to

FSSFA =

bK�2Â

m=0

Bm+1Â

k=Bm+1

bK�1Â

i=m+1Fi� ri

l

�kbK

’j=i+1

� r jl

�B j

bK�2Â

m=0

Bm+1Â

k=Bm+1

bK�1Â

i=m+1

� ril

�kbK

’j=i+1

� r jl

�B j

(A.8)

where we have set B0 = 0.

Let

ai =

8

>

>

<

>

>

:

bK’

j=i+1

� r jl

�B j�1 � ril

�k, i = m+1,m+2, . . . , bK�1,

0, otherwise.

Since r j/l 1, B j � 1, j = 1,2, . . . , bK, k = Bm + 1,Bm + 2, . . . ,Bm+1, and m =

0,1, . . . , bK�2, we obtain ai+1 � ai for i = 1,2, . . . , bK�1.

Since Fi+1 �Fi, and ai+1 � ai, i = 1,2, . . . , bK�1, we obtain

bK�1Â

i=m+1Fi� ri

l

�kbK

’j=i+1

� r jl

�B j

bK�1Â

i=m+1

� ril

�kbK

’j=i+1

� r jl

�B j

=

bK�1Â

i=1Fi

bK’

j=i+1

� r jl

�

ai

bK�1Â

i=1

bK’

j=i+1

� r jl

�

ai

�

bK�1Â

i=1Fi

bK’

j=i+1(

r jl

)

bK�1Â

i=1

bK’

j=i+1(

r jl

)

. (A.9)

Or equivalently,

FSSFA �

bK�2Â

m=0

Bm+1Â

k=Bm+1

bK�1Â

i=1Fi� ri

l

�kbK

’j=i+1

� r jl

�

bK�2Â

m=0

Bm+1Â

k=Bm+1

bK�1Â

i=1

� ril

�kbK

’j=i+1

� r jl

�

= FE⇤(bK)A .

This proves the lemma.

134



Proof. Since Ff

B = ÂbKj=1 e j/r

bK when f =SSF, E⇤(bK), (3.71) is proved. Equation

(3.72) can be written as

Ân2N SSF

C

qSSFn FSSF(n)

Ân2N SSF

C

qSSFn

=

Ân2N E⇤(bK)

C

qE⇤(bK)n FE⇤(bK)(n)

Ân2N E⇤(bK)

C

qE⇤(bK)n

(A.10)

where qf

n is defined in (3.65). Since N SSFC = N E⇤(bK)

C with qSSFn = qE⇤(bK)

n and

FSSF(n) = FE⇤(bK)(n) for all n 2 N SSFC (n 2 N E⇤(bK)

C ), (A.10) holds true. This

proves the lemma.


Proof. According to the definition of qf

n in (3.65), we obtain

pf

Bbn=

1

ÂBn=0 qf

n, (A.11)

since ÂBn=0 pf

n = 1. Based on the definition (3.65), we also obtain

qSSFn

8

>

<

>

:

= qE⇤(bK)n , B

bK�1 +1 n B,

qE⇤(bK)K , 0 n B

bK�1,(A.12)

where qSSFBbK= qE⇤(bK)

BbK

= 1. Thus ÂBn=0 qSSF

n ÂBn=0 qE⇤(bK)

n . Together with (A.11), this

implies pSSFBbK� pE⇤(bK)

BbK

. Then, by (A.12) and (3.65), we get pSSFBbK

qSSFn � pE⇤(bK)

BbK

qE⇤(bK)n

135


for BbK +1 n B. This also reads as

pSSF(r j) =B j

Ân=B j�1+1

pSSFn �

B j

Ân=B j�1+1

pE⇤(bK)n = pE⇤(bK)(r j), r j 2 RC, j = 1,2, . . . ,K

and this proves the lemma.


For clarity of presentation, we define

Pf

q

= Ân2N f

q

pf

n . (A.13)

Lemma A.3. Consider the system defined in Section 3.2, if B j � 2, l �p

5+12 r

bK�1,

j = 1,2, . . . , bK�1, then

l

r j

l

r j

B j+1�1Â

i=0

⇣

l

r j+1

⌘i

B j�1Â

i=0

� r jl

�i, j = 1,2, . . . , bK�1. (A.14)

Proof. We rewrite (A.14) as

B j�1

Âi=0

⇣r j

l

⌘i

B j+1�1

Âi=0

✓

l

r j+1

◆i. (A.15)

For the case where B j+1 = 2, B j is sufficiently large, and r j = r j+1 =p

5�12 l

(l =p

5+12 r

bK�1), (A.15) can be specified as

1

1�p

5�12

1+p

5+12

, (A.16)

136


which clearly holds true. Also, (A.16) is also a sufficient condition for (A.15). This

proves this lemma.

Lemma A.4. Consider the system defined in Section 3.2. If B j � 2, j = 1,2, . . . ,K,

l �p

2+12 r

bK�1, and PSSFA PE⇤(bK)

A , then

FSSFA PSSF

A �FE⇤(bK)A PE⇤(bK)

A �ÂbK�1

j=1 e j

rbK�1

(PSSFB �PE⇤(bK)

B ). (A.17)

Proof. Based on definitions (3.64) and (3.67), for j = 1,2, . . . , bK�2, we have

pE⇤(bK)(r j+1) = pE⇤(bK)j+1 = pE⇤(bK)

jl

r j= pE⇤(bK)(r j)

l

r j, (A.18)

and

pSSF(r j+1) =B j+1

Âi=B j+1

pSSFi =

l

r j

B j+1�1

Âi=0

✓

l

r j+1

◆ipSSF

B j=

l

r j

B j+1�1Â

i=0

⇣

l

r j+1

⌘i

B j�1Â

i=0

� r jl

�ipSSF(r j).

(A.19)

According to Lemma A.3, it holds

pE⇤(bK)(r j+1)

pE⇤(bK)(r j)

pSSF(r j+1)

pSSF(r j). (A.20)

Thus, if pSSF(r j+1) pE⇤(bK)(r j+1), then, pSSF(r j+1) pE⇤(bK)(r j); while if pSSF(r j)>

pE⇤(bK)(r j), then, pSSF(r j+1)> pE⇤(bK)(r j+1). Therefore there exists j⇤ 2 {1,2, . . . , bK�

2} satisfying

pSSF(r j)

8

>

<

>

:

> pE⇤(bK)(r j), j⇤+1 j bK�1,

pE⇤(bK)(r j), 1 j j⇤.(A.21)

137


That is

FSSFA PSSF


A =bK�1

Âj=1

F j

⇣

pSSF(r j)� pE⇤(bK)(r j)⌘

=j⇤

Âj=1

F j

⇣


+bK�1

Âj= j⇤+1

F j

⇣


= FbK�1j⇤+1

bK�1

Âj= j⇤+1

⇣


�F j⇤1

j⇤

Âj=1

⇣

pE⇤(bK)(r j)� pSSF(r j)⌘

(A.22)

where we recall F j = Â ji=1 ei/r j, and define

F ji =

jÂ

j0=iF j0 |pSSF(r j0)� pE⇤(bK)(r j0)|

jÂ

j0=i|pSSF(r j0)� pE⇤(bK)(r j0)|

, i = 1,2, . . . , j.

Then, according to (A.22), we obtain

FSSFA PSSF


A �F j⇤1

bK�1

Âj=1

⇣


. (A.23)

SincebK�1Âj=1

⇣


0, (A.23) implies

FSSFA PSSF


A �Fbk�1

bK�1

Âj=1

⇣


. (A.24)


A proof of Proposition 3.7.

138

A.8 Lemmas for Proposition 3.6

Proof. Since FSSFB = FE⇤(bK)

B and FSSFC = FE⇤(bK)

C by Lemma 3.3, the condition (3.73)

of our feasible region can be written as

FSSFA PSSF


A +FSSFB (PSSF

B �PE⇤(bK)B )+FSSF

C (PSSFC �PE⇤(bK)

C )� 0.

Based on Lemma A.4,

FSSFA PSSF


A +FSSFB (PSSF



C )

�FbK�1(P

SSFA �PE⇤(bK)

A )+FSSFB (PSSF



C )

= (FSSFB �F

bK�1)(PE⇤(bK)A �PSSF

A )+(FSSFC �FSSF

B )(PSSFC �PE⇤(bK)

C )

� (PSSFB �PE⇤(bK)

B )(FSSFB �FE⇤(bK)

A )� 0. (A.25)

The identity is obtained based on PFA +Pf

B +Pf

C = 1 for f = SSF,E⇤(bK). Since

PSSFC � PE⇤(bK)

C by Lemma 3.4, we obtain the second inequality. Note that PSSFA <

PE⇤(bK)A (equality is not achieved) requires bK < K. Thus, (3.73) holds true, which is

equivalent to {2,3, . . . ,K�1}✓FbK .


Lemma A.5. For a general system, ifp

2�12 µ3 � µ1, e3/µ3 �

p2e2/µ2, then

µ2e3� e2µ3 � 2(µ1e2� e1µ2). (A.26)

Proof. For clarity of presentation, let a1 = e1/µ1, a2 = e2/µ2 and a3 = e3/µ3.

Inequality (A.26) is equivalent to

a3µ2µ3�a2µ2µ3 � 2(a2µ1µ2�a1µ1µ2), (A.27)

139


ora3�a2

a2�a1� 2µ1

µ3. (A.28)

Using a3 �p

2a2, we obtain the inequality

a3�a2

a2�a1� 2�

p2p

2�2a1a3

. (A.29)

A sufficient condition for (A.28) is then that

2�p

2p2� 2µ1

µ3, (A.30)

orµ1

µ3p

2�12

.

As the latter holds true, the lemma is proven.

Lemma A.6. For an extended policy with k 2 [1,B1], if l � µ1 +µ2 +µ3, then

⇣

l

µ1

⌘

k

⇣

l

r2

⌘B2⇣

l

r1

⌘B1l

r3

✓

⇣

l

r3

⌘B3�1◆

l

r3�1

2

4

l

r2

✓

⇣

l

r2

⌘B2+B1�k

�1◆

l

r2�1

�⇣

l

r2

⌘B1�k

⇣

l

r2

⌘

✓

⇣

l

r2

⌘B2�1◆

⇣

l

r2

⌘

�1

3

5

+ l

µ1

⇣

l

r2

⌘B2l

r3

✓

⇣

l

r3

⌘B3�1◆

⇣

l

r3

⌘

�1

2

4

⇣

l

µ1

⌘B1⇣

l

µ1

⌘

k

�1l

µ1�1�⇣

l

r2

⌘B1�k

⇣

l

µ1

⌘

k�1 l

µ1

✓

⇣

l

µ1

⌘B1�1◆

⇣

l

µ1

⌘

�1

3

5� 0.

(A.31)

Proof. The inequality (A.31) is the same as

⇣

l

µ1

⌘

k

⇣

l

r2

⌘B2⇣

l

r1

⌘B1l

r3

✓

⇣

l

r3

⌘B3�1◆

l

r3�1

0

@

l

r2

✓

⇣

l

r2

⌘B1�k

�1◆

l

r2�1

1

A

+ l

µ1

⇣

l

r2

⌘B2 l

r3

⇣

l

r3

⌘B3�1l

r3�1

2

4

⇣

l

µ1

⌘B1⇣

l

µ1

⌘

k

�1l

µ1�1�⇣

l

r2

⌘B1�k

⇣

l

µ1

⌘

k�1 l

µ1

✓

⇣

l

µ1

⌘B1�1◆

l

µ1�1

3

5� 0,

140


or

l

µ1

⇣

l

r2

⌘B2 l

r3

⇣

l

r3

⌘B3�1l

r3�1

2

4

⇣

l

r1

⌘B1+k�1 l

r2

✓

⇣

l

r2

⌘B1�k

�1◆

l

r2�1

+⇣

l

µ1

⌘B1⇣

l

µ1

⌘

k�1�1

l

µ1�1

�⇣

l

r2

⌘B1�k

⇣

l

µ1

⌘

k�1 l

µ1

✓

⇣

l

µ1

⌘B1�1◆

l

µ1�1

1

A� 0,

or

Âk�1i=1

⇣

l

µ1

⌘B1+i�1�⇣

l

r2

⌘B1�k

⇣

l

µ1

⌘i+k�1�

+ÂB1i=k

⇣

l

µ1

⌘B1+k�1⇣l

r2

⌘i�k

�⇣

l

r2

⌘B1�k

⇣

l

µ1

⌘i+k�1�

� 0,

or

k�1

Âi=1

✓

l

µ1

◆i+k�1"

✓

l

µ1

◆B1�k

�✓

l

r2

◆B1�k

#

+B1

Âi=k

✓

l

µ1

◆i+k�1✓l

r2

◆i�k+1"

✓

l

µ1

◆B1�i�✓

l

r2

◆B1�i�1#

� 0. (A.32)

As the inequality (A.32) holds, Lemma A.6 is proven.

Lemma A.7. For a general system and a given integer k 2 [1,B1], if l � µ1 +µ2 +

µ3, µ3 � 1B3�1(µ1 +µ2) and B3 � 2, then,

⇣

l

r2

⌘B1+B2�k

0

@

⇣

l

r3

⌘

✓

⇣

l

r3

⌘B3�1◆

l

r3�1

1

A�l

r2

✓

⇣

l

r2

⌘B1+B2�k

�1◆

l

r2�1

. (A.33)

Proof. The inequality (A.33) is equivalent to

l

r3

✓

⇣

l

r3

⌘B3�1

◆

⇣

l

r2

⌘B1+B2�k

⇣

l

r2�1

⌘

� l

r2

✓

⇣

l

r2

⌘B1+B2�k

�1◆

⇣

l

r3�1

⌘

,

141


or

l

r3

✓

⇣

l

r3

⌘B3�1

◆

⇣

l

r2

⌘B1+B2�k+1�⇣

l

r3�1

⌘⇣

l

r2

⌘B1+B2�k+1

� l

r3

✓

⇣

l

r3

⌘B3�1

◆

⇣

l

r2

⌘B1+B2�k

+⇣

l

r3�1

⌘

l

r2� 0,

or

✓

⇣

l

r3

⌘B3+1�2 l

r3+1

◆

⇣

l

r2

⌘B1+B2�k+1

� l

r3

✓

⇣

l

r3

⌘B3�1

◆

⇣

l

r2

⌘B1+B2�k

+⇣

l

r3�1

⌘

l

r2� 0,

or

⇣

l

r2

⌘B1+B2�k

⇣

l

r2�1

⌘⇣

l

r3

⌘B3+1�⇣

2 l

r2�1

⌘

l

r3+ l

r2

�

+⇣

l

r3�1

⌘

l

r2� 0.

Setting

f (l ) =✓

l

r2�1

◆✓

l

r3

◆B3+1�✓

2l

r2�1

◆

l

r3+

l

r2.

142


We compute the derivatives. It is a sample matter to check that

8

>

>

>

>

>

>

>

>

>

>

>

>

>

>

>

>

>

>

>

>

>

>

>

>

>

>

<

>

>

>

>

>

>

>

>

>

>

>

>

>

>

>

>

>

>

>

>

>

>

>

>

>

>

:

f (l ) = 1r2rB3+1

3l

B3+2� 1rB3+13

l

B3+1� 2r2r3

l

2 +⇣

1r3+ 1

r2

⌘

l ,

f (1)(l ) = B3+2r2rB3+1

3l

B3+1� B3+1rB3+13

l

B3� 4r2r3

l + 1r3+ 1

r2,

f (2)(l ) = (B3+2)(B3+1)r2rB3+1

3l

B3� (B3+1)B3

rB3+13

l

B3�1� 4r2r3

,

f (3)(l ) = (B3+2)(B3+1)B3

r2rB3+13

l

B3�1� (B3+1)B3(B3�1)rB3+1

3l

B3�2,

= (B3+1)B3l

B3�2

rB3+13

⇣

(B3 +2) l

r2�B3 +1

⌘

.

For l � r3 > r2, f (l ), f (1)(l ), f (2)(l ), and f (3)(l ) are straightforwardly positive.

Since µ3 � 1B3�1r2, we obtain r3

r2� B3

B3�1 , hence, f (2)(r3) =(B3+2)(B3+1)

r2r3�

(B3+1)B3r2

3� (B3+1)(B3+2)

r23

⇣

r3r2� B3

B3+2

⌘

> 0, f (1)(r3) =B3�1

r2� B3

r3� 0 and f (r3) = 0,

so that f (l )� 0 is true for all l � r3. This establishes Lemma A.7.

Lemma A.8. For a general system and a given integer k 2 [1,B1� 1], if l �

µ1 +µ2 +µ3 and µ1 µ2, then,

l

µ1

2

4

⇣

l

µ1

⌘B1

⇣

l

µ1

⌘

k

�1l

µ1�1

!

l

r2

✓

⇣

l

r2

⌘B2�1◆

l

r2�1

�⇣

l

µ1

⌘

k�1 l

µ1

✓

⇣

l

µ1

⌘B1�1◆

l

µ1�1

l

r2

✓

⇣

l

r2

⌘B1+B2�k

�1◆

l

r2�1

3

5

+ l

µ1

⇣

l

r2

⌘B2l

r3

✓

⇣

l

r3

⌘B3�1◆

l

r3�1

2

4

⇣

l

µ1

⌘B1⇣

l

µ1

⌘

k

�1l

µ1�1�⇣

l

r2

⌘B1�k

⇣

l

µ1

⌘

k�1 l

µ1

✓

⇣

l

µ1

⌘B1�1◆

l

µ1�1

3

5

+2⇣

l

µ1

⌘

k

⇣

l

r2

⌘B2⇣

l

r1

⌘B1l

r3

✓

⇣

l

r3

⌘B3�1◆

l

r3�1

2

4

l

r2

✓

⇣

l

r2

⌘B1+B2�k

�1◆

l

r2�1

�⇣

l

r2

⌘B1�k

l

r2

✓

⇣

l

r2

⌘B2�1◆

l

r2�1

3

5

� 0.(A.34)

143


Proof. The inequality (A.34) is the same as

l

µ1

2

4

⇣

l

µ1

⌘B1

⇣

l

µ1

⌘

k

�1l

µ1�1

!

l

r2

✓

⇣

l

r2

⌘B2�1◆

l

r2�1

�l

µ1

✓

⇣

l

µ1

⌘B1�1◆

l

µ1�1

⇣

l

µ1

⌘

k�1 l

r2

✓

⇣

l

r2

⌘B1+B2�k

�1◆

l

r2�1

3

5

+ l

µ1

⇣

l

r2

⌘B2l

r3

✓

⇣

l

r3

⌘B3�1◆

l

r3�1

"

2⇣

l

r1

⌘B1+k�1⇣

l

r2

⌘B1�k+1�1

l

r2�1

�⇣

l

r1

⌘B1+k�1�⇣

l

r2

⌘B1�k

⇣

l

µ1

⌘

k�1 l

µ1

✓

⇣

l

µ1

⌘B1�1◆

l

µ1�1

+⇣

l

µ1

⌘B1⇣

l

µ1

⌘

k�1�1

l

µ1�1

3

5� 0.

Based on Lemma A.7, a sufficient condition for (A.8) to occur is

l

µ1

⇣

l

µ1

⌘B1

⇣

l

µ1

⌘

k

�1l

µ1�1

!

2

4

l

r2

✓

⇣

l

r2

⌘B2�1◆

l

r2�1

3

5

+ l

µ1

⇣

l

r2

⌘B2l

r3

✓

⇣

l

r3

⌘B3�1◆

l

r3�1

"

2⇣

l

r1

⌘B1+k�1⇣

l

r2

⌘B1�k+1�1

l

r2�1

�⇣

l

r1

⌘B1+k�1�2

⇣

l

r2

⌘B1�k

⇣

l

µ1

⌘

k�1 l

µ1

✓

⇣

l

µ1

⌘B1�1◆

l

µ1�1

+⇣

l

µ1

⌘B1⇣

l

µ1

⌘

k�1�1

l

µ1�1

3

5� 0.

The expression in the square parenthesis

2⇣

l

r1

⌘B1+k�1⇣

l

r2

⌘B1�k+1�1

l

r2�1

�⇣

l

r1

⌘B1+k�1

�2⇣

l

r2

⌘B1�k

⇣

l

µ1

⌘

k�1 l

µ1

✓

⇣

l

µ1

⌘B1�1◆

l

µ1�1

+⇣

l

µ1

⌘B1⇣

l

µ1

⌘

k�1�1

l

µ1�1

,

144


is equal to

Âk�1i=1

⇣

l

µ1

⌘B1+i�1�2

⇣

l

r2

⌘B1�k

⇣

l

µ1

⌘i+k�1�

+⇣

l

µ1

⌘B1+k�1

�⇣

l

r2

⌘B1�k

⇣

l

µ1

⌘2k�1+2ÂB1

i=k+1

⇣

l

µ1

⌘B1+k�1⇣l

r2

⌘i�k

�⇣

l

r2

⌘B1�k

⇣

l

µ1

⌘i+k�1�

.

(A.35)

We can rewrite (A.35) as

k�1Â

i=1

⇣

l

µ1

⌘i+k�1

⇣

l

µ1

⌘B1�k

�2⇣

l

r2

⌘B1�k

�

+⇣

l

µ1

⌘2k�1

⇣

l

µ1

⌘B1�k

�⇣

l

r2

⌘B1�k

�

+2B1Â

i=k+1

⇣

l

µ1

⌘i+k�1⇣l

r2

⌘i�k

⇣

l

µ1

⌘B1�i�⇣

l

r2

⌘B1�i�

.

(A.36)

Since r1 = µ1, l � µ1 +µ2 +µ3, µ2 � µ1, (A.36) is no less than zero. Thus (A.8)

is true and Lemma A.8 is proven.

Lemma A.9. For a general system, ifp

2�12 µ3 � µ1 and e3/µ3 �

p2e2/µ2, then

µ1e3 +µ2e3 +µ1e4 +µ2e4� e1µ3� e1µ4� e2µ3� e2µ4

� µ1e2 +µ1e3 +µ1e4� e1µ2� e1µ3� e1µ4.

(A.37)

Proof. Inequality (A.37) is equivalent to

µ2e3 +µ2e4� e2µ3� e2µ4 � µ1e2� e1µ2. (A.38)

According to Lemma A.5, we have µ2e3� e2µ3 � µ1e2� e1µ2, so that a sufficient

condition for (A.38) is

µ2e4� e2µ4 � 0. (A.39)

As this last inequality holds true, Lemma A.9 is proven.

145



Proof. For clarity of presentation, Let E = E E⇤k ,L = L E⇤

k and E 0 = E SSF,L 0 =

L SSF. Note that bK = 2 is set through all this proof. Also, let p

0n and pn be the steady

state probability of state n under SSF and E⇤k

, respectively. The reciprocal values of

energy efficiency for SSF and E⇤k

are given by

E 0

L 0 =p

00

L 0

0

B

B

@

l

µ1

✓

⇣

l

µ1

⌘B1�1

◆

l

µ1�1

e1 +

✓

l

µ1

◆B1l

r2

✓

⇣

l

r2

⌘B2�1

◆

l

r2�1

(e1 + e2)

+

✓

l

µ1

◆B1✓

l

r2

◆B2l

r3

✓

⇣

l

r3

⌘B3�1

◆

l

r3�1

(e1 + e2 + e3)

+

✓

l

µ1

◆B1✓

l

r2

◆B2✓

l

r3

◆B3l

r4

✓

⇣

l

r4

⌘B4�1

◆

l

r4�1

(e1 + e2 + e3 + e4)

1

C

C

A

, (A.40)

and

E

L=

p0

L

0

B

B

@

l

µ1

⇣⇣

l

µ1

⌘

k

�1⌘

l

µ1�1

e1 +

✓

l

µ1

◆

k

l

r2

✓

⇣

l

r2

⌘B1+B2�k

�1◆

l

r2�1

(e1 + e2)

+

✓

l

µ1

◆

k

✓

l

r2

◆B1+B2�k

l

r3

✓

⇣

l

r3

⌘B3�1

◆

l

r3�1

(e1 + e2 + e3)

+

✓

l

µ1

◆

k

✓

l

r2

◆B1+B2�k

✓

l

r3

◆B3l

r4

✓

⇣

l

r4

⌘B4�1

◆

l

r4�1

(e1 + e2 + e3 + e4)

1

C

C

A

, (A.41)

where ri = Âij=1 µ j, as defined before. In addition, according to the definitions of

policies MEESF and E⇤ for our server farms,

146


L 0

p

00=

l

µ1

✓

⇣

l

µ1

⌘B1�1

◆

l

µ1�1

·µ1 +

✓

l

µ1

◆B1l

r2

✓

⇣

l

r2

⌘B2�1

◆

l

r2�1

· (µ1 +µ2)

+

✓

l

µ1

◆B1✓

l

r2

◆B2l

r3

✓

⇣

l

r3

⌘B3�1

◆

l

r3�1

· r3

+

✓

l

µ1

◆B1✓

l

r2

◆B2✓

l

r3

◆B3l

r4

✓

⇣

l

r4

⌘B4�1

◆

l

r4�1

· r4, (A.42)

and

L

p0=

l

µ1(⇣

l

µ1

⌘

k

�1)l

µ1�1

·µ1 +

✓

l

µ1

◆

k

l

r2(( l

r2)B1+B2�k �1)

l

r2�1

· (µ1 +µ2)

+

✓

l

µ1

◆

k

✓

l

r2

◆B1+B2�k

l

r3

✓

⇣

l

r3

⌘B3�1

◆

l

r3�1

· r3

+

✓

l

µ1

◆

k

✓

l

r2

◆B1+B2�k

✓

l

r3

◆B3l

r4

✓

⇣

l

r4

⌘B4�1

◆

l

r4�1

· r4. (A.43)

We want to show that E 0/L 0 �E /L � 0, which is equivalent to

E 0/L 0

p

00/L

0 ·L

p0� E /L

p0/L· L

0

p

00� 0. (A.44)

Substituting factors in (A.44) by (A.40)-(A.43), we obtain a long expression for

E 0/L 0 �E /L , which is the sum of the four terms, S1, . . . ,S4, given by

S1 =

l

µ1

⇣⇣

l

µ1

⌘

k

�1⌘

l

µ1�1

· µ1 · E 0/L 0

p

00/L

0 �l

µ1

⇣⇣

l

µ1

⌘

k

�1⌘

l

µ1�1

· e1 · L 0

p

00, (A.45)

S2 =

✓

l

µ1

◆

k

·

l

r2

✓

⇣

l

r2

⌘B1+B2�k

�1◆

l

r2�1

r2 ·E 0/L 0

p

00/L

0

147


�✓

l

µ1

◆

k

·

l

r2

✓

⇣

l

r2

⌘B1+B2�k

�1◆

l

r2�1

(e1 + e2)L 0

p

00, (A.46)

S3 =

✓

l

µ1

◆

k

✓

l

r2

◆B1+B2�k

l

r3

✓

⇣

l

r3

⌘B3�1

◆

l

r3�1

r3 ·E 0/L 0

p

00/L

0

�✓

l

µ1

◆

k

✓

l

r2

◆B1+B2�k

l

r3

✓

⇣

l

r3

⌘B3�1

◆

l

r3�1

(e1 + e2 + e3)L 0

p

00, (A.47)

and

S4 =

✓

l

µ1

◆

k

✓

l

r2

◆B1+B2�k

✓

l

r3

◆B3l

r4

✓

⇣

l

r4

⌘B4�1

◆

l

r4�1

r4 ·E 0/L 0

p

00/L

0

�✓

l

µ1

◆

k

✓

l

r2

◆B1+B2�k

✓

l

r3

◆B3l

r4

✓

⇣

l

r4

⌘B4�1

◆

l

r4�1

(e1 + e2 + e3 + e4)L 0

p

00.

(A.48)

Recall that

S1 +S2 +S3 +S4 =E 0/L 0

p

00/L

0 ·L

p0� E /L

p0/L· L

0

p

00,

and that we want to show that S1 +S2 +S3 +S4 � 0. But the sum S1 +S2 +S3 +S4

is also equal to the sum of x1 + x2 + x3 + x4 + x5 where

x1 = (µ1e2� e2µ2)l

µ1

2

6

6

4

✓

l

µ1

◆B1

⇣

l

µ1

⌘

k

�1l

µ1�1

l

r2

✓

⇣

l

r2

⌘B2�1

◆

l

r2�1

�✓

l

µ1

◆

k�1l

µ1

✓

⇣

l

µ1

⌘B1�1

◆

l

µ1�1

·

l

r2

✓

⇣

l

r2

⌘B1+B2�k

�1◆

l

r2�1

3

7

7

5

, (A.49)

148


x2 = (µ1e2 +µ1e3� e1µ2� e1µ3)l

µ1·✓

l

r2

◆B2l

r3

✓

⇣

l

r3

⌘B3�1

◆

l

r3�1

·

2

6

6

4

✓

l

µ1

◆B1

⇣

l

µ1

⌘

k

�1l

µ1�1

�✓

l

r2

◆B1�k

✓

l

µ1

◆

k�1l

µ1

✓

⇣

l

µ1

⌘B1�1

◆

l

µ1�1

3

7

7

5

,

(A.50)

x3 = (µ1e3 +µ2e3� e1µ3� e2µ3)

✓

l

µ1

◆

k

✓

l

r2

◆B2✓

l

r1

◆B1

·

l

r3

✓

⇣

l

r3

⌘B3�1

◆

l

r3�1

·

2

6

6

4

l

r2

✓

⇣

l

r2

⌘B1+B2�k

�1◆

l

r2�1

�✓

l

r2

◆B1�k

l

r2

✓

⇣

l

r2

⌘B2�1

◆

l

r2�1

3

7

7

5

,

(A.51)

x4 = (µ1e2 +µ1e3 +µ1e4� e1µ2� e1µ3� e1µ4)l

µ1

✓

l

r2

◆B2✓

l

r3

◆B3

·

l

r4

✓

⇣

l

r4

⌘B4�1

◆

l

r4�1

2

6

6

4

⇣

l

µ1

⌘

k

�1l

µ1�1

✓

l

µ1

◆B1

�✓

l

r2

◆B1�k

✓

l

µ1

◆

k�1l

µ1

✓

⇣

l

µ1

⌘B1�1

◆

l

µ1�1

3

7

7

5

,

(A.52)

and

x5 =(µ1e3 +µ2e3 +µ1e4 +µ2e4� e1µ3� e1µ4� e2µ3� e2µ4)

✓

l

µ1

◆B1+k

✓

l

r2

◆B2

✓

l

r3

◆B3l

r4

✓

⇣

l

r4

⌘B4�1

◆

l

r4�1

2

6

6

4

l

r2

✓

⇣

l

r2

⌘B1+B2�k

�1◆

l

r2�1

�✓

l

r2

◆B1�k

l

r2

✓

⇣

l

r2

⌘B2�1

◆

l

r2�1

3

7

7

5

.

(A.53)

149


We shall prove S1 + S2 + S3 + S4 � 0 by showing x1 + x2 + x3 + x4 + x5 � 0. A

sufficient condition for this inequality is that x1 + x2 + x3 � 0 and x4 + x5 � 0.

As, we want to show that x1 + x2 + x3 � 0; we rewrite x1 + x2 + x3 as

[(µ2e3� e2µ3)y3 +(µ1e2� e1µ2)(y1 + y2)]+(µ1e3� e1µ3)(y3 + y2)

where8

>

>

>

>

<

>

>

>

>

:

y1 = x1/(µ1e2� e1µ2) ,

y2 = x2/(µ1e2 +µ1e3� e1µ2� e1µ3) ,

y3 = x3/(µ1e3 +µ2e3� e1µ3� e2µ3) .

But, by Lemma A.5, we have

µ2e3� e2µ3 � 2(µ1e2� e1µ2) ,

where µ3 � µ1 and e3µ3�p

2 e2µ2

, and a sufficient condition for x1 + x2 + x3 � 0 is

therefore

2y3 + y1 + y2 � 0,andy3 + y2 � 0.

These inequalities were established as Lemma A.6 and Lemma A.8, and therefore

x1 + x2 + x3 � 0.

To show x4 + x5 � 0, we proceed as follows: Based on Lemma A.9, we obtain

µ1e3 +µ2e3 +µ1e4 +µ2e4� e1µ3� e1µ4� e2µ3� e2µ4

� µ1e2 +µ1e3 +µ1e4� e1µ2� e1µ3� e1µ4.

Thus, a sufficient condition for x4+x5 � 0 is y4+y5 � 0, where y4,y5 are defined by

y4 =⇣

l

µ1

⌘B1+k

⇣

l

r2

⌘B2⇣

l

r3

⌘B3l

r4

✓

⇣

l

r4

⌘B4�1◆

l

r4�1

2

4

l

r2

✓

⇣

l

r2

⌘B1+B2�k

�1◆

l

r2�1

�⇣

l

r2

⌘B1�k

l

r2

✓

⇣

l

r2

⌘B2�1◆

l

r2�1

3

5 ,

150


and

y5 =l

µ1

⇣

l

r2

⌘B2⇣

l

r3

⌘B3l

r4

✓

⇣

l

r4

⌘B4�1◆

l

r4�1

2

4

⇣

l

r1

⌘B1⇣

l

µ1

⌘

k

�1l

µ1�1�⇣

l

r2

⌘B1�k

⇣

l

µ1

⌘

k�1 l

r1

✓

⇣

l

r1

⌘B1�1◆

l

r1�1

3

5 .

Note that y4 + y5 � 0 is the same as

⇣

l

µ1

⌘B1+k�10

@

l

r2

✓

⇣

l

r2

⌘B1+B2�k

�1◆

l

r2�1

�⇣

l

r2

⌘B1�k

⇣

l

r2

⌘B2�1l

r2�1

1

A

+⇣

l

r1

⌘B1⇣

l

µ1

⌘

k

�1l

µ1�1�⇣

l

r2

⌘B1�k

⇣

l

µ1

⌘

k�1 l

r1

✓

⇣

l

r1

⌘B1�1◆

l

r1�1

� 0,

or

⇣

l

µ1

⌘B1+k�1 l

r2

✓

⇣

l

r2

⌘B1�k

�1◆

l

r2�1

+⇣

l

r1

⌘B1⇣

l

µ1

⌘

k

�1l

µ1�1�⇣

l

r2

⌘B1�k

⇣

l

µ1

⌘

k�1 l

r1

✓

⇣

l

r1

⌘B1�1◆

l

r1�1

� 0,

or

k�1

Âi=1

✓

l

µ1

◆i+k�1"

✓

l

µ1

◆B1�k

�✓

l

r2

◆B1�k

#

+B1

Âi=k

✓

l

µ1

◆i+k�1✓l

r2

◆i�k

"

✓

l

µ1

◆B�i�✓

l

r2

◆B�i#

� 0 (A.54)

But (A.54) is clearly true, then x4 + x5 � 0. In summary, we have x1 + x2 + x3 + x4 +

x5 � 0 as deserved.

All the lemmas used in above proof are given in Appendix A.8.


For clarity of presentation, we define a new system associated with the system

defined in Section 3.2. We refer to the system defined in Section 3.2 as the original

151


system and to the new system as a surrogate system. Given an original system

characterized by given values of bK for E⇤, l , µ j, and e j, j = 1, . . . ,K, we construct

a surrogate system, where the average arrival rate of this surrogate system is set to

l . This surrogate system is constructed as follows. Four servers are defined in the

surrogate system, for which the service rates and power consumptions are denoted

by n1, n2, n3 and n4, and e1, e2, e3 and e4, respectively. These service rates and

power consumptions are set to satisfy following equations.

8

>

>

>

>

>

>

>

>

>

<

>

>

>

>

>

>

>

>

>

:

ÂBbK�1

i=1�

n1l

�i=

bK�2Â

i=1

⇣

ÂBij=1

� ril

� j⌘

·⇣

’bK�1j=i+1

� r jl

�B j⌘

+BbK�1Âj=1

(rbK�1l

) j,

n1+n2l

= ÂbKi=1 µil

,

n3l

=µ

bK+1l

,

n4l

=µ

bK+2l

,

(A.55)

and

8

>

>

>

>

>

>

>

<

>

>

>

>

>

>

>

:

e1 = FSSFA n1,

e2 + e2 = ÂbKj=1 e j,

e3 = e

bK+1,

e4 = e

bK+2,

(A.56)

where µ j, l , and B j are the parameters in the original system, ri = Âij=1 µ j, and bK

is the parameter of E⇤ in the original system. The buffer size of the first server in

the surrogate system is BbK�1, and buffer sizes of the second, third and fourth servers

in the surrogate system are equal to BbK , B

bK+1, and BbK+2, respectively. For clarity,

we denote the policy E⇤ and the policy E⇤k

with parameter bK as E⇤(bK) and E⇤k

(bK),

respectively. We will analyze the original system under SSF and E⇤ by considering

the corresponding surrogate system under SSF and E⇤bK(2) in the surrogate system.

For a given bK, consider the original system and the corresponding surrogate

system. We divide the states of both systems into three periods denoted by A, B and

152


C. Remember that q , q 2 {A,B,C}, is the time period during which the total service

rate of the system is in set Rq

defined in (3.69).

We again recall the definition of pf

n in (3.64) with Ân2N pf

n = 1, f 2 F. To

clarify our two systems, we denote pf

n in the original system and the surrogate system

as pf

n and pSS,fn , respectively. In a similar way, we denote Pf

q

and Ff

q

, which are

defined in (A.13) and (3.70), as Pf

q

, PSS,fq

, Ff

q

and FSS,fq

for the original and the

surrogate system, respectively.

Lemma A.10. Consider the system defined in Section 3.2 and its associated surro-

gate system, of which the parameters are defined by (A.55) and (A.56). If K = bK +2,

then

PSSFq

= PSS,SSFq

, q 2 {A,B,C}. (A.57)

Proof. For q 2 {A,B,C} and f 2 {SSF,E⇤(bK),E⇤bK(2)}, we define

qf

q

= Âµ2R

q

pf (µ)

pf

BbK

,

qSS,fq

= Âµ2R

q

pSS,f (µ)

pSS,fBbK

.

(A.58)

Definition (A.55) is equivalent to qSSFq

= qSS,SSFq

, and we obtain

PSSFq

=q

q

SSF

Âq

02{A,B,C}qSSF

q

0=

qSS,SSFq

Âq

02{A,B,C}qSS,SSF

q

0

= PSS,SSFq

, q 2 {A,B,C}.



gate system, of which the parameters are defined by (A.55) and (A.56). If Condition

3.2 hold true for m = bK, thene3

n3�p

2e2

n2. (A.59)

153


Proof. Lete2

n2= F(n1,FA,SSF) =

ÂbKj=1 e j�FA,SSFn1

ÂbKj=1 µ j�n1

. (A.60)

Then, since FA,SSF � e1µ1

and n1 rbK�1,

F(n1,FA,SSF)F(n1,e1

µ1)F(r

bK�1,e1

µ1), (A.61)

where rbK�1 = ÂbK�1

j=1 µ j. Note that, in (A.61), the equality is achieved when e jµ j

= e1n1

for j = 1,2, . . . , bK�1 and µ1� µ j for j = 2,3, . . . , bK�1.

We rewrite (A.61) as

e2

n2= F(n1,FA,SSF)F(r

bK�1,e1

µ1) =

e

bKµ

bK+

ÂbK�1j=2 e j

µ

bK 1p

2

e

bK+1µ

bK+1=

e3p2n3

where the second inequality is based on Condition 3.2. This proves the lemma.


gate system, if K = bK +2, l � rbK+1, and Condition 3.2 hold true for m = bK, then,

BbK

Ân=0

pE⇤(bK)n �

BbK

Ân=0

pSS,E⇤

bK(2)

n �⇣r

bKl

⌘BbK�bK�1 bK

Âi=1

⇣

n1

l

⌘i� 0. (A.62)

Proof.

BbK

Ân=0

pE⇤(bK)n �

BbK

Ân=0

pSS,E⇤

bK(2)

n

=⇣r

bKl

⌘BbK�bK

bK�1

Âi=1

bK�1

’j=i

r j

l

+BbK�bK

Âi=0

⇣rbK

l

⌘i�⇣r

bKl

⌘BbK�bK�1 bK

Âi=1

⇣

n1

l

⌘i�

BbK�bK�1

Âi=0

⇣rbK

l

⌘i

=⇣r

bKl

⌘BbK�bK

+⇣r

bKl

⌘BbK�bK�1

"

rbK

l

bK�1

Âi=1

bK�1

’j=i

r j

l

�bK

Âi=1

⇣

n1

l

⌘i#

�⇣r

bKl

⌘BbK�bK�1

"

rbK

l

�bK

Âi=1

⇣

n1

l

⌘i#

. (A.63)

154


Since rbK � (1+

p3)r

bK�1 � (1+p

3)n1, we obtain

rbK

l

�bK

Âi=1

⇣

n1

l

⌘i� (p

3+1)rbK�1l

�rbK�1l

1� rbK�1l

�

(p

3+1)✓

1�rbk�1l

◆

�1�

rbK�1l

1� rbK�1l

.

Together with l � µ

bK+1 + rbK � (

p3+2)r

bK�1, we obtain

rbK

l

�bK

Âi=1

⇣

n1

l

⌘i�

(p

3+1)✓

1�rbk�1l

◆

�1�

rbK�1l

1� rbK�1l

�rbK�1l

1� rbK�1l

�bK

Âi=1

⇣

n1

l

⌘i,

namely

BbK

Ân=0

pE⇤(bK)n �

BbK

Ân=0

pSS,E⇤

bK(2)

n �⇣r

bKl

⌘BbK�bK�1

"

rbK

l

�bK

Âi=1

⇣

n1

l

⌘i#

�⇣r

bKl

⌘BbK�bK�1 bK

Âi=1

⇣

n1

l

⌘i.


Lemma A.13. Consider the system defined in Section 3.2 and its associated sur-

rogate system, if K = bK + 2, l � rbK+1, and Condition 3.2 hold true for m = bK,

then

✓

PSS,E⇤

bK(2)

A �PE⇤(bK)A

◆

⇣

FSS,SSFA �FSS,SSF

B

⌘

�✓

PSS,E⇤

bK(2)

C �PE⇤(bK)C

◆

⇣

FSS,SSFC �Fss,SSF

B

⌘

� 0. (A.64)

Proof. Let

Qf

q

=Pf

q

pf

BbK

, (A.65)

and

QSS,fq

=PSS,f

q

pSS,fBbK

, (A.66)

for q = A,B,C and f = SSF,E⇤(bK),E⇤bK(2). Since QE(bK)

C = QSS,E⇤

bK(2)

C , we get

155


PSS,E⇤

bK(2)

A �PE⇤(bK)A

PSS,E⇤

bK(2)

C �PE⇤(bK)C

=

QSS,E⇤

bK(2)

A

⇣

QE⇤(bK)A +QE⇤(bK)

B +QE⇤(bK)C

⌘

�QE⇤(bK)A

✓

QSS,E⇤

bK(2)

A +QSS,E⇤

bK(2)

B +QSS,E⇤

bK(2)

C

◆

QSS,E⇤

bK(2)

C

⇣


B +QE⇤(bK)C

⌘

�QE⇤(bK)C

✓

QSS,E⇤

bK(2)

A +QSS,E⇤

bK(2)

B +QSS,E⇤

bK(2)

C

◆

=Q

SS,E⇤bK(2)

A QE⇤(bK)B +Q

SS,E⇤bK(2)

A QE⇤(bK)C �QE⇤(bK)

A QSS,E⇤

bK(2)

B �QE⇤(bK)A Q

SS,E⇤bK(2)

C

QE⇤(bK)C

✓


B �QSS,E⇤

bK(2)

A �QSS,E⇤

bK(2)

B

◆ .

(A.67)

According to definition (A.65) and (A.66), we can rewrite (A.62) as


B �QSS,E⇤

bK(2)

A �QSS,E⇤

bK(2)

B � QSS,E⇤

bK(2)

A .

Then, based on Lemma A.12 and (A.67), we obtain

PSS,E⇤

bK(2)

A �PE⇤(bK)A

PSS,E⇤

bK(2)

C �PE⇤(bK)C

Q

SS,E⇤bK(2)

A QE⇤(bK)B +Q

SS,E⇤bK(2)

A QE⇤(bK)C �QE⇤(bK)

A QSS,E⇤

bK(2)

B �QE⇤(bK)A Q

SS,E⇤bK(2)

C

QE⇤(bK)C Q

SS,E⇤bK(2)

A

.

=

QE⇤(bK)B +QE⇤(bK)

C �QE⇤(bK)A

QSS,E⇤

bK(2)

B +QSS,E⇤

bK(2)

C

QSS,E⇤

bK(2)

A

!

QE⇤(bK)C

.

=QE⇤(bK)

B

QE⇤(bK)C

+1�QE⇤(bK)

A

QSS,E⇤

bK(2)

A

0

@

QSS,E⇤

bK(2)

B

QSS,E⇤

bK(2)

C

+1

1

A . (A.68)

Based on definition (A.65) and (A.66), we can obtain

QE⇤(bK)A

QSS,E⇤

bK(2)

A

� 0,Q

SS,E⇤bK(2)

B

QSS,E⇤

bK(2)

C

� 0 andQE⇤(bK)

B

QE⇤(bK)C

1BbK+1 +B

bK+2. (A.69)

156


Then, according to (A.68), we obtain

PSS,E⇤

bK(2)

A �PE⇤(bK)A

PSS,E⇤

bK(2)

C �PE⇤(bK)C

1BbK+1 +B

bB+2+1 5

4. (A.70)

According to Lemma A.12 and (A.70), a sufficient condition for (A.64) is

FSS,E⇤

bK(2)

C �FSS,E⇤

bK(2)

B

FSS,E⇤

bK(2)

B �FSS,E⇤

bK(2)

A

�F

SS,E⇤bK(2)

C �FSS,E⇤

bK(2)

B

FSS,E⇤

bK(2)

B

� 54,

or

FSS,E⇤

bK(2)

C �

bK+1Âj=1

e j

bK+1Âj=1

µ j

� 94

FSS,E⇤

bK(2)

B =94

bKÂj=1

e j

bKÂj=1

µ j

,

which is given in Condition 3.2. The lemma is proven.

Lemma A.14. Consider the system defined in Section 3.2 and its associated sur-

rogate system, if K = bK + 2, and Conditions 3.1-3.2 hold true for m = bK, then,

Âq2{A,B,C}

FE⇤(bK)q

PE⇤(bK)q

Âq2{A,B,C}

FSS,E⇤

bK(2)

q

PSS,E⇤

bK(2)

q

. (A.71)

Proof. Since FE⇤(bK)A FSSF

A (Lemma 3.2), FE⇤(bK)B = FSSF

B and FE⇤(bK)C = FSSF

C

(Lemma 3.3), we obtain

Âq2{A,B,C}

FE⇤(bK)q

PE⇤(bK)q

Âq2{A,B,C}

FSSFq

PE⇤(bK)q

. (A.72)

By the definition (A.56), FSSFq

= FSS,SSFq

forq = A,B,C, hence

Âq2{A,B,C}

FE⇤(bK)q

PE⇤(bK)q

Âq2{A,B,C}

FSSFq

PE⇤(bK)q

= Âq2{A,B,C}

FSS,SSFq

PE⇤(bK)q

. (A.73)

157


With Lemma A.13, we now obtain

Âq2{A,B,C}

FSS,E⇤

bK(2)

q

PSS,E⇤

bK(2)

q

= Âq2{A,B,C}

PE⇤(bK)q

FSS,SSFq

+

✓

PSS,E⇤

bK(2)

A �PE⇤(bK)A

◆

⇣

FSS,SSFA �FSS,SSF

B

⌘

+

✓

PSS,E⇤

bK(2)

C �PE⇤(bK)C

◆

⇣

FSS,SSFC �FSS,SSF

B

⌘

� Âq2{A,B,C}

PE⇤(bK)q

FSS,SSFq

� Âq2{A,B,C}

PE⇤(bK)q

FE⇤(bK)q

. (A.74)


Lemma A.15. Consider the system defined in Section 3.2, with K = bK +2, we have

FbK \ F 0

bK ✓ FbK,

where F 0bK✓ {1,2, . . . ,K} has the property that m 2 F 0

bKif and only if Conditions 3.1

and 3.2 hold true.

Proof. We consider the original system as defined in Section 3.2, and its associated

surrogate system. When equations in (A.55) hold true and K = bK +2, Lemma A.10

implies

PSSFA = Â

BbK�1

n=0 pSSFn = Â

BbK�1

n=0 pSS,SSFn = PSS,SSF

A ,

PSSFB = Â

BbK

n=BbK�1+1 pSSF

n = ÂBbK

n=BbK�1+1 pSS,SSF

n = PSS,SSFB ,

and

PSSFC = ÂB

n=BbK+1 pSSF

n = ÂBn=B

bK+1 pSS,SSFn = PSS,SSF

B .

158


Using definitions (A.55) and (A.56), we obtain

e4

n4=

e

bK+2µ

bK+2�

e

bK+1µ

bK+1=

e3

n3(A.75)

ande1 + e2

n1 +n2=

ÂbKj=1 e j

ÂbKj=1 µ j

�FA,SSF =e1

n1. (A.76)

We can rewrite (A.76) ase2

n2� e1

n1, (A.77)

since e1, e2, n1 and n2 are all positive.

By Conditions 3.1-3.2 for m = bK, according to Lemma A.11, we obtain

e3

n3�p

2e2

n2. (A.78)

Therefore, under Conditions 3.1-3.2, the surrogate system following (A.55) and

(A.56) is a special case for the system analyzed in Proposition 3.6.

Moreover, based on (A.10) and (A.56), we obtain

Âq2{A,B,C}

Fq ,SSFP

q ,SSF = Âq2{A,B,C}

FSS,SSFq

PSS,SSFq

. (A.79)

According to Lemma A.14, if K = bK +2, l � ÂbK+1j=1 µ j and Conditions 3.1-3.2

hold true for m = bK, then,

Âq2{A,B,C}

FE⇤(bK)q

PE⇤(bK)q

Âq2{A,B,C}

FSS,E⇤

bK(2)

q

PSS,E⇤

bK(2)

q

. (A.80)

159


Since the surrogate system follows Proposition 3.6, we obtain

Âq2{A,B,C}

FSSFq

PSSFq

= Âq2{A,B,C}

FSS,SSFq

PSS,SSFq

� Âq2{A,B,C}

FSS,E⇤

bK�1(2)

q

PSS,E⇤

bK�1(2)

q

� Âq2{A,B,C}

FE⇤(bK)q

PE⇤(bK)q

,

(A.81)

which is equivalent to E SSF

L SSF � E E⇤(bK)

L E⇤(bK)for the original system. This proves the

lemma.

The proof of Proposition 3.8.

Proof. According to Lemma A.15 and Corollary 3.1, if K � bK + 2, l � ÂbK+1j=1 µ j

and then Conditions 3.1-3.2 hold true for m = bK, then, E SSF

L SSF � E E⇤(bK)

L E⇤(bK).

This proves Proposition 3.8.

160

Appendix B

Theoretical Proofs for Non-Jockeying

Job-Assignments

The system model and notation used in this appendix has been defined in Chapter 4.

B.1 Proof for Proposition 4.1

Lemma B.1. For the system defined in Section 4.2 with exponentially distributed job

sizes, let an ,gj (n j) denote the action taken in state n j under the policy which optimizes

(4.11) for a given n . The following statements are equivalent:

1. an ,gj (n j) = 1,

2. n l

⇣

V n

j (n j +1,Rgj)�V n

j (n j,Rgj)⌘

,

where n j = 1,2, . . . ,B j�1, j = 1,2, . . . ,K, n 2 R.

Proof. According to (4.11) and (4.12), observe that, for a given Rgj , n j = 1,2, . . . ,B j�

1, j 2J , there exists a value n

⇤j (n j,R

gj) 2 R, satisfying

an ,gj (n j) =

8

>

<

>

:

1, n n

⇤j (n j,R

gj),

0, otherwise.

161

Theoretical Proofs for Non-Jockeying Job-Assignments

We rewrite (4.11) as

V n

j (n j,Rgj) = max

⇢

µ j� e⇤e j�gµ j

+V n(n j�1,R⇤j),

µ j� e⇤e j�gl +µ j

� n

l +µ j+

l

l +µ jV n(n j +1,R⇤j)+

µ j

l +µ jV n(n j�1,R⇤j)

�

,

(B.1)

where 1l+µ j

and 1µ j

are the average lengths of one sojourn in state n j with and without

new arrivals (tagged and untagged), respectively. We obtain

n

⇤j (n j,R

gj) = l

⇣

V n


j (n j�1,Rgj)⌘

�l (µ j� e⇤e j�g)

µ j(B.2)

Again, by (B.1), if an ,gj (n j) = 1, then

V n

j (n j,Rgj)�V n

j (n j�1,Rgj)

=l

µ j

⇣

V n


j (n j,Rgj)⌘

+µ j� e⇤e j�g

µ j� n

µ j. (B.3)

Recall that an ,gj (n j) = 1 when n n

⇤j (n j,R

gj), where the identity indicates no dif-

ference between an ,gj (n j) = 0 and an ,g

j (n j) = 1. We conclude that n

⇤j (n j,R

gj) =

l

⇣

V n


j (n j,Rgj)⌘

for each n j = 1,2, . . . ,B j � 1. This proves the

lemma.

Lemma B.2. For the system defined in Section 4.2 with exponentially distributed job

sizes, we have V n


j (n j,Rgj)�

µ j�e⇤e j�gµ j

for all n j = 1,2, . . . ,B j�1

and j = 1,2, . . . ,K,

Proof. For j = 1,2, . . . ,K and n j = B j�1, V n

j (B j,Rgj)�V n

j (B j�1,Rgj) =


.

According to (B.1), if an ,gj (n j) = 0, n j = 1,2, . . . ,B j � 1, then, V n

j (n j,Rgj)�

V n

j (n j � 1,Rgj) =


. Also, if an ,gj (n j) = 1, n j = 1,2, . . . ,B j � 1, then we

162


obtain (B.3). Based on Lemma B.1, we find

V n

j (n j,Rgj)�V n

j (n j�1,Rgj)

�µ j� e⇤e j�g

µ j+

l

µ j

⇣

V n


j (n j,Rgj)⌘

� 1µ j

l

⇣

V n


j (n j,Rgj)⌘

=µ j� e⇤e j�g

µ j.

(B.4)


Next we start the proof of Proposition 4.1.

Proof. For j = 1,2, . . . ,K, if an ,gj (n j) = 0 then, according to Lemmas B.1 and B.2,

we see that

n > l

⇣

V n(n j +1,Rgj)�V n(n j,R

gj)⌘

�l (µ j� e⇤e j�g)

µ j.

It remains to prove that if n >l (µ j�e⇤e j�g)

µ j, then an ,g

j (n j) = 0.

From the definition, V n

j (B j,Rgj)�V n

j (B j�1,Rgj) =


, and according to

Lemma B.1, n

⇤j (B j�1,Rg

j) =l (µ j�e⇤e j�g)

µ j.

Finally, we complete the proof by induction. Assume that n

⇤j (n,R

gj)=

l (µ j�e⇤e j�g)µ j

,

for all n � n j, n j = 2,3, . . . ,B j� 1. If n >l (µ j�e⇤e j�g)

µ j, then an ,g

j (n) = 0 for all

n� n j; that is, V n

j (n,Rgj)�V n

j (n�1,Rgj) =


, for all n� n j. Together with

Lemma B.1, n

⇤j (n j� 1,Rg

j) = l

⇣

V n

j (n j,Rgj)�V n

j (n j�1,Rgj)⌘

=l (µ j�e⇤e j�g)

µ j. In

other words, if n >l (µ j�e⇤e j�g)

µ j, then n > n

⇤j (n j�1,Rg

j); that is, an ,gj (n j�1) = 0.


163



Proof. We now dicuss the action made in state 0; that is, af jj (0). Let p j,n j be the

steady state distribution of state n j 2N j under policy f

⇤j over the process PH

j . We

consider the following problem.

max

(

�e⇤e0j , (�e⇤e0

j )p j,0 +(1�p j,0)(µ j� e⇤e j)�B j�1

Ân j=1

an

j (n j)p j,n jn�p j,0n

)

.

(B.5)

It follows that an

j (0) = 1 is equivalent to

n (1�p j,0)(µ j� e⇤e j + e⇤e0

j )

B j�1Â

n j=1an

j (n j)p j,n j +p j,0

. (B.6)

Based on Proposition 4.1, if n l (µ j�e⇤e j�g)µ j

where g is a given value, then

an ,gj (n j) = 1 for all n j = 1,2, . . . ,B j � 1; otherwise, an ,g

j (n j) = 0. According to

our definitions and Corollary 4.4.3, an

j (n j) = an ,g⇤j (n j), n j = 1,2, . . . ,B j�1, where

g⇤ > 0 is the average reward of process PHj under policy f

⇤j 2FH

j . Now we split

the establishment of (B.6) into two cases.

If n l (µ�e⇤e j�g⇤)µ j

, then, (B.6) is equivalent to

n

B j

Âi=1

⇣

l

µ j

⌘i(µ j� e⇤e j + e⇤e0

j )

B j�1Â

i=0

⇣

l

µ j

⌘i=

l

µ j

�


. (B.7)

According to the definition of our model in Section 4.2, (B.7) is valid when e

0j � 0

and n l (µ j�e⇤e j+e⇤e0j )

µ j.

If n >l (µ j�e⇤e j�g⇤)

µ j, then (B.6) is equivalent to

n l

µ j

�


. (B.8)

164


As a consequence,

n

⇤j (0) =

l

µ j

�


. (B.9)


Proof. By virtue of Proposition 4.2, Proposition 4.3 is proved for n j = 0. We assume

without loss of generality that n l (µ j�e⇤e j+e⇤e0j )

µ j, i.e., an ,g(0) = 1.

According to Corollary 4.4.3, we obtain

g⇤ = Ân j2N j

p j,n j

⇣

R j(n j)�an ,g⇤(n j)n⌘

. (B.10)

Together with Proposition 4.1, we rewrite n n

⇤j (n j,R

g⇤j ) for n j = 1,2, . . . ,B j�1 as

n l

µ j

�

µ j� e⇤e j� (1�p j,0)(µ j� e⇤e j)�p j,0(�e

0j )+n(1�p j,B j))

�

=l

µ j

�

p j,0(µ j� e⇤e j + e⇤e0j )+n(1�p j,B j)

�

, (B.11)

where

n l

µ j(µ j� e⇤e j + e⇤e0

j ). (B.12)

As was the case for n n

⇤(n j,Rg⇤j ), for n j = 1,2, . . . ,B j� 1, we rewrite n >

n

⇤(n j,Rg⇤j ) as

n >l

µ j(µ j� e⇤e j + e⇤e0

j ). (B.13)

Equations (B.12) and (B.13) together prove the proposition.

165


B.4 Consequences of the averaging principle

For K0i , i = 1,2, . . . , K with ÂK

i=1 K0i = K0, define random variables xt as follows.

Let t0,k, k = 1,2, . . . and ti,k, i = 1,2, . . . ,K0, k = 1,2, . . ., be the times of the kth

arrival and of the kth departure from server i, respectively. We assume without loss

of generality that K = K0. For the server farm, inter-arrival and inter-departure times

are positive with probability one and, also with probability one, two events will not

occur at the same time. Define a random vector x

x

x t = (x0,t ,x1,t , . . . ,xK0,t) as follows.

For j = 0,1, . . . ,K0,

x j,t =

8

>

>

>

>

>

>

>

>

>

>

<

>

>

>

>

>

>

>

>

>

>

:

t j,k� t j0,k0 , t j,k = mink00=1,2,...

{t j,k00 |t j,k00 > t},

t j0,k0 = maxj00=0,1,...,K0,k00=1,2,...

{t j00,k00 |t j00,k00 < t j,k},

t j0,k0 t < t j,k,

0, otherwise.

(B.14)

Here, the x

x

x t is almost surely continuous except for a finite number of discontinuities

of the first kind in any bounded interval of t > 0.

For x 2 RI , and i = 1,2, . . . , I, we define the action for the Whittle’s index policy

as

aindexi (x) =

8

>

>

<

>

>

:

1 i = min{i|x�i > 0, i = 1,2, . . . , I},

0 otherwise,(B.15)

where x�i = Âik=1 xk, and the action for the optimal solution of the relaxed problem

is

aOPTi (x) =

8

>

>

<

>

>

:

1 n

⇤i � n , xi > 0,

0 otherwise,(B.16)

for given state indices n

⇤i , i 2 ˜N {0,1}[ ˜N {0}, and n .

166


We define a function, Qf (i, i0,x,xxx ), where f 2 F, i, i0 2 Ñ {0,1} [ Ñ {0}. For

given x

x

x = (x0,x1, . . . ,xK0) 2 RK0+1 and x = (x1,x2, . . . ,xI) 2 RI , Qf (i, i0,x,xxx ) is

given, for i�1, i, i+1 2 Ñ {0,1}j [ Ñ {0,1}

j , j = 1,2, . . . , K by

Qf (i, i+1,x,xxx ) = af

i (x)1x0+ f 0

i,a(x,xxx ),

Qf (i, i�1,x,xxx ) =dx�i eÂ

j=dx�i�1e+1

1x j+ fi,a(x,xxx ),

Qf (i, i0,x,xxx ) = 0, otherwise,

(B.17)

where f is set to be either index or OPT , x�i =Âik=1 xk, and, with 0< a< 1, f 0

i,a(x,xxx )

and fi,a(x,xxx ) are appropriate functions to make Q(i, i0,x,xxx ) smooth in x for all given

x

x

x 2 RK0+1 and 0 < a < 1. Here a is a parameter controlling the Lipschitz constant.

Then, for any given 0 < a < 1, Qf (i, i0,x,xxx ) is bounded and satisfies a Lipschitz

condition over any bounded set of x 2 RI and x

x

x 2 RK0+1.

For 0 < a < 1 and e > 0, we define Xf ,et to be a solution of the differential

equation

Xf ,et = bf (Xe

t ,xxx t/e

)

= Âi02 Ñ {0,1}[ Ñ {0}

Qf (i0, i,Xe

t ,xxx t/e

)�Qf (i, i0,Xe

t ,xxx t/e

), (B.18)

where f is set to be either index or OPT . It follows that bf (Xe

t ,xxx t/e

) also satisfies a

Lipschitz condition on bounded sets in RI⇥RK0+1.

From the above definitions, for any x 2 RI , d > 0, there exists bf

(x) satisfying

limT!+•

P⇢

�

�

�

�

1T

Z t+T

tbf (x,xs)ds�bf

(x)�

�

�

�

> d

�

= 0, (B.19)

uniformly in t > 0. Let xf

t be the solution of xf

t = bf

(xt), xf

0 = Xf ,e0 .

167


Now we invoke [114, Chapter 7, Theorem 2.1]: if (B.19) holds, and E�

�bf (x,xxx t)�

�

2<

+• for all x 2 RI , then, for any T > 0, d > 0,

lime!0

P

(

sup0tT

�

�

�

Xf ,et �xf (t)

�

�

�

> d

)

= 0. (B.20)

We scale the time line of the stochastic process by e > 0. As e tends to zero,

time is speeded up, and in this way the stochastic process {Xf ,et }, driven by the

random variable x

x

x t/e

, converges to the deterministic process {xf (t)} defined by the

differential equation bf

(x).

Now we interpret the scaling by e in another way. Along similar lines, for a

positive integer n and a scaled system, where Ki = nK0i replaces K0

i , i = 1,2, . . . , K,

and K = nÂKi=1 K0

i , we define tnj,k, j = 0,1, . . . ,K, k = 1,2, . . . and the random vari-

ables x

x

x

nt by analogy with the unscaled systems. Then the random variables x

nj,t ,

j = 1,2, . . . ,K, and x

n0,t are exponentially distributed with rate l

0i , i= b( j�1)/nc+1

and nl

00 , respectively, where l

0i , i = 0,1, . . . ,K0, are the corresponding rates for

random variables xi,t . We then define, Qf ,n(i, i0,x,xxx n) for x 2 RI and x

x

x

n 2 RnK0+1

as in Equation (B.17), with appropriate modifications for the change in dimension,

where again the functions f 0,ni,a (x,xxx n) and f n

i,a(x,xxxn) are apprppriately defined to

guarantee smoothness of Qf ,n(i, i0,x,xxx ) x for all given x

x

x

n 2 RnK0+1 and 0 < a < 1.

Here, again, a is a parameter controlling the Lipschitz constant, and f is set to be

either index or OPT , x�i = Âik=1 xk,

In the same vein, for x 2 RI and x

x

x

n 2 RnK0+1, a differential equation is given by

bf ,n(x,xxx n)

= Âi02 ˜N {0,1}[ ˜N {0}

Qf ,n(i0, i,x,xxx n)�Qf ,n(i, i0,x,xxx n). (B.21)

If we set e = 1/n then, for any x 2 RI , n > 0 and T > 0,R T

0 bf (x,xt/e

)dt andR T

0 (bf ,n(nx,x nt )/n)dt are equivalently distributed with f set to be either index or

168


OPT . We define Zf ,e0 = Zf ,n

0 = x0/(K0 +1) (there is a zero-reward virtual server),

and

Zf ,nt =

1n(K0 +1)

bf ,n(n(K0 +1)Zf ,nt ,x n

t ),

and

Zf ,et =

1K0 +1

bf ((K0 +1)Zf ,et ,xt/e

).

From (B.20), we obtain

limn!+•

P

(

sup0tT

�

�

�

Zf ,nt �xf (t)/(K0 +1)

�

�

�

> d

)

= 0, (B.22)

for f set to be either index or OPT . Therefore the scaling of time by e = 1/n is

equivalent to scaling of system size by n.

Because of the Lipschitz continuity of Zf ,nt and xf

(t) on 0< a< 1, lima!0 dZnt /da=

0 and lima!0 dxf

(t)/da = 0, equation (B.22) holds true in the limiting case as a! 0.

Also, if Zf ,n0 = Zf (0), and xf (0)/(K0 +1) = zf (0), then lima!0 xf (t)/(K0 +1)!

zf (t) and lima!0 Zf ,nt ! Zf (t), where f is set to be either index or OPT , Zf (t)

represents the proportions of servers under policy f at time t and zf (t) is given by

(4.19) (as defined in Section 4.4.4). This leads to (4.21).

169

file

Documents