1866 ieee transactions on parallel and distributed …

DGLB: Distributed Stochastic GeographicalLoad Balancing over Cloud Networks

Tianyi Chen, Student Member, IEEE, Antonio G. Marques, Senior Member, IEEE,

and Georgios B. Giannakis, Fellow, IEEE

Abstract—Contemporary cloud networks are being challenged by the rapid increase of user demands and growing concerns about

global warming, due to their substantial energy consumption. This requires future data centers to be both energy efficient and

sustainable, which calls for leveraging cutting-edge features and the flexibility provided by the modern smart grids. To fulfill those goals,

this paper puts forward a systematic approach to designing energy-aware traffic-efficient geographical load balancing schemes for

data-center networks that are not only optimal, but also computationally efficient and amenable to distributed implementation. Under

this comprehensive approach, workload and power balancing schemes are designed jointly across the network, both delay-tolerant and

interactive workloads are accommodated, novel smart-grid features such as energy storage units are incorporated to cope with

renewables, and incentive pricingmechanisms are adopted in the design. To further account for the spatio-temporal variation of

demands, energy prices and renewables, the task is formulated as a two-timescale stochastic optimization. Leveraging dual stochastic

approximation and the fast iterative shrinkage-thresholding algorithm (FISTA), the proposed optimization is decomposed across time

slots (first-stage) and data centers (second-stage). While the resultant online algorithm is strictly feasible and provably optimal under a

Markovian assumption for the underlying random processes, extensive numerical tests further demonstrate that it also works well in

real-data scenarios, where the underlying randomness is highly correlated across time.

Index Terms—Data center, renewables, energy storages, incentive payment, network resource allocation, stochastic programming

Ç

1 INTRODUCTION

THE recent trend towards cloud computing has created anew class of computing systems, known as warehouse-

scale computers, or, data centers (DCs) [1], which are rap-idly proliferating all over the world. DCs are nowadaysessential to provide Internet services such as web search,video distribution or data analytics. To enhance reliabilityand quality-of-service (QoS), cloud-service providers usu-ally deploy DCs across different geographical regions. Asan example, Apple is undertaking its biggest European DCproject to date, with an investment of around $1.9 billion ontwo massive DCs, one in Ireland and one in Denmark [2].Along with their growth in number and scale, the consider-able amount of energy consumed by DCs not only chal-lenges DC operators’ budgets, but also raises globalwarming and climate-change concerns [3].

Optimal energy and workload management for setupswith a single DC have been thoroughly investigated [4], [5],[6], [7], [8]. The issues of speed scaling and dynamic resizingin a server farm were considered in [4] and [5], while

optimal power control with energy storage devices was dis-cussed in [6]; however, [4], [5], [6] did not account forrenewable energy sources (RES), which have been investi-gated separately in [9] and [10]. Schemes to minimize thecooling consumption when future state information isknown were reported in [7]; see also [8] for stochastic for-mulation. However, [7] and [8] process user requests locallyor presume that optimal routing has been performed, thusmissing to leverage the spatio-temporal diversity of RES,data demand and energy prices.

Fewer works have dealt with the geographical load balancingtask over a DC network [11], [12], [13], [14], [15]. To distributedelay-tolerant workloads (DWs), [11] developed an onlinealgorithmwith a desirable tradeoff among energy cost, work-load fairness and latency. A Lyapunov-optimization-basedapproachwas proposed in [12] for joint workload routing andthermal storage management for geo-distributed DCs, while[13] adopted a two-timescale scheme to solve the workloadmanagement at a fast timescale and the server activation at aslow timescale. However, the approaches in [11], [12], [13]were tailored to schedule DWs, and the generalization tointeractive workloads (IWs) is not straightforward. Toaccount for IWs, alternating direction method of multipliers(ADMM)-based schemes were employed in [14] and [15] todesign distributed algorithms that solved the cost minimiza-tion over a single time slot. Yet, the approaches in [14] and[15] mainly deal with a deterministic workload allocationproblem, rather than considering the spatio-temporal uncer-tainty inherent to data demands, energy prices and renew-ables, which challenges the design of schemes incorporatingbothDWand IW traffic.

� T. Chen and G.B. Giannakis are with the Department of Electrical and Com-puter Engineering, and the Digital Technology Center, University ofMinne-sota,Minneapolis,MN 55455. E-mail: {chen3827, georgios}@umn.edu.

� A.G. Marques is with the Department of Signal Theory and Communica-tions, Rey Juan Carlos University, Madrid 28943, Spain.E-mail: [email protected].

Manuscript received 12 Mar. 2016; revised 15 Nov. 2016; accepted 3 Dec.2016. Date of publication 6 Dec. 2016; date of current version 14 June 2017.Recommended for acceptance by B. Urgaonkar.For information on obtaining reprints of this article, please send e-mail to:[email protected], and reference the Digital Object Identifier below.Digital Object Identifier no. 10.1109/TPDS.2016.2636210

1866 IEEE TRANSACTIONS ON PARALLEL AND DISTRIBUTED SYSTEMS, VOL. 28, NO. 7, JULY 2017

1045-9219� 2016 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission.See http://www.ieee.org/publications_standards/publications/rights/index.html for more information.

Different from [4], [5], [6], [7], [8], [9], [10], the present paperdeals with online joint workload and energy management fora cloud network consisting of multiple geo-distributed map-ping nodes (MNs) and DCs. The intended resource allocationtask is formulated as an infinite time horizon optimizationtask to minimize the time-average network-wide cost, subjectto various operational constraints. Compared to [11], [12],[13], [14], [15], the proposedworkload routing and schedulingpolicy considers both DWs and IWs, while the energy manage-ment scheme integrates renewables, storage units and two-way energy trading, to minimize the total energy cost fromcooling and information technology (IT) operating systems.Furthermore, leveraging the flexibility provided by the newdemand-response programs [16] and [17], a simple yet effi-cient incentive payment mechanism is introduced, whichmodulates the peaks of IW demand, while guaranteeing QoS;see also [15] and references therein. Major research challengesinclude: i) accounting for and exploiting the spatio-temporalvariations of the state variables such as RES, IT demand andenergy prices, even when their joint distribution is unknown;ii) joint consideration of costs and constraints that couple opti-mization variables across time and space; and iii) develop-ment of low-complexity distributed algorithms that can beimplemented in real time.

The main contribution of this paper is the development ofDGLB, a two-timescale algorithm for online distributed geo-graphical load balancing over cloud networks. DGLB is obtainedas the solution of a rigorously formulated nonlinear con-strained optimization; it accounts for the inherent spatio-tem-poral uncertainty in the network (including renewables,energy prices and users demands); entails a low computa-tional complexity; can be implemented distributedly; doesnot require statistical knowledge of the random variablesinvolved; and has provable convergence and optimality. Wesummarize the contributions of this paper as follows.

C1) Targeting a unified framework for geographical loadbalancing in DCs, we take a holistic view and encompass anumber of factors related to workload and energy manage-ment in a model more comprehensive than that adopted bystate-of-the-art approaches. Compared to [4], [5], [6], [7], [8],[9], [10], [11], [12], [13], [14], [15], online resource allocationincorporating both DWs and IWs is tackled, which chal-lenges the development of fast and distributed solvers.

C2)We decouple the optimization variables across time byjudicious relaxation, and reformulate the dynamic problem asa stationary stochastic program. Leveraging a two-stageLagrange relaxation, we develop a novel two-timescale sto-chastic resource allocation scheme termedDGLB. Specifically,i) a stochastic dual gradient method is run at a slow timescaleto deal with the long-term constraints; and ii) a diagonallyweighted FISTA is run at a fast timescale to ensure fast real-time coordination and decentralized implementation. DBLGcan operate without knowing the distribution of the involvedrandom variables, and requires each agent in the network tocollect information only from its one-hop neighbors.

C3)While most existing works in this setting (e.g., [6], [8],[12], [13]) adopt an i.i.d. assumption for the underlying ran-dom state variables, we analytically establish feasibility andoptimality of our two-timescale online approach under amore general Markovian assumption, which readilyincludes the i.i.d. setting and can accommodate a wider

range of real-world applications. We further demonstrate,by extensive simulations, that our DGLB also works well inmore challenging real-world scenarios, where the underly-ing randomnesses are highly correlated over time.

Paper Outline. Modeling preliminaries are given inSection 2. The stochastic geographical management schemeis proposed in Section 3, while its distributed implementa-tion is developed in Section 4. Performance analysis is pre-sented in Section 5. Numerical tests are provided inSection 6, followed by conclusions in Section 7.

2 MODELING PRELIMINARIES

Our system operates on discrete time slots indexed by t, withan infinite scheduling horizon T :¼ f0; 1; . . .g. A networkwith J :¼ f1; 2; . . . ; Jg MNs, and I :¼ f1; 2; . . . ; Ig heteroge-neous DCs is considered. MNs collect user requests over ageographical area (e.g., a city or a state) and forward the corre-sponding workloads to one or more DCs, which are distrib-uted across a large area (e.g., a country). In addition to the ITsystem present to process the assigned workloads, each DC isequippedwith a cooling system to remove the heat generatedby the IT system, and a power supply system supporting theIT and cooling infrastructure. MNs make forwarding deci-sions based on the user requirements, the communication andnetworking costs, the load of different DCs, and their mar-ginal energy price. The goal is to leverage the spatio-temporalvariation of communication costs, energy prices, RES andcooling supplies, to obtain amore efficient network operation.

The ensuing sections describe the detailed operation ofeach MN and DC, including workload, network, powersupply and power demand models, as well as the differentsystem costs and incentive payment mechanisms that canbe used to modulate the users’ demand.

2.1 Traffic Workloads and Network Constraints

Suppose that each MN collects two types of workloads:delay-sensitive interactive and delay-tolerant workloads[13]. The IWs such as instant messaging and voice servicesare real-time requests that need to be served immediately.DWs are relatively time insensitive and deferrable withingiven slots. Typical examples include system updates anddata backup. This provides ample optimization opportuni-ties for workload allocation based on the dynamic variationof energy prices and RES availabilities. Table 1 summarizesall the notation introduced in this section.

For IWs, let Vj;t denote the workload requested (arrivalrate) to MN j at time t, and vi;j;t the amount of workload dis-tributed fromMN j to DC i at time t. Per slot t, MN j shoulddispatch all arrived IWs to a set of DCs physically con-nected to it. If I j � I denotes the set of DCs connected toMN j, the following constraints must be satisfied

P

i2Ij vi;j;t ¼ Vj;t; 8j; t: (1)

Although multiple IW types can be considered, since all mustbe served immediately, to simplify notation we aggregatethem to Vj;t. In contrast, multiple types of DW are collected inthe setQ :¼ f1; 2; . . . ;Qg. The reason for consideringmultipleclasses of DWs is twofold: i) the utility generated by each ofthe services can be different, and ii) since this type of

CHEN ET AL.: DGLB: DISTRIBUTED STOCHASTIC GEOGRAPHICAL LOAD BALANCING OVER CLOUD NETWORKS 1867

workloads is deferrable, the developed algorithms can givedifferent priority to each of the services. In this case, let Wj;q;t

and ~wi;j;q;t denote the amount of DW q arriving at MN j at slott and the amount of DW q routed fromMN j to DC i at slot t,respectively. Since DWs are deferrable, the fraction ofunrouted workload is buffered in queues (one per class ofDW) obeying the following dynamic recursion

Y mnj;q;tþ1 ¼

h

Y mnj;q;t þWj;q;t �

P

i2Ij ~wi;j;q;t

i1

0; 8j; q; t; (2)

where Y mnj;q;t is the queue length of DW q in MN j at the

beginning of slot t, and ½��ba :¼ maxfa;minfb; �gg.At the DC side, IWs must be processed once received,

while DWs are deferrable. With wi;q;t denoting the amountof DW q processed by DC i during slot t, the unserved por-tion of the workloads are buffered at the DC using separatequeues. This leads to the following dynamic recursion

Y dci;q;tþ1 ¼

h

Y dci;q;t � wi;q;t þ

P

j2J ~wi;j;q;t

i1

0; 8i; q; t; (3)

where Y dci;q;t is the queue length of DW q in DC i at the begin-

ning of slot t. Queue dynamics slightly different from the

one in (2) and (3) can also be considered [12], [13], but suchdifferences are not relevant for the subsequent analysis.

The total IT demand of DC i in slot t, is thus the superpo-sition of IWs and DWs, which is given by

di;t ¼P

j2J vi;j;t þP

q2Q wi;q;t; 8i; t: (4)

Lastly, to account for the bandwidth of the MN-to-DClinks, the total workload distribution rate per link from MNj to DC i is upper bounded by the time-invariant constant

vi;j;t þP

q2Q ~wi;j;q;t � Bj;i; 8j; i; t; (5)

with Bj;i ¼ 0 if MN j and DC i are not connected. To sim-plify derivations, each MN-to-DC link is modeled here as asingle-hop channel with a dedicated bandwidth constraint.However, the proposed formulation can be modified toaccommodate multi-hop communications among MNs andDCs. Although this will require incorporation of routingvariables and flow conservation constraints, the basic struc-ture of the problem and the approach to solve it will remainthe same; see, e.g., [18] for details.

A workload allocation diagram summarizing the varia-bles introduced in Section 2.1 is presented in Fig. 1.

2.2 Power Demand and Supply Models

The main cost when operating a DC is due to its power con-sumption. In this section, we describe the relation betweenthe load served by a DC, and the corresponding power con-sumption, as well as the different sources of energy avail-able at each DC.

We start by modeling P si , the power consumed by a single

server in DC i. Let Ps

i denote the peak power consumption ofa server in DC i, and let c 2 ½0; 1� denote the speed of theserver, oftentimes referred to as CPU usage (processed workdividedby the server capacity). PowerP s

i can be then approxi-

mated as P si ðcÞ ¼ P

s

i %cs þ 1� %ð Þ, where the fraction of peak

consumption 1� % represents the power consumed in idlestate (i.e., c ¼ 0), which is around 0.4, and constant s � 1 istypically set to 2 in state-of-the-art servers [4]. Assume alsothat the Mi servers in DC i are all identical. Then, given thetotal IT demand di;t in DC i at time t, it follows from the con-vexity of P s

i ðcÞ that the most energy-efficient allocation is todivide di;t uniformly across servers. In this way, with Di

denoting the per-server capacity of DC i, all servers are run-ning at the same speed ðdi;t=MiÞ= Di 2 ½0; 1�, and the totalpower consumption at DC i can be expressed as

TABLE 1Notation List

Symbol Definition

i, I, I Index, number, and set of DCs.j, J , J Index, number, and set of MNs.q,Q,Q Index, number, and set of DWs.Vj;t Amount of arrived IW at MN j per slot t.vi;j;t Amount of IW routed fromMN j to DC i per slot t.Wj;q;t Amount of arrived DW q at MN j per slot t.~wi;j;q;t Amount of DW q routed fromMN j to DC i per slot t.wi;q;t Amount of DW q being processed at DC i per slot t.Y mnj;q;t Queue length of DW q in MN j at the beginning of slot t.

Y dci;q;t

Queue length of DW q in DC i at the beginning of slot t.

di;t Total IT demand of DC i during slot t.Bj;i Limit of distribution rate per link fromMN j to DC i.

Ps

iPeak power consumption of a server in DC i per slot.

Mi Total number of servers in DC i.Di Computing capacity of a single server in DC i.ei;t Power coefficient reflecting environment factors.

P gi;t; P

g

iCG generation in DC i per slot t and its upper-limit.

P ri;t; P

r

iRG generation in DC i per slot t and its upper-limit.

P bi;t

Power (dis-)charged to the battery in DC i per slot t.

P bi ; P

b

iLower- and upper-limits of P b

i;t.

Y bi;t

SOC of the battery in DC i at the beginning of slot t.

Pmi;t Power that DC i buys/sells from/to the market at time t.

api;t;a

si;t Price of buying/selling energy by DC i per slot t.

pj;t Price that MN j pays for demand curtailment at time t.�Vj;t IW amount that users in MN jwill reduce per slot t.~Vj;t Actual amount of IW arriving at MN j per slot t.

P si ð � Þ Power consumption of a single server in DC i per slot.

P iti ð � Þ Power consumption of all servers in DC i per slot.

P aci ð � Þ Power consumption of cooling facilities in DC i per slot.

Uvj ð � Þ Revenue from IWs at MN j per slot.

Uwi;qð � Þ Revenue from DW q at DC i per slot.

Gdi;jð � Þ Cost of distributing loads fromMN j to DC i per slot.

Gei ð � Þ Energy transaction cost in DC i per slot.

Gci ð � Þ Cost of CG in DC i per slot.

Gbi ð � Þ Cost of (dis-)charging the battery in DC i per slot.

Guj ð � Þ Cost of user dissatisfaction at MN j per slot.

Fig. 1. A geographical load balancing system diagram.


P iti ðdi;tÞ ¼ MiP

si

di;tMiDi

� �

¼%d2i;tMiD2

i

Ps

i þ ð1� %ÞMiPs

i : (6)

Clearly, function P iti ð � Þ is increasing and convex with

respect to (w.r.t.) di;t. Here, the number of active servers Mi

is assumed to be the same across the scheduling horizon.The reason for this is that the so-termed “switching cost”incurred from toggling a server in and out of a power-sav-ing mode (including the delay, energy, and wear-and-tearcosts) is substantial, so that frequently changing Mi is notbeneficial. Additional details as well as specific research ondynamic sizing of DCs can be found in [5].

Along with the increasing density of IT equipment inDCs, a considerable amount of electricity is consumed bythe cooling system [3]. We assume for simplicity that thecooling consumption is proportional to the total IT powerconsumption as P ac

i ðdi;tÞ ¼ ei;tPiti ðdi;tÞ, where ei;t is time-

varying and depends on a variety of environment factors(e.g., humidity, temperature). A typical value is around 0.3with advanced cooling facilities [7]. In any case, we willassume henceforth that at time t the value of ei;t is determin-istically known. Note finally that although a simple coolingconsumption model is adopted, our framework can easilyinclude more advanced convex cooling consumption mod-els; see, e.g., [7], [8].

The next step is to describe the power supply model. Inparticular, we assume that each DC is supplied by a renew-able-integrated (micro-)grid consisting of a conventionalgenerator (CG) (e.g., a fuel generator), an on-site renewablegenerator (RG) (e.g., wind or solar), and an energy storageunit (e.g., a battery). Specifically,

� P gi;t stands for the energy generated at time t by the

CG in DC i, which is upper bounded by Pg

i , so that

0 � P gi;t � P

g

i ; 8t: (7)

� P ri;t is the renewable energy generated at the begin-

ning of slot t by the RG in DC i, which is also

bounded in 0 � P ri;t � P

r

i , 8t.� P b

i;t is the power delivered to or drawn from the battery(storage unit) in DC i at slot t, which amounts to either

charging (P bi;t > 0) or discharging (P b

i;t < 0) the bat-

tery. Let Y bi;0 and Y b

i;t denote the initial amount of stored

energy and the state of charge of the storage unit inDCi at the beginning of time slot t. Each unit has a finite

capacity Yb

i as well as a minimum level Y bi . The

dynamics of the storage unit are described as

Y bi � Y b

i;t � Yb

i ; 8i; t (8)

Y bi;tþ1 ¼ Y b

i;t þ P bi;t; 8i; t (9)

P bi � P b

i;t � Pb

i ; 8i; t; (10)

where the bounds on the (dis)charging amount

P bi < 0 and P

b

i > 0 in (10) are dictated by physicallimits.

In addition to the energy resources within the microgrid,the DCs can resort to the external wholesale electricity

market in an on-demand manner. To be specific, Pmi;t denotes

the energy that DC i buys from the market at time t. Since atwo-way energy trading facility is considered, if negative,Pmi;t denotes the energy sold by the DC.

With these notational conventions, at each time t, thepower demand and supply at each of the DCs has to be bal-anced. Mathematically, this amounts to requiring

Pmi;t þ P g

i;t þ P ri;t ¼ P it

i ðdi;tÞ þ P aci ðdi;tÞ þ P b

i;t: (11)

Under constraints (7), (8), (9), (10), and (11), P ri;t is the state

variable, while fP gi;t; P

bi;t; P

mi;tg are optimization variables.

2.3 Revenues and Operation Costs

Starting with the service and distribution of the workloads,we consider the revenue for IWs at MN j to be given by anincreasing and concave utility function Uv

j ð � Þ, and the revenue

for DW q at DC i to be given by the increasing and concaveutility Uw

i;qð � Þ. On the other hand, distribution of workloads

across the network generates bandwidth costs. To this end, we

will use the convex function Gdi;jð � Þ to denote the cost of dis-

tributing workloads from MN j to DC i, which, among otherfactors, will depend on the distance between them.

Regarding power supply sources, each DC can buy energyfrom external energy markets in period t at price a

pi;t (if

Pmi;t > 0), or, sell energy to the markets at price as

i;t if

(Pmi;t < 0). Clearly, the shortage energy that needs to be pur-

chased by the DC is ½Pmi;t �

þ; while the surplus energy that can

be sold is ½Pmi;t �

�. Notwithstanding, we shall always consider

that api;t � as

i;t. This prevents less relevant buy-and-sell activi-

ties of the DC for profit and guarantees that, for every t, eitherthe shortage or the surplus energy is zero, so that at most oneof them can be positive. Those prices can be used to define theenergy transaction cost between the DC microgrid and theexternalmarket per time t

Gei ðPm

i;tÞ :¼ api;t½Pm

i;t �þ � as

i;t½Pmi;t �

�: (12)

Moreover, we will use the convex function Gci ð � Þ to denote

the cost of CG during time t, which typically is smooth qua-dratic [19]. Finally, to model the potential battery degenera-tion during the charging/discharging cycle, a strongly

convex (dis)charging costGbi ð � Þ can be employed to prevent

fast and frequent (dis)charging of batteries [20].

2.4 Incentive Payment Models

While most existing works (e.g., [5], [7], [8], [14]) assume thatIWs are fixed and inelastic, a number of interactive servicestolerate their partial execution [15] (a.k.a. workload curtail-ment). This motivates MNs to offer incentive prices for end-users to curtail their instantaneous demand or accept partialexecution, so that the peak demand is reduced under theguaranteed QoS [21]. These incentive prices are usuallyoffered when the local marginal price is high, or when thegrid operator sends emergency demand response signalssuch as a power outage [16].

Mathematically, let pj;t denote the incentive price that attime t MN j pays to users willing to reduce their IW [17].

This way, if users reduce their demand by an amount �Vj;t,


the MNwill pay pj;t �Vj;t. It is further assumed that users reactto the given price in a simple non-strategic manner. Specifi-

cally, when reducing their workload demand by an amount �Vj;t,

users incur a strictly convex unsatisfactory cost Guj ð �Vj;tÞ ¼

kjð �Vj;tÞ2, where the coefficient kj > 0 can be learned fromhistorical data. Rational users set then their demand bysolving the following optimization problem

max0� �Vj;t�hVj;t

pj;t �Vj;t �Guj ð �Vj;tÞ; (13)

where Vj;t is the total interactive workload demand for MN jwithout incentive payment [cf. (1)], and h is the threshold ofmaximum workload reduction.

Note that (13) admits a closed-form solution, namely

�Vj;t ¼ �Vjðpj;tÞ ¼ ðrGuj Þ

�1ðpj;tÞ ¼ pj;t=2kj� �hVj;t

0; (14)

where rGuj denotes the gradient of G

uj w.r.t. �Vj;t, ðrGu

j Þ�1 is

the inverse function of rGuj , and ½ � �hVj;t0 stands for the pro-

jection onto the interval ½0; hVj;t�. Hence, for a given incen-tive price pj;t, the actual workload demand of MN j

becomes ~Vj;t ¼ Vj;t � �Vjðpj;tÞ, which can be written as a con-

vex (linear) function of pj;t as ~Vj;t ¼ Vj;t � pj;t=ð2kjÞ� �1

ð1�hÞVj;t.

As a result, the IW revenue Uvj ð ~Vj;tÞ can written as Uv

j

ðpj;t;Vj;tÞ; i.e., a function of pj;t parameterized by Vj;t.

3 STOCHASTIC LOAD BALANCING

Section 2 identified the variables, costs and constraints thatmust be accounted for in our network optimization problem,which is rigorously formulated here. In particular, we aim topursue online energy and workload management for theconsidered MN-DC network. At each time t, the systemoperator in each DC and MN performs real-time schedulingto optimize routing fvi;j;t; ~wi;j;q;tg, workloads fwi;q;tg, DC data

demand fdi;tg, incentive prices fpj;tg, CG generation fP gi;tg,

battery charging energy fP bi;tg, and external power supply

fPmi;tg. The goal is to minimize the limiting average network

cost, subject to IT operational constraints, as well as CG andstorage constraints. It is instructive to collect all sources ofrandomness into the state vector sst :¼ fap

i;t;asi;t;Wj;q;t;

Vj;t; Pri;t; 8i; j; qg, and also all the optimization variables into

xt :¼ fvi;j;t; wi;q;t; ~wi;j;q;t; di;t; pj;t; Pmi;t; P

gi;t; P

bi;t; 8i; j; qg. Strictly

speaking, not all variables in xt are free. Indeed, using someof the equality constraints in Section 2, the value of variablessuch as di;t can be found using vi;j;t and wi;q;t via (4). The rea-son for writing them as optimization variables and forcingthe equality through a constraint, which will also turn outout to facilitate distributed solvers, will be apparent later.The resultant aggregated network cost for the consideredMN-DC network at time t is

Ct ¼ Cðxt; sstÞ :¼X

i2IGe

i ðPmi;tÞ þGc

i ðPgi;tÞ þGb

i ðP bi;tÞ

� �

�X

i2I

X

q2QUwi;qðwi;q;tÞ þ

X

j2Jpj;t �Vjðpj;tÞ � Uv

j ðpj;t;Vj;tÞ� �

þX

i2I

X

j2JGd

i;j

�

vi;j;t þX

q2Q~wi;j;q;t

�

:

(15)

Defining also Yt :¼ fY bi;t; Y

mnj;q;t; Y

dci;q;t; 8i; j; qg, the optimal

scheduling is obtained as the solution to the following long-term network-optimization problem:

C :¼ min

fxt;Yt; 8tglimT!1

1

T

X

T

t¼1

E Cðxt; sstÞ½ � (16a)

s.t. Y bi;tþ1 ¼ Y b

i;t þ P bi;t; 8i; t (16b)

Y bi � Y b

i;t � Yb

i ; 8i; t (16c)

P bi � P b

i;t � Pb

i ; 8i; t (16d)

0 � P gi;t � P

g

i ; 8i; t (16e)

0 � di;t � MiDi; 8i; t (16f)X

i2Ijvi;j;t ¼ Vj;t � pj;t=ð2kjÞ; 8j; t (16g)

0 � pj;t � 2hkjVj;t; 8j; t (16h)

vi;j;t þX

q2Q~wi;j;q;t � Bj;i; 8j; i (16i)

di;t ¼X

j2Jvi;j;t þ

X

q2Qwi;q;t; 8i; t (16j)

Pmi;t þ P g

i;t þ P ri;t ¼ P it

i ðdi;tÞ þ P aci ðdi;tÞ þ P b

i;t; 8i; t (16k)

Y mnj;q;tþ1 ¼

h

Y mnj;q;t þWj;q;t �

X

i2Ij~wi;j;q;t

i1

0; 8j; q; t (16l)

Y dci;q;tþ1 ¼

h

Y dci;q;t � wi;q;t þ

X

j2J~wi;j;q;t

i1

0; 8i; q; t (16m)

Y mnj;q;t < 1; 8j; q; t; Y dc

i;q;t < 1; 8i; q; t; (16n)

where the objective considers all time instants jointly (i.e.,the entire scheduling horizon), and the expectation is takenover all sources of randomness (i.e., all variables in sst). Twoadditional remarks on (16) are in order. First, feasibility of(16) is assumed throughout the paper. However, some ofthe constraints could render the problem infeasible, and thiscould be detected by tracking the corresponding multi-pliers. Second, although strictly speaking the problem in(16) is convex, the battery dynamics in (16b) as well as thedelay-tolerant workload queues in (16l) and (16m) couplethe optimization variables over the infinite time horizon.This requires running a joint optimization scheme with aprohibitively high dimensionality. Even worse, for the prac-tical case where the knowledge of sst is causal, finding theoptimal solution while accounting for the coupling acrosstime calls for dynamic programming tools, which are gener-ally intractable. Our approach to circumventing this obsta-cle is to relax (16b), (16l) and (16m), by replacing them withaverage constraints, and employ dual decomposition tech-niques to separate the solution across time. This is elabo-rated in the next section.

3.1 Problem Relaxation

Combining (16l), (16m) and (16n), it follows that in the longterm the average workload arrival and departure rates mustsatisfy the following necessary conditions


limT!1

1

T

X

T

t¼1

E Wj;q;t

� �

� limT!1

1

T

X

T

t¼1

E

"

X

i2Ij~wi;j;q;t

#

; 8j; q (17a)

and

limT!1

1

T

X

T

t¼1

E

"

X

j2J~wi;j;q;t

#

� limT!1

1

T

X

T

t¼1

E wi;q;t

� �

; 8i; q: (17b)

In words, in the long term all buffered DWs should beserved. Upon observing that the batteries in (16b) and (16c)exhibit dynamics very similar to those of the workloadqueues, we use the same relaxation and require

limT!1

1

T

X

T

t¼1

E P bi;t

h i

¼ 0; 8i: (17c)

As before, (17c) guarantees that in the long term the energystored into the battery and the energy taken from it are equal.Using (17a), (17b), and (17c), a relaxed version of (16) is

~C :¼minfxtg limT!1

1

T

X

T

t¼1

E Cðxt; sstÞ½ �

s.t. ð16dÞ � ð16jÞ; ð17aÞ � ð17cÞ:(18)

Compared to (16), variables Yt :¼ fY bi;t; Y

mnj;q;t; Y

dci;q;t; 8i; j; qg

are not present in (18), and the time-coupling constraints(16b) and (16l) and (16m) are replaced with (17a), (17b), and(17c).

The problem in (18) has a number of interesting proper-ties, including: a) since (18) is a relaxed version of (16), it fol-lows that ~C � C

; b) if fsstg is stationary, the solution isstationary too and easy to characterize—this will be furtherdiscussed in the next paragraph; and c) as will argued inSection 5, there exist low-complexity solvers that approxi-mate the solution of (18) while being feasible for (16).

Regarding property b), using arguments similar to thosein, e.g., [6], [22], it can be shown that if the random processfsstg is stationary, there exists a time-invariant control policyx : sst ! xt inducing a solution xt ¼ xtðsstÞ ¼ xðsstÞ, which canbe shown to: satisfy the constraints (16d), (16e), (16f), (16g),(16h), (16i), and (16j); achieve optimal performance E½CðxðsstÞ; sstÞ� ¼ ~C; and satisfy the long-term constraints E½Wj;q

ðsstÞ �P

i2Ij ~wi;j;qðsstÞ� � 0; 8j; q, E½wi;qðsstÞ �P

j2J ~wi;j;q ðsstÞ� �0; 8i; q and E½P b

i ðsstÞ� ¼ 0; 8i [cf. (17a), (17b), and (17c)]. Acritical implication of this is that all the expectations insidethe limiting time averages in (18) yield the same result.Therefore, the time averages can be removed and the prob-lem can be tackled using “standard” convex stochastic pro-gramming tools. To handle the coupling acrossoptimization variables introduced by the expectations in(17a), (17b), and (17c), we will dualize the long-term con-straints (17a), (17b), and (17c), and use a decompositionapproach in the dual domain. As explained in detail in thenext section, after the dualization, the optimal solution foreach t can be computed separately across time.

3.2 Dual Decomposition

Let f�mnj;q g, f�dc

i;qg and f�bi g denote the Lagrange multipliers

associated with constraints (17a), (17b), and (17c), respec-tively. With xT :¼ fxtgt2T ¼ fxðsstÞgt2T , and �� collecting all

the multipliers, the partial Lagrangian function of (18) is

LðxT ; ��Þ :¼ E½CðxðsstÞ; sstÞ� þX

i2I

X

q2QE

"

�dci;q

�

X

j2J~wi;j;qðsstÞ � wi;qðsstÞ

�

#

þX

j2J

X

q2QE

"

�mnj;q

�

Wj;qðsstÞ �X

i2I j~wi;j;qðsstÞ

�

#

þX

i2IE �b

i P bi ðsstÞ

� � �

;

(19)where the expectation is taken over the steady-state distri-bution of fsstg. Since this distribution is time-invariant, notethat the index t in sst could be dropped.

With X t :¼ XðstÞ denoting feasible set defined by theinstantaneous constraints (16d), (16e), (16f), (16g), (16h),(16i), and (16j), which are the ones not dualized in (19), theLagrange dual function is

Dð��Þ :¼ minfxðstÞ2XðstÞgt2T

LðxT ; ��Þ (20)

and the dual problem of (18) is

max��

Dð��Þ: (21)

For the dual problem (21), a standard iterative dual subgra-dient algorithm (DSA) can be employed to obtain the optimal��. Namely, with k being an iteration index, the multipliersat iteration kþ 1, denoted by ��ðkþ 1Þ, are found as

�dci;qðkþ 1Þ ¼

�

�dci;qðkÞ þ mg�dc

i;qðkÞ�10; 8i; q (22a)

�mnj;q ðkþ 1Þ ¼

�

�mnj;q ðkÞ þ mg�mn

j;qðkÞ�10; 8j; q (22b)

�bi ðkþ 1Þ ¼ �b

i ðkÞ þ mg�biðkÞ; 8i; (22c)

where m > 0 is a constant stepsize that, if convenient, canbe rendered different for each multiplier, andg��ðkÞ :¼ fg�b

iðkÞ; g�dc

i;qðkÞ; g�mn

j;qðkÞ; 8i; j; qg denote the subgra-

dients of Dð��Þ in (20) w.r.t. the corresponding dual

variables. These can be expressed as g�dci;qðkÞ ¼ E

�P

j2J

~wi;j;qðsst; kÞ � wi;qðsst; kÞ�

, g�mnj;qðkÞ ¼ E

�

Wj;qðsst; kÞ �P

i2Ij ~wi;j;qðsst; kÞ�

,

and, g�biðkÞ ¼ E

�

P bi ðsst; kÞ

�

, with xT ðkÞ ¼ fxtðkÞgt2T ¼ fxðsst; kÞgt2Tstanding for the primal minimizers of the Lagrangian forthe kth iteration of the subgradient method, i.e., xT ðkÞ :¼argminxT LðxT ; ��ðkÞÞ subject to (16d), (16e), (16f), (16g),

(16h), (16i), and (16j).Due to the linearity of the E½�� operator, the minimization

w.r.t. the primal variables in (20) can be performed sepa-rately across time. Hence, the primal minimizers xT ðkÞ ¼fxðsst; kÞgt2T can be found by solving the following (infi-nitely many) instantaneous sub-problems (one per st)

xðsst; kÞ 2 argminxt

Cðxt; sstÞ þX

i2I

X

q2Q�dci;qðkÞ

�

X

j2J~wi;j;q;t

� wi;q;t

�

þX

j2J

X

q2Q�mnj;q ðkÞ

�

Wj;q;t �X

i2I~wi;j;q;t

�

þX

i2I�bi ðkÞP b

i;t

s.t. ð16dÞ � ð16jÞ;(23)


where the operator 2 accounts for cases that the Lagrangianhas more than one minimizer. The problem in (23) is convexand has a small-to-moderate dimensionality, so that in mostpractical cases, it is not difficult to solve. In fact, for a num-ber of relevant cost and utility functions (including qua-dratic and logarithmic), closed-form solutions for many ofthe primal variables can be found.

3.3 Stochastic DSA

The standard DSA in (22) involves taking the expectationover the stationary distribution of sst to obtain the subgra-dient g��ðkÞ. This can be challenging not only for numericalreasons, but also because such distributions can be difficultto characterize or estimate when unknown. To circumventthis challenge, we will resort to stochastic approximation[23]. The benefits are multiple, including: a) considerablyreduced computational complexity; b) the distribution of sstneed not be known; and, c) the resultant algorithms arerobust to noise and non-stationary environments. Specifi-cally, the iterations in (22) are replaced with

�bi;tþ1 ¼ �b

i;t þ mP bi;t; 8i (24a)

�dci;q;tþ1 ¼

�dci;q;t þ m

�

X

j2J~wi;j;q;t � wi;q;t

�

�1

0

; 8i; q (24b)

�mnj;q;tþ1 ¼

�mnj;q;t þ m

�

Wj;q;t �X

i2Ij~wi;j;q;t

�

�1

0

; 8j; q; (24c)

where fP bi;t; ~wi;j;q;t; wi;q;tg are found by solving

minxt

Fðxt; sstÞ :¼ Cðxt; sstÞ þX

i2I

X

q2Q�dci;q;t

�

X

j2J~wi;j;q;t � wi;q;t

�

þ �bi;tP

bi;t

�

þX

j2J

X

q2Q�mnj;q;t

�

Wj;q;t �X

i2Ij~wi;j;q;t

�

s.t. ð16dÞ � ð16jÞ:(25)

As will be shown in Section 5, the stochastic iterations in(24) and (25) come with two additional benefits critical forthe problem at hand. First, there are performance and feasi-bility guarantees establishing that the solution provided by(25) is a tight approximation to the solution of (18). Second,if properly initialized, the solution to (25) can be shown tobe feasible for the original problem in (16). Last but notleast, links between the stochastic estimates in (24) and thebattery and queue lengths can be established; see [24] and[25] for a rigorous discussion.

Remark 1. In practice, it can be useful to re-scale the subgra-dient in (24) so that each dual variable is updated with adifferent stepsize. First, if the order of magnitude of thebattery (dis)charging and the workload arrival rate arevery different, stepsize adjustment facilitates numericalconvergence. Second, within one class of constraints—e.g.,flow conservation at the MN side—using different stepsizesoffers as amechanism to effect delay or queuing priorities.

4 REAL-TIME DISTRIBUTED LOAD BALANCING

Though the online optimization (16) was separated acrosstime instants, the instantaneous convex problem (25) still

requires a centralized solver optimizing the variables of allDCs and MNs jointly, which can be challenging if, e.g., thenumber of variables is very large. Our next goal is to developan algorithm that, for each time slot t, finds the optimal solu-tion distributedly across the network entities (MNs and DCs)using only local exchanges. Distributed algorithms exhibit anumber of attractive features in networked setups, includinglow computational complexity, robustness and privacy [26].

Toward these objectives, wewill again rely on dual decom-position based approaches. Specifically, we further dualizethe constraint (16j) in (25), which couples the optimizationvariables amongMNs and DCs. The fact that the constraint isinstantaneousmeans that it has to be satisfied per time instantt (or equivalently per realization st). As a result, the algo-rithms developed in this section have to run several iterationsper time instant (those can be thought of as micro-slots), andthe overall network optimization algorithm operates in two

timescales.1 With pp :¼ ½p1; . . . ;pI �> 2 RI denoting the instan-

taneous Lagrange multipliers associated with (16j) in (25), thepartial Lagrangian of the instantaneous problem in (25) can bewritten as [cf.Fðxt; stÞ in (25)]

~Lðx;ppÞ :¼ FðxÞ þX

i2Ipi

X

j2Jvi;j þ

X

q2Qwi;q � di

!

; (26)

where the stochastic dual variable ��t is implicit inFðxÞ since��t is updated in a slower time scale and can be regarded as

constant when solving (25). With ~X denoting feasible setdefined by the constraints (16d), (16e), (16f), (16g), (16h),

and (16i), the Lagrange dual function of (25) is ~DðppÞ :¼minx2~X

~Lðx;ppÞ, and the dual problem of (25) is

maxpp

~DðppÞ: (27)

It is worth stressing that different from the dual formulationin (21), which facilitates the implementation of stochasticapproximation schemes, the goal of the dual relaxation in(27) is to obtain a fully distributed algorithm, which impliesthat the computation and communication tasks can be car-ried out at each MN and DC.

To this end, we propose two gradient methods for solv-ing (27): a DSA that can be used for any convex formulation,and a dual accelerated gradient method that requires someadditional assumptions.

4.1 DSA-Based Solution

We consider first DSA, which is the workhorse method tofind the optimal Lagrange multipliers [27]. With ‘ denotingthe iteration (micro-slot) index, the optimal pp is foundupon running

ppð‘þ 1Þ ¼ ppð‘Þ þ bð‘Þr~Dðppð‘ÞÞ; (28)

where bð‘Þ is the stepsize, and the gradient evaluated at ppð‘Þis given by

r~Dðpið‘ÞÞ ¼X

j2Jvi;jð‘Þ þ

X

q2Qwi;qð‘Þ � dið‘Þ; 8i: (29)

1. As all problems here are instantaneous (for a certain t and st).Time index twill be dropped throughout Section 4 for brevity.


As before, xð‘Þ stands for the minimizer of the Lagrangian~Lðx;ppÞ when pp ¼ ppð‘Þ. Partitioning minx2 ~X

~Lðx;ppÞ, per itera-tion ‘, each DC i needs to obtain a tentative power allocation

fdið‘Þ; P gi ð‘Þ; P b

i ð‘Þg by solving

mindi;P

gi;Pb

i

Gei ðPm

i Þ þGci ðP

gi Þ þGb

i ðP bi Þ þ �b

i Pbi � pið‘Þdi

s.t. ð16dÞ � ð16fÞ(30)

and a delay-tolerant workload schedule fwi;qð‘Þg by solving

minwi;q

pið‘Þ � �dci;q

� �

wi;q � Uwi;qðwi;qÞ (31)

while each MN j has to obtain fpjð‘Þ; vi;jð‘Þ; ~wi;j;qð‘Þg via

minpj;vi;j; ~wi;j;q

X

i2Ij

X

q2Q�dci;q � �mn

j;q

� �

~wi;j;q þ pið‘Þvi;j

þGdi;j

�

vi;j þX

q2Q~wi;j;q

��

þðpjÞ22kj

� Uvj ðpj;VjÞ

s.t. ð16gÞ � ð16iÞ:

(32)

The DSA enjoys convergence guarantees if a sequence ofnon-summable diminishing stepsizes is chosen to satisfylim‘!1 bð‘Þ ¼ 0 and

P1‘¼0 bð‘Þ ¼ 1 [27]. Since (25) is convex,

the duality gap is zero, and the minimizer of the Lagrangianyields the optimal solution to the primal problem (25). Alter-natively, if a constant stepsize b is adopted, the subgradientiterations (28) are guaranteed to converge to a neighborhoodof the optimal pp for the dual problem (27), and the runningaverage of the primal variables will converge to the optimalsolution [27]. In practice, the iterations can be stopped once apre-specified tolerance (or duality gap) is met. It is also worthnoting that the DSA is fairly robust and exhibits a number offeatures that are attractive for networked setups, includingthe fact of converging to a near-optimal solution even whenthe information exchanges (e.g., the multipliers) are noisy orsporadically lost. This can happen in the presence of noise inthe communication links across the network; see, e.g., [28].

4.2 Accelerated FISTA-Based Solution

Though universally applicable and widely used, DSA doesnot leverage properties that are specific to the problem athand, including differentiability of the dual function andLipschitz continuity of its gradient. In this section, weimprove the solver for the problem at the fast timescale (i.e.,(25)) by advocating an alternative approach based on FISTA[29], which exhibits faster convergence rate than that ofDSA in (28). This is important, because for any t a newinstance of (25) needs to solved in real time.

Algorithm 1. Dual FISTA Iteration for (27)

1: Initialize:with proper ppð0Þ, ppð1Þ, upð0Þ and stepsize b2: for ‘ ¼ 1; 2 . . .do3: Update upð‘þ 1Þ via (33b).4: Update �ppð‘Þ via (33a).5: DCs send �ppð‘Þ to MNs.6: Each MN and DC locally solve (30), (31), and (32) using

pp ¼ �ppð‘Þ to obtain the virtual decision xð‘Þ.7: Update Lagrange multipliers ppð‘Þ via (33c).8: end for

Inheriting Nesterov’s acceleration scheme [30], FISTAwas originally developed in the context unconstrained pri-mal optimization [29], and has been recently applied todual problems for network utility maximization [31]. Unlikethe dual gradient iteration (28), which relies only on the cur-rent iterate, FISTA leverages the memory of one previousiterate aiming to improve convergence. Per iteration (micro-slot) ‘, FISTA constructs an intermediate iterate �ppð‘Þ byusing an affine combination of the two most recent iteratesppð‘Þ and ppð‘� 1Þ, that is

�ppð‘Þ ¼ 1� 1� upð‘� 1Þupð‘Þ

� �

ppð‘Þ þ 1� upð‘� 1Þupð‘Þ

ppð‘� 1Þ;

(33a)

where the weights are “optimally” updated as [29]

upð‘Þ ¼1

21þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1þ 4u2pð‘� 1Þq

� �

: (33b)

By computing the dual gradient at the intermediate iter-ate �ppð‘Þ, the dual variable ppð‘Þ is updated using a gradientascent step based on �ppð‘Þ, that is

pið‘Þ ¼ �pið‘Þ þ b

�

X

j2Jvi;jð‘Þ þ

X

q2Qwi;qð‘Þ � dið‘Þ

�

; 8i; (33c)

where b is a proper stepsize. The dual FISTA iteration is sum-marized in Algorithm 1. Note that the key difference betweenthe DSA iteration (28) and the dual FISTA iteration (33) is theintermediate iterate �ppð‘Þ alongwith the gradient at �ppð‘Þ. Intui-tively, by combining the previous two dual iterates, the“smoothed” iterates �ppð‘Þ can mitigate the undesirable oscilla-tion of the simple gradient ascent iteration, thus achieving fastconvergence. A more detailed interpretation of Nesterov’sacceleration and FISTA can be found in, e.g., [32].

Convergence of FISTA requires: a) the dual function~DðppÞ to be differentiable; and b)r ~D to be Lipschitz continu-ous. Hence, in the remainder of the section we first elaborateon these two conditions, and then assert the convergence ofFISTA formally in Proposition 3.

To satisfy a) in our setup, the following assumption isrequired: (as1) for a given t, the network cost CðxÞ is stronglyconvex w.r.t. x. From an engineering perspective, assumingstrong convexity is reasonable. Oftentimes in practice themarginal rewards (costs) are monotonically decreasing(increasing), which does guarantee strong convexity. Buteven if they are not, one can approximate FðxÞ in (25) by the

regularized strongly convex cost function FðxÞ þ ð�=2Þkxk2,with � > 0. While the regularizer may introduce a smalloptimality loss, the solution is feasible and the loss is pro-portional to �, which is typically selected small (see Appen-dix A, which can be found on the Computer Society DigitalLibrary at http://doi.ieeecomputersociety.org/10.1109/TPDS.2016.2636210, for details). Indeed, since our ultimategoal for DGLB is to have an OðmÞ-optimal online solution,we will show that it suffices to set � ¼ OðmÞ. Once (as1)

holds, differentiability of the dual function ~DðppÞ followsreadily, as formally stated next [33, Lemma 1].


Proposition 1. For a given pp, the partial Lagrangian (26) has a

unique minimizer; thus, the dual function ~DðppÞ is continu-ously differentiable.

Proof. See [33, Lemma 1]. tuHowever, finding the Lipschitz constant of r ~D required

in b) is nontrivial due to the coupling among primal varia-bles. To circumvent this impasse, we introduce an equiva-lence between differentiability of a convex function andstrong convexity of its conjugate to facilitate the derivationof the Lipschitz constant; see [31, Lemma II.1].

Lemma 1. Let h:Rn ! ð�1;1� be a proper, lower semicontinu-ous, convex function, and constant s > 0. The following state-ments are equivalent: S1) Function h is differentiable and its

gradient rh is Lipschitz continuous with constant 1s; S2) The

conjugate function h:Rn ! ð�1;1� is s-strongly convex.Leveraging Lemma 1, the Lipschitz constant of r ~D can

be found by analyzing the convexity of the primal objective.The precise result is given in the next proposition.

Proposition 2. The Lipschitz constant ofr ~D is

L :¼ ðJ þQþ 1Þ

maxq;j;i1

sðUwi;qÞ

;1

sðGbi Þ;

1

sðGdi;jÞ

;1

sðGci Þ;MiD

2i

2asi%

;2k2j

2kj þ sðUvj Þ

( )

;

(34)

where sðUwi;qÞ, sðGb

i Þ, sðGdi;jÞ, sðGc

i Þ, and sðUvj Þ are defined in

Proposition 6 in Appendix B, available in the online supple-mental material.

Proof. See Appendix B, available in the online supplemen-tal material. tuWith the definition of L, we are ready to establish the

convergence result [30], [31], which closes this section.

Proposition 3. If L denotes the Lipschitz constant of r ~D in

(34), and the step-size b 2 ð0; 1L�, then Algorithm 1 converges

to optimal dual variable pp. And for ‘ � 1, it satisfies

~DðppÞ � ~Dðppð‘ÞÞ � 2kpp � ppð0Þk2

bð‘þ 1Þ2: (35)

Proof. See the proof of [29, Theorem 4.4]. tu

4.3 Diagonally Weighted FISTA

Computing the Lipschitz constant L, whose value has to beknown to set b in (33), requires in general communicationamong all the DCs and MNs [cf. (34)], which may be diffi-cult (or costly). In this section, we consider the scaled ver-sion of FISTA, where each dual variable pi is updated usinga different stepsizewith limited information exchanges.

Collect the I stepsizes in the I I diagonal matrix Lwhose ith diagonal element is given by

Lii ¼X

j2Jmaxi2Ij

1

sðGdi;jÞ

;2k2j

2kj þ sðUvj Þ

( )

þX

q2Q

1

sðUwi;qÞ

þmax1

sðGbi Þ;

1

sðGci Þ;MiD

2i

2asi%

�

:

(36)

Compared to the standard FISTA in Section 4.2, here eachDC only needs to know the global information

P

j2J

maxi2Ij 1=sðGdi;jÞ; 2k2j=

�

2kj þ sðUvj Þ

n o

, which can be

obtained from the subset of MNs the particular DC is con-nected to.

With the diagonal scaling matrix L defined in (36), wecan consequently establish the next proposition.

Proposition 4. If the update for pið‘Þ in (33c) is replaced with

pið‘Þ ¼ �pið‘Þ þ L�1ii

X

j2Jvi;jð‘Þ þ

X

q2Qwi;qð‘Þ � dið‘Þ

!

; 8i

then Algorithm 1 converges to the optimal dual variable pp.And for ‘ � 1, it holds that

~DðppÞ � ~Dðppð‘ÞÞ � 2kpp � ppð0Þk2L

ð‘þ 1Þ2: (37)

Proof. See Appendix C, available in the online supplemen-tal material. tuNotice that since L � LI, it follows that

2kpp � ppð0Þk2L

ð‘þ 1Þ2� 2Lkpp � ppð0Þk2

ð‘þ 1Þ2� 2kpp � ppð0Þk2

bð‘þ 1Þ2:

This implies that along with the reduction of the communi-cation overhead, the scaled FISTA also enjoys a faster con-vergence rate.

4.4 Real-Time Distributed Implementation

Integrating the stochastic DSA in Section 3.3 with the dualFISTA algorithm in Section 4.3 gives rise to the DGLB algo-rithm proposed in this paper, whose steps are described indetail in Algorithm 2.

Algorithm 2. Distributed Geographical Load Balancing

1: Initialize Lagrange multipliers ��0, and stepsizes m;L.2: Per slot t, observe the state sst, obtain ��t from the step 7 at

iteration t� 1, and then run the following tasks.3: Signaling exchange. Obtain pp

t using either the DSA (28) orthe FISTA updates (33).

4: MN pricing and routing. Each MN solves (32) using ppt , and

obtains fpj;t; vi;j;t; ~wi;j;q;tg. Offer incentive payment pj;t to theend users nearby, and perform workload routingfvi;j;t; ~wi;j;q;tg based on the actual arrival rates.

5: DC workload schedule. Obtain fwi;q;tg by solving (31) usingpi;t. Process IWs based on fvi;j;tg, and schedule DWs in each

class according to fwi;q;tg.6: DC energy schedule and trading. Obtain fdi;t; P g

i;t; Pbi;tg by

solving (30). Perform energy transaction with the main grid;

that is, buy the energy amount ½Pmi;t �

þ with price atb;t upon

energy deficit, or, sell the energy amount ½Pmi;t�

� with price

ats;t upon energy surplus. Perform battery (dis)charging

according to P bi;t, and plan CG generations.

7: Update the stochastic multipliers ��tþ1. With fP bi;t; ~wi;j;q;t;

wi;q;tg available, DCs update Lagrange multipliers f�bi;tþ1g

and f�dci;q;tþ1g via (24a) and (24b), and MNs update Lagrange

multipliers f�mnj;q;tþ1g via (24c).


Regarding steps 4 and 5, if the real-time energy purchaseand selling prices are identical at each DC [34], i.e.,api;t ¼ as

i;t; 8t, the DC subproblems (30) and (31) can be

solved in closed-form, as formalized next.

Proposition 5. For DC subproblem (30), the optimal solutionscan be expressed in a closed-form

P gi ð‘Þ ¼ ðrGg

i Þ�1ðap

i Þh iP

gi

0; P b

i ð‘Þ ¼ ðrGbi Þ

�1ð��bi � a

pi Þ

h iPbi

Pbi

and likewise

dið‘Þ ¼ ðMiD2ipið‘ÞÞ=ð2%ð1þ eiÞap

i Þ� �MiDi

0:

For subproblem (31), the optimal solution is

wi;qð‘Þ ¼ ðrUwi;qÞ

�1ðpið‘Þ � �dci;qÞ

h i1

0:

Proof. See Appendix D, available in the online supplemen-tal material. tuNotice that the communication overhead of DGLB is

fairly low. While DC sub-problems have closed-form solu-tions, MN sub-problems (32) can be solved in parallel,reducing the per-iteration complexity. In addition, leverag-ing the accelerated convergence offered by FISTA, the algo-rithm usually finds the solution within tens of iterations.

5 PERFORMANCE GUARANTEES

To arrive at our main claim, we begin by quantifying theoptimality gap of the proposed DGLB. Based on the results[6], [24], the following lemma holds true.

Lemma 2. If the random state sst is either i.i.d. or follows a finitestate ergodic Markov chain, and the random duration of the

renewal interval of the Markov chain DTn satisfies E½DT 2n � <

1, then the limiting time-averaged net-cost under the proposedonline algorithm satisfies

limT!1

1

T

X

T�1

t¼0

E C xt; stð Þ½ � � C þ mM

E½DT 2n �

E½DTn�;

where DTn ¼ 1 for the i.i.d. case, the constantM is defined as

M :¼ 1

2

X

j2J

X

q2Q

max

(

W j;q;X

i2IBj;i

)!2

þ 1

2

X

i2I

X

q2Q

�

maxfMiDi;X

j2JBj;ig

�2

þ ðmaxfP b

i ;�P bi gÞ

2�

andC is the optimal value of (16) under any feasible control.

Proof. See Appendix E, available in the online supplemen-tal material. tuLemma 2 asserts that our DGLB can achieve a near-optimal

objective value for (16). However, sinceDGLB approximates arelaxation of (16) [cf. (18)], the resultant dynamic control pol-icy is not guaranteed to be feasible. In the sequel, we willestablish that, when properly initialized, DGLB indeed yieldsa feasible policy for (16). To achieve this, we start by character-izing a pair of properties of the optimal policy.

Lemma 3. If api :¼ maxfap

i;t; 8tg and asi :¼ minfas

i;t; 8tg, thereal-time battery (dis)charging decisions P b

i;t generated by the

DGBL algorithm satisfy: i) P bi;t ¼ P b

i , if �bi;t > �as

i � @Gbi ; and,

ii)P bi;t ¼ P

b

i , if �bi;t < �a

pi � @G

b

i .

Lemma 3 nicely entails the economic interpretation ofLagrange multipliers. Indeed, the stochastic multiplier �b

i;t

can be viewed as the instantaneous charging price. When

the price �bi;t is high enough, the optimal decision is to fully

discharge the battery; and when the price �bi;t is sufficiently

low, it will be optimal to charge the battery as much as pos-sible. Based on this salient optimal structure, Lemma 3allows us to further establish the following result.

Lemma 4. If the stepsize satisfies m � m, where

m :¼ maxi�

api þ @G

b

i � asi � @Gb

i

=�

Yb

i � Y bi þ P b

i � Pb

i

n o

;

then the stochastic multipliers generated by DGLB satisfy

�api � @G

b

i þ mP bi � �b

i;t � mYb

i � mY bi � a

pi � @G

b

i þ mP bi ; 8i; t.

These two lemmas are generalizations of [8, Lemma 4]and [8, Lemma 5]; their proof is omitted here for brevity.

Consider now the linear mapping

Y bi;t ¼

�

�bi;t þ a

pi þ @G

b

i

=mþ Y bi � P b

i ; 8i: (38)

It can be readily seen from Lemma 4 that Y bi � Y b

i;t � Yb

i

holds for all i and t; i.e., (16c) are always satisfied under theproposed online scheme. With the battery (dis)chargingdynamics (16b) naturally performed, feasibility of the con-trol actions xð��tÞ can be maintained for the original prob-lem, provided that we select a stepsize m � m.

Using Lemmas 2 and 4, the following theorem assessingthe feasibility and optimality of DGLB, which is the mainresult of this section, can be established.

Theorem 1. Upon setting �bi;0 ¼ mY b

i;0 � mY bi � a

pi � @G

b

iþmP b

i , 8i, and selecting a stepsize m � m, the DGLB algorithm

yields a feasible dynamic control scheme for (16), which satisfies

limT!1

1

T

X

T�1

t¼0

E C xt; stð Þ½ � � C þ mM

E½DT 2n �

E½DTn�;

where DTn,M and m are specified in Lemmas 2, 3 and 4.

Proof. Theorem follows from Lemmas 2 and 4 readily. tuThe theorem asserts that with a proper initialization and

m � m, DGLB is feasible for the original problem in (16), and

incurs a bounded optimality loss. Since this loss is increasingwith m, setting m ¼ mminimizes the gap and limits the perfor-

mance loss tomME½DT 2n �=E½DTn�. Scenarioswhere both thedif-

ference between purchase and selling prices ðapi � as

iÞ and thedifference between marginal charging and discharging costs

@Gb

i � @Gbi approach zero allow for selecting m very small [cf.

Lemma 4], so that the optimality loss is practically null. The

same is true in scenarios where the battery capacities Yb

i arevery large. This makes sense intuitively because as both

ðapi � as

iÞ and @Gb

i � @Gbi approach zero, purchasing extra

energy to charge the batteries (if they are close to empty) or


selling it to discharge them (if they are close to full) is alwaysprofitable. Similarly, when batteries have large capacity, theupper bounds in (16c) do not hold as equalities, and stationarypolicies obeying the long-term energy conservation constraintare optimal. Note finally that selecting m > m can be used to

reach the close-to-optimal (steady-state) operation point morequickly, but the incurred optimality losswill be higher [8], [24].

Remark 2. While feasibility of the battery dynamics holds forarbitrary sample paths of fsstg, near optimality of DGLB isguaranteed under the assumption that the random state sstis either i.i.d. or follows an ergodic Markov chain. Marko-vianity is widely used in wireless networks and power sys-tem applications to model the stochastic demand,renewable generation, and price processes [35]. Numericalresults will further demonstrate that the DGLB can obtain adesirable performance even in real data scenarios.

Remark 3. Readers familiarwith the so-called Lyapunov-opti-mization framework can recognize similarities between thestochastic DSA proposed here, and the tools in [6], [22]. Thedifferences between them can be summarized as follows:

D1) The Lyapunov-optimization solver relies on the so-called “virtual queues” to ensure that long-termaverage constraints are met, where the tuningparameter V in [6], [22] corresponds to the inverseof the stepsize m in the stochastic DSA. In contrast,“virtual queues” emerge naturally as Lagrangemultiplier iterates in our stochastic DSA;

D2) Leveraging duality and stochastic approximationtechniques, the multiplier update in the stochasticDSA is also easy to interpret. The multipliers forinstance, can be viewed as the instantaneous charg-ing prices, revealing the intuition behind workloadrouting, scheduling and real-time (dis)chargingdecisions, as discussed after Lemma 4; and

D3) Results from duality theory, including sensitivityand weak duality, can be used to characterize theperformance of our stochastic DSA (cf. Lemma 3).

6 NUMERICAL TESTS

This section presents numerical tests to confirm the analyticalclaims in Section 5, and demonstrate the merits of the pro-posed approach. We start by describing the simulation setup.The network considered has I ¼ 4 DCs and J ¼ 4 MNslocated in the eastern, central, mountain and western parts ofthe US. The number of servers at each DC isfMig ¼ f1;000; 750; 750; 1;000g. One unit of workload isassumed to require the computing resources of five servers(Di ¼ 0:2; 8i), and the IW curtailment ratio is h ¼ 0:2. For sim-plicity, the cooling coefficients at each DC are consideredtime-invariant with values fei;tg ¼ f0:2; 0:3; 0:4; 0:5g. Thebandwidth limits fBj;ig are generated from a uniform

distribution within ½20; 300�, and the communication cost is

Gdi;j ðf ~wi;j;q;tgq2Q; vi;j;tÞ ¼ cdi;jðvi;j;t þ

P

q2Q ~wi;j;q;tÞ2 þ v2i;j;t þP

q2Q ~w2i;j;q;t, with cdi;j inversely proportional to Bj;i. We con-

siderQ ¼ 2 types of DW tasks, and the revenue for each type

is the same at all DCs Uwi;qðwi;q;tÞ ¼ �uqðwi;q;tÞ2 þ 50uqwi;q;t

with uq being uniformly distributedwithin ½1; 3� cents/(unit)2.The coefficient kj in Gu

j ð �Vj;tÞ is generated from a uniform dis-

tribution within ½1; 3� cents/(unit)2. Each DC is connected to amicrogrid consisting of a CG, an RG, a battery, and facilitiesfor two-way trading with the external market. The power-related parameters are set identically across all DCs as listed

in Table 2. The CG cost is Gci ðP

gi;tÞ ¼ cgðP g

i;tÞ2 þ 10cgP gi;t with

cg ¼ 0:5 cents/(kWh)2, while the battery (dis)charging cost is

Gbi ðP b

i;tÞ ¼ cbðP bi;tÞ

2 with cb ¼ 1 cents/(kWh)2. In addition, the

energy purchase price is set equal to the selling price; i.e.,api;t ¼ as

i;t; 8i; t, and the duration of a scheduling period (time

slot) is one hour.Two sets of numerical results are presented: one to demon-

strate convergence and optimality in synthetic scenarioswhere the random variables are drawn either from a givendistribution, or, an ergodic Markov chain (Section 6.1); andthe other one to illustrate performance in a practical scenariousing real data (Section 6.2). Note that workload arrivals,energy prices, and renewable generations in Case study 2 arehighly correlated over time, so it also serves to assess theapplicability of DGLB to non-stationary setups. To benchmarkperformance of the proposed algorithm, three baselineschemes are tested including both the local load balance (LLB)aswell as the geographical load balancing schemes (GLB).

1) ALG 1 (LLB, with incentive payment, DW scheduling,RES and storages): ALG 1 is similar to the algorithmsin [6], [8], where MNs only route workloads to theclosest DC.

2 ALG 2 (GLB, without DW scheduling, nor incentive pay-ment, with RES and storages): ALG 2 is similar to themethod in [14], that is widely used in practical net-work systems, where no incentive pricing is used,and all the DWs are processed once they arrive,without any delay.

2 ALG 3 (GLB, without RES and storages, with DWscheduling and incentive payment): ALG 3 mimics thealgorithm in [13], but with only one-timescale

TABLE 2DC Power-Related Parameters

Pg

i Y bi Y

b

iY bi;0 P b

i Pb

iP

s

i

100 5 100 5 �20 20 1

The units are kW or kWh.

Fig. 2. The left panel shows the empirical CDF of the number of itera-tions needed to converge (T ¼ 100 realizations are considered). For oneof those realizations, the right panel plots the evolution of the primalobjective residual for the noise-free case. In the legend, “D” denotes dis-tributed, “F” FISTA, “G” gradient, “N” the presence of noise, and “L” thatsome of the multipliers are lost.


operation, where DCs are only powered by CG orpower from the spot market, without consideringRES and storage units.

6.1 Case Study 1: Convergence and Optimality

We start by running an experiment where sst is i.i.d. Specifi-cally, the purchase price a

pi;t is uniformly distributed within

½10; 30� $/kWh, samples of renewables fP ri;tg are generated

from a uniform distribution within ½1; 300� kWh, IWs fVj;tgand class-q DWs fWj;q;tg arrive at each MN according to aPoisson process, all with average arrival rates 50 units/slot.The convergence results when solving the real-time prob-lems in (25) are compared in Fig. 2, where T ¼ 100 slots areconsidered, each consisting of up to 200 micro-slots. A

sequence of stepsizes bð‘Þ ¼ 0:5=ffiffi

‘p

, ‘ ¼ 1; . . . ; 200 areemployed for the subgradient iteration in (28), the diagonal-scaled stepsize (36) is used for the dual FISTA iteration, andthe stopping criteria is either the primal objective residualbeing smaller than 0.01, or, the number of iterations beinggreater than 200. In the noise-free case (red lines in Fig. 2a),the dual FISTA (D-F) converges within 60 iterations in allslots, while the dual gradient (D-G) needs more than 150iterations on average, and fails to converge within 200 itera-tions in some cases. The accelerated convergence of D-F isfurther illustrated in Fig. 2b, where the residual reductionper update (micro-slot) is considerably larger for D-F.Regarding robustness, the blue and black lines in Fig. 2arepresent the empirical CDFs of iteration complexity whenthe exchanged multipliers are noisy (under zero-mean

Gaussian noise with variance s2 ¼ 1) or sporadically lost(under link outage probability2 0.2), respectively. The con-clusion is that the number of iterations required for D-G toconverge ranges from 70 to 200, while in most cases D-Fconverges within as few as 45-55 iterations.

The second experiment models fsstg as a Markovchain, where a

pi;t takes values from a three-state set

f10; 20; 30g, P ri;t from f10; 150; 300g, and Vj;t as well as

Wj;q;t are drawn from f25; 50; 75g. The transition matrixof this Markov chain is

P ¼0:2 0:3 0:50:3 0:3 0:40:2 0:4 0:4

0

@

1

A:

Under this setting, the optimality loss of DGLB relative tothe optimal solution of the relaxed problem (18) is

depicted in Fig. 4 for different stepsizes. The results confirmthat the optimality gap vanishes as the stepsize decreases,and also illustrates that larger stepsizes give rise to fasterconvergence. This nicely matches the theoretical characteri-zation of the optimality loss provided in Theorem 1.

6.2 Case Study 2: Scenario with Real Data

In this test case, the purchase prices fapi;tg at DCs 1 and 3� 4

are re-scaled from the local real-time data in PJM (eastern),MISO (central), and CAISO (western) during Oct. 01-25,2015, while the renewable generations fP r

i;tg are based on

the data during Oct. 01-25, 2012 [36], [37], [38]. Real-timeprices and renewable generation in the mountain region arehard to obtain, so that we generate them by averaging andre-scaling the data from central and western areas. As work-load traces are not available from public sources, IWs aregenerated by duplicating the Wikipedia load trace over a24-hour period [14], while DWs are generated by repeatingthe hourly MapReduce trace over a day [39]. In both cases,white Gaussian noise with variance randomly drawnbetween 3�5 dB was added to the original values, whichwere also re-scaled to model regional differences. The West-ern Time Zone (UTC-8) was used for time-keeping, and alldata was shifted to show the effect of time zone differences.To facilitate the interpretation of the results, the values offap

i;tg and fP ri;tg over a week are shown in Figs. 3a and 3b,

those of IWs and DWs over a day are shown in Figs. 3c and3d, and their average is listed in Table 3.

Fig. 5 depicts the running average of the network cost(primal objective) of DGLB and ALGs 1-3. Over T ¼ 600

Fig. 3. Time variation of the RES generation, local marginal prices, and IW and DW arrivals used in Test Case 2 [14], [36], [37], [38], [39].

Fig. 4. Optimality gap of DGLB for different stepsizes (m ¼ 0:35).2. In the case of a link outage, the subproblems (30), (31), and (32) at

iteration ‘ are solved using the outdated Lagrange multipliers pptð‘� 1Þ.


slots, the average network cost of DGLB is 17 percent lowerthan that of ALGs 1-2, and around 46 percent lower thanthat of ALG 3. Recall that ALGs 1-2 are vulnerable to highfluctuations of prices, RES, and workload demands due tothe lack of geographical allocation capabilities, or,“workload smoothing” tools (e.g., the incentive payment,the workload delay), and ALG 3 is sensitive to the energyprices since neither RES nor storage units are accounted for.As corroborated by Fig. 5, ALGs 1-3 incur a higher cost sincethey have to buy more (expensive) energy from the spotmarket on the peaks of user demand. In contrast, DGLBtakes advantage of incentive payments, workload queues aswell as RES and storage devices, so it can smooth the work-load curvatures and leverage RES and stored energy toavoid future purchases at high prices.

The average energy cost and the ratio of RES to the totalenergy consumption are shown in Fig. 6. While DGLB gen-erally incurs lower energy cost, ALG 2 (LLB) has the

smallest energy cost and the largest RES usage in DCs 2-3.This makes sense because the workload demands in DCs 2-3 are relatively low [cf. Figs. 3c and 3d]. As GLB schemessmooth the load profile by allocating remote loads to MNs2-3, LLB only uses local resources, leading to a higher costand lower RES utilization at DC 4.

To better understand the role of Lagrange multipliers inworkload and power balancing, the trajectories of multi-pliers associated with MN 1 and DC 1 are depicted inFigs. 7a and 7b. The (negative) Lagrange multiplier ��b

1;t inthe central panel of Fig. 7a serves as the stochastic discharg-ing price, in the sense that it always increases when the spot

market price increases; the Lagrange multiplier �b1;t pre-

cisely maps the evolution of the battery level C1;t corrobo-rating the affine mapping in (38); and the battery, tomitigate the variability of RES, will always discharge whenthe price a

p1;t is very high. The Lagrange multipliers reflect-

ing workload queue lengths in Fig. 7b are related to the ser-vice delay (the Little’s law), or, the congestion price at eachMN or DC. Per Fig. 7b, the multipliers for DGLB and GLB-based ALG 3 follow almost the same trajectory. Differently,ALG 1 exhibits larger delay, especially at the MN side. Intu-itively, this is because in ALG 1, MNs allocate all their work-loads to the nearest DC, incurring higher delay when theinstantaneous workload arrival rate is very high, or, whenthe nearest DC is overloaded. Although not shown in

Fig. 5. Comparison of time-average network costs in the DC network.

Fig. 6. Comparison of energy cost and RES usage at each DC. The RESusage is the ratio of consumed RES to the total energy consumption.

Fig. 7. The left panel shows the evolutions of the price ap1;t, the battery level C1;t, and the Lagrange multiplier �b

1;t. The middle panel plots the evolu-

tions of Lagrange multipliers �mn1;t and

�

�dc1;1;t þ �dc

1;2;t

=2 for DGLB. The dashed lines are the running average of instantaneous multipliers. The right

panel compares the evolutions of network costs for DGLB using CVX [40] to centrally solve (25), and for DGLB using the (distributed) diagonallyweighted FISTA running 20 and 40 iterations.

TABLE 3Averages of the Time Series used to Run Test Case 2

Index for DC i or MN j 1 2 3 4

Mean (api;t), cent/kWh 9.77 6.47 8.32 12.15

Mean (P ri;t), kWh 484.93 290.78 405.54 189.06

Mean (Vj;t), unit 51.81 34.69 43.32 65.10Mean (Wj;1;t), unit 26.68 19.47 19.50 31.37Mean (Wj;2;t), unit 29.10 21.26 21.29 34.2064


Fig. 7b, ALG 2 indeed experiences a close-to-zero delay, atthe expense of high network cost [cf. Fig. 5].

Since the stopping criteria of FISTA requires a-prioriknowledge of the optimal value of (25) that is not available toDGLB in practice, we finally assess the performance of DGLBwhen running a fixed number of iterations in Fig. 7c. Interest-ingly, the performance of DGLB with 40 iterations is veryclose to that of the centralized solver (10�4 relative optimalityloss), and the performance of DGLB running 20 iterations is

also good enough in practice (10�3 relative optimality loss).As a secondary comment, note that Fig. 7c demonstrates thatthe distributed DGLB with 20 iterations leads to a slightlylower cost than its more complex counterparts. The reason isthat for a very small number of iterations, the constraints (16j)are marginally violated, resulting in a better objective value.In any case, the results in this experiment (together with thosein Fig. 2) corroborate the merits of DGLB, and its suitabilityfor distributed real-time implementation.

7 CONCLUSION

Optimal schemes were designed in this paper for real-timegeographical load balancing tailored to the forthcoming sus-tainable cloud networks. Accounting for the spatio-tempo-ral variability of workloads, renewables, and electricityprices, a stochastic optimization problem was formulated tominimize the long-term aggregate cost of the MN-to-DCnetwork. Leveraging the celebrated dual decompositionmethodology, the task was decomposed across time andspace, enabling an online distributed implementation.Using a two-timescale approach, a stochastic dual gradientscheme was first implemented to handle the long-term con-straints and decouple the optimization across time slots.Then, for each time slot, an accelerated dual gradientmethod was adopted to tackle the short-term constraintscoupling the optimization across DCs. It was analyticallyestablished that by properly choosing constant stepsizesand initializations, the novel schemes attain near-optimalperformance while respecting the battery capacity andremaining operating constraints, even without knowing thedistributions of the underlying stochastic processes.Numerical tests using both synthetic and real data corrobo-rated the effectiveness and merits of the novel approaches.

ACKNOWLEDGMENTS

Work in this paper was supported by US National ScienceFoundation grants 1509040, 1508993, 1423316, 1442686, andby Spanish MINECO Grant TEC2013-41604-R and CAMGrant S2013/ICE-2933.

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Tianyi Chen (S’14) received the BEng degree(with highest honors) in communication scienceand engineering from Fudan University, in 2014,and the MSc degree in electrical and computerengineering from the University of Minnesota(UMN), in 2016, respectively. Since July 2016, hehas beenworking toward the PhD degree at UMN.His research interests lie in online convex optimi-zation, data-driven network optimization withapplications to smart grids, sustainable cloud net-works, and green communications. He received

the Student Travel Grant from the IEEE Communications Society in2013, a National Scholarship from China in 2013, and the UMN ECEDepartment Fellowship in 2014. He is a student member of the IEEE.

Antonio G. Marques (SM’13) received the Tele-communications Engineering and Doctoratedegree, both with highest honors, from the CarlosIII University of Madrid, Spain, in 2002 and 2007,respectively. In 2007, he became a faculty in theDepartment of Signal Theory and Communica-tions, King Juan Carlos University, Madrid, Spain,where he currently develops his research andteaching activities as an associate professor. From2005 to 2015, he held different visiting positionswith the University of Minnesota, Minneapolis. In

2015 and 2016, he was a visiting scholar with the University ofPennsylvania. His research interests lie in the areas of communication the-ory, signal processing, and networking. His current research focuses onstochastic resource allocation for wireless networks and smart grids, non-linear network optimization, and signal processing for graphs. He hasserved the IEEE in a number of posts (currently, he is an associate editor ofthe IEEE Signal Processing Letters), and his work has been awarded inseveral conferences andworkshops. He is a seniormember of the IEEE.

Georgios B. Giannakis (F’97) received theDiploma degree in electrical engineering from theNational Technical University of Athens, Greece,in 1981. He received the MSc degree in electricalengineering, MSc degree in mathematics, andthe PhD degree in electrical engineering from theUniversity of Southern California (USC), in 1983,1986, and 1986, respectively. From 1982 to1986, he was with the USC. He was with the Uni-versity of Virginia from 1987 to 1998, and since1999 he has been a professor with the University

of Minnesota, where he holds an endowed chair in Wireless Telecommu-nications, a University of Minnesota McKnight presidential chair in ECE,and serves as a director of the Digital Technology Center. His generalinterests span the areas of communications, networking and statisticalsignal processing—subjects on which he has published more than 400journal papers, 680 conference papers, 25 book chapters, two editedbooks and two research monographs (h-index 119). Current researchfocuses on learning from big data, wireless cognitive radios, and networkscience with applications to social, brain, and power networks withrenewables. He is the (co-) inventor of 28 patents issued, and the (co-)recipient of eight best paper awards from the IEEE Signal Processing(SP) and Communications Societies, including the G. Marconi PrizePaper Award in Wireless Communications. He also received TechnicalAchievement Awards from the SP Society (2000), from EURASIP(2005), a Young Faculty Teaching Award, the G. W. Taylor Award forDistinguished Research from the University of Minnesota, and the IEEEFourier Technical Field Award (2015). He is a fellow of the EURASIP,and has served the IEEE in a number of posts, including that of a distin-guished lecturer for the IEEE-SP Society. He is a fellow of the IEEE.

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