exact analysis of some split-merge queues

Exact Analysis of Some Split-Merge Queues

Lester LipskyComputer Science & Engineering

University of Connecticut

Pierre FioriniCF Search

Portsmouth, NH

Contents• Introduction

– Synchronized Queues• Fork-Join Queues• Split-Merge Queues

• Related Work– Research Strategies– Harrison & Zertal (2003)– Lebrecht & Knottenbelt (2007)– Tsimashenka & Knottenbelt (2014)

• Background– Order Statistics

• Homogenous• Heterogeneous

– Matrix Exponential Distributions• Matrix Exponential Functions• Homogenous Order Statistics• Heterogeneous Order Statistics• Joint Distributions

– M/G/1 Queues• PK Formula• Stationary Queue Length

Distribution• Response Time Distribution

• Examples of Split-Merge Queues– Homogeneous Split–Merge

Queues– Heterogeneous Split–Merge

Queues– Split–Merge Queues with

Subtask Failure & Repair– Split–Merge Queues with a

Variable Number of Subtasks

Introduction – Split-Merge Queues

• On arrival a job is split into n sub-tasks which are serviced in parallel. • Only when all the tasks finish servicing and have rejoined can the

next job start• A type of “synchronized” queue

Introduction – Fork-Join Queues

• Incoming jobs are split on arrival for service by numerous servers and joined before departure

• Jobs can arrive at any time• Exact results only exist for N = 2

𝑇=12− 𝜌8

𝑥1− 𝜌

Nelson, R. & Tantawi, A. N. (1988).

Introduction – Difference Between Fork-Join & Split-Merge Queues

• FJ – new jobs can arrive at any time

• SM – new jobs can only arrive after the last subtask finishes service

• This means SM provide upper-bound response time for FJ queues

• SM queues are “more synchronized”

• Most researchers have used the EMOS (Expected Maximum Order Statistic) technique for the service time distribution

• Usually computed via numerical integration

• They have applied this to the M/G/1 queue and used the PK Formula to generate performance measures

Related Work – Research Strategies

𝑅E[max ( 𝑋1 , 𝑋 2,…, 𝑋𝑛)2]

1−𝜌

Related Work - Harrison & Zertal (2003)

• “Queueing Models with Maxima of Service Times”

• Found recursive way to compute the moments of o.s.– Identical + Non-Identical– Exact for exponential– Approximate for non-

exponential

• Applied results to analyze performance of RAID subsystems (exponential) for M/G/1 queues

Related Work - Lebrecht & Knottenbelt (2007)

• “Response Time Approximations in Fork-Join Queues”

• A response time approximation of the fork-join queue is presented using EMOS

• Used to model performance of RAID

• Used various distributions for M/G/1 queues & heterogeneous servers

Related Work – Tsimashenka & Knottenbelt (2014)

• “Trading off Subtask Dispersion and Response Time in Split-Merge Systems”

• Describe a methodology for managing the trade off between subtask dispersion and task response time

• That is, the time between the arrival of the first and last subtasks originating from a given task in the output buffer– The “Range”

• Used M/G/1 queue to generate performance measures

ContributionsResearch Queueing

ModelsHomogenous

Order StatisticsHeterogeneous Order Statistics

Stationary Queue Length Distribution

Response Time Distribution

Unreliable SM Queues

Variable Subtasks

Harrison & Zertal (2003)

M/G/1 (PK Formula)

Exact for Exponential, Approximations


No results No results No results No results

Lebrecht & Knottenbelt (2007)

M/G/1 (PK Formula)

Numerical Integration

Approximations & Numerical Integration


Tsimashenka & Knottenbelt (2014)

M/G/1 (PK Formula)

Not discussed Numerical Integration


Fiorini & Lipsky (2015)

M/G/1, M/G/1/N, M/G/1//N, M/G/C, M/G/C//N, G/G/1//N

Markov Chain Markov Chain Explicit ME representation

Explicit ME representation

ME rep ME rep

Background – Homogenous Order Statistics

F(x)X1

X2

X3

X4 X5

X(1)≤ X(2) ≤ X(3) ≤ X(4) ≤ X(5)

Randomly sample F(x)

Order sample by size…Parent Distribution

o.s. Distributions


• Interested in the Extremes• Maximum• Minimum

Background – Heterogeneous Order Statistics

F1(x)X1

X2

X3

X4 X5

X(1)≤ X(2) ≤ X(3) ≤ X(4) ≤ X(5)

Randomly sample Fi(x)

Order sample by size…

F2(x)

Fn(x)Parent Distribution

Non-Identical o.s. Distributions

Background – Heterogeneous Order Statistics

Let X1, X2, X3, … be the order statistics

The joint distribution is (Bapet & Beg 1989)

Background – LAQTAny arbitrary pdf can be represented by an m-dimensional vector-matrix pair < p, B >

Let X be a matrix-exponential (ME) r.v. greater than or equal to 0. The cdf is

Let density function is

Moments can be computed by

Background – LAQT (M/G/1)For the open M/G/1 queue, the stationary queue length probabilities are given by

where

Background – LAQT (M/G/1)For the open M/G/1 queue, the mean queue length can be calculated by

or

Background – LAQT (M/G/1)For the open M/G/1 queue, the response time distribution can be calculated by

• Intuition…– Think of 4 tasks (i.e., “r.v.’s”) running concurrently…

• The 1st task finishes at t1

• The 2nd finishes at t2

– The maximum happens at X(4)

ME Order Statistics - Max

X(1)

X(2)

X(3)

X(4)

t1 t2 t3 t4

ME Order StatisticsState Transition Diagram

max{B1, B2, B3, B4} 𝑩1⊕𝑩2⊕𝑩3⊕𝑩4

𝑩1⊕𝑩2 𝑩3⊕𝑩4𝑩1⊕𝑩3𝑩1⊕𝑩4𝑩2⊕𝑩3𝑩2⊕𝑩4

𝑩1 𝑩2 𝑩3 𝑩4

𝑩4

𝑩3 𝑩3

𝑩3𝑩2 𝑩3

𝑩4 𝑩2𝑩1𝑩2

𝑩3 𝑩4

𝑩2

𝑩2 𝑩2𝑩1

𝑩1

𝑩1𝑩3𝑩3 𝑩3

𝑩4 𝑩4 𝑩4

𝑩1

𝑩4

𝑩3

𝑩1

ME Order Statistics – Markov Chainmax{B1, B2, B3, B4}

𝑩=¿


Let X1, X2,…, Xn be independent (and possibly non-identical) ME distributed r.v.’s having CDF


Need to construct the M & P matrices…

• Examples…

– Homogeneous split-merge queues – Heterogeneous split-merge queues – Split-Merge queues with unreliable subtasks– Variable number of subtasks

Examples of Split-Merge Queues

• In this example, all subtasks are iid and have the same parent distribution,

• For now, we assume the following:– n = 2 subtasks (but, n can by any number)

– Subtasks are exponentially distributed with parameter m

Homogeneous Split-Merge Queues

Homogeneous Split-Merge Queues

• Process rate matrix

• Service time matrix

• Density function, which is maximum of 2 exp r.v. with rate µ

Homogeneous Split-Merge Queues• Mean response time (using the

PK Formula for M/G/1 queues)

• Response time distribution (pdf)

Split-Merge vs. Fork-Join (n = 2)

(Nelson & Tantawi, 1988)

(Fiorini & Lipsky, 2015)

Mean upper-bound for FJ Queue, n = 2

• Upper-Bound Response Time Distribution for Fork-Join queues where n = 2

Split-Merge vs. Fork-Join (n = 2)

• Here we assume all subtasks are iid and have different parent distributions,

• We assume the following:– n = 2 subtasks (but, n can by any number)– Subtasks are exponentially distributed, where

Heterogeneous Split-Merge Queues


• Process rate matrix

• Service time matrix

• Density function of maximum o.s. of 2 non-identical exponential r.v.’s


• Mean response time (using the PK Formula) for M/G/1 queue

• Suppose a subtask fails at rate a and is repaired at rate b

• Assume when subtask is in upstate, it completes at rate m

• The generator of this process is

Unreliable Split-Merge Queues

• For n = 2, we have


• The mean response time turns out to be (using the PK formula) for the M/G/1 queue


• Other modeling factors…

– Subtasks can have different failure rates, – Subtasks can have different repair rates, – Subtasks can be a mixture of subtasks that fail and

those that don’t– Subtasks can have different recovery polices • prd, prs, etc.


Split-Merge Queues with Variable Number of Forked Subtasks

• There is an a1p1 vector probability of 1 task being forked, a2p2 vector probability of 2 tasks, and so on…

Split-Merge Queues with Variable Number of Forked Subtasks

SummaryResearch Queueing

ModelsHomogenous

Order StatisticsHeterogeneous Order Statistics

Stationary Queue Length Distribution

Response Time Distribution

Unreliable SM Queues

Variable Subtasks

Harrison & Zertal (2003)

M/G/1 (PK Formula)




Lebrecht & Knottenbelt (2007)

M/G/1 (PK Formula)

Numerical Integration

Approximations & Numerical Integration


Tsimashenka & Knottenbelt (2014)

M/G/1 (PK Formula)

Not discussed Numerical Integration


Fiorini & Lipsky (2015)

M/G/1, M/G/1/N, M/G/1//N, M/G/C, M/G/C//N, G/G/1//N

Markov Chain Markov Chain Explicit ME representation

Explicit ME representation

ME rep ME rep

• Study performance bounds of homogenous and heterogeneous fork-join queues– Use split-merge queues– Heavy-Tails?– Not well understood

• Unreliable split-merge queues– Subtasks can have different failure rates, – Subtasks can have different repair rates, – Subtasks can be a mixture of subtasks that fail and those that don’t– Subtasks can have different recovery polices

• prd, prs, etc.

Future Work

Questions???