1

Throughput-Optimal Scheduling Design withRegular Service Guarantees in Wireless Networks

Bin Li, Ruogu Li and Atilla Eryilmaz

Abstract—Motivated by the regular service requirements ofvideo applications for improving Quality-of-Experience (QoE) ofusers, we consider the design of scheduling strategies in multi-hop wireless networks that not only maximize system throughputbut also provide regular inter-service times for all links. Since theservice regularity of links is related to the higher-order statisticsof the arrival process and the policy operation, it is challengingto characterize and analyze directly. We overcome this obstacleby introducing a new quantity, namely the time-since-last-service(TSLS), which tracks the time since the last service. By combiningit with the queue-length in the weight, we propose a novelmaximum-weight type scheduling policy, called Regular ServiceGuarantee (RSG) Algorithm. The unique evolution of the TSLScounter poses significant challenges for the analysis of the RSGAlgorithm.

To tackle these challenges, we first propose a novel Lyapunovfunction to show the throughput optimality of the RSG Algorithm.Then, we prove that the RSG Algorithm can provide service reg-ularity guarantees by using the Lyapunov-drift based analysis ofthe steady-state behavior of the stochastic processes. In particular,our algorithm can achieve a degree of service regularity within afactor of a fundamental lower bound we derive. This factor is afunction of the system statistics and design parameters and can beas low as two in some special networks. Our results, both analyticaland numerical, exhibit significant service regularity improvementsover the traditional throughput-optimal policies, which reveals theimportance of incorporating the metric of time-since-last-serviceinto the scheduling policy for providing regulated service.

Index Terms—Wireless scheduling, throughput-optimality, ser-vice regularity, Quality-of-Experience (QoE), real-time traffic.

I. INTRODUCTION

During the past years, there has been increasing deploymentof a variety of real-time applications over the wireless net-works, especially streaming multi-media applications. Unlikeits non-real-time counterpart, the real-time traffic often hasvarious quality-of-service (QoS) requirements besides through-put. Such requirements usually include end-to-end delay con-straints, packet delivery ratio requirements, and the regularityof the inter-service times. Unlike the traditional long-term mean

This work is supported by the NSF grants: CAREER-CNS-0953515 andCCF-0916664. Also, the work of A. Eryilmaz was in part supported by theQNRF grant number NPRP 09-1168-2-455.

Bin Li is with the Department of Electrical and Computer Engineeringand the Coordinated Science Laboratory, University of Illinois at Urbana-Champaign, Urbana, IL 61801 USA (e-mail: lib@illinois.edu).

Ruogu Li is with ASSIA, Inc., Redwood City, CA 94065 USA (e-mail:lirg03@gmail.com).

Atilla Eryilmaz is with the Department of Electrical and ComputerEngineering, The Ohio State University, Columbus, OH 43210 USA (e-mail:eryilmaz.2@osu.edu).

throughput based requirements, these QoS requirements oftenhave a complex dependence on the higher-order statistics ofthe arrival process as well as the system operation. Thus, thecanonical optimization-based approaches that aim to optimizethe throughput performance (e.g., [24], [4], [15], [20], [16]) donot apply.

Recently, valuable efforts have been exerted in the design ofalgorithms that improve various aspects of the QoS, especiallyon the delay performance of the algorithms. For example,some works focus on designing algorithms with low end-to-end delay performance, such as [1], [28], [26]. Constant delaybounds (e.g. [18]) and delivery ratio requirements for deadline-constrained traffic (e.g. [7], [8], [9], [10], [12]) are some of theother QoS metrics considered in the literature.

However, these QoS metrics do not fully characterize theQuality-of-Experience (QoE) of users in video applications inwireless networks. To see it, we can envision the networkscenario where each individual user wants to download itsvideo from the base station, as shown in Fig. 1. Each mobile

Video 1

Video 2

Video L

Mobile user 1

Mobile user 2

Mobile user L

Playback buffer 1

Playback buffer 2

Playback buffer L

Base station

Fig. 1: Cellular network with a single base station and L users

user would like to receive the data from the base stationregularly. Indeed, the QoE of users is highly related to theaverage Perceived Video Quality (PVQ) across the sequenceof scenes forming the video, where the PVQ traditionally is alocal quality measure associated with a particular scene or ashort period of time. In [27], [11], the authors point out thatthe variance in PVQ leads to the worse QoE than the constantquality video with even smaller average PVQ. Yet, both thetime-varying nature of wireless channels and the schedulingpolicy significantly affect the variance of the received dataof each mobile user. Traditional scheduling policies aiming tomaximize the system throughput or minimize the delay at the

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base station side do not take users’ experience into account andthus lead to the high variance of the received data of users.

This motivates us to reduce variability of arrivals to themobile users, which can be achieved by providing regulatedinter-service times for the arriving flows at the base stationend. However, the inter-service time characteristics are difficultto analyze directly due to: its complex dependence on the high-order statistics of the arrival and service processes, and itsnon-Markovian evolution. To overcome this, we need to findnew approaches to study the inter-service time behavior. To thebest of our knowledge, this is the first work that rigorouslystudies the service regularity of the scheduling policies. Ourcontributions in this work can be summarized as follows:• We introduce a new quantity (cf. Section II), namely thetime-since-last-service, that has a tight relationship with theservice regularity performance, and hence enables novel designstrategies. Yet, this new parameter has its unique evolution,drastically different from a queue, which poses new challengesfor its analysis.• We develop a novel maximum-weight type scheduling policythat combines the time-since-last-service parameter and thequeue-length in its weight measure (cf. Section III). Then,we show that the proposed scheduling policy possesses thedesirable throughput optimality property by using a novelLyapunov function.• We derive lower and upper bounds on the service regularityperformance (cf. Section IV) by utilizing a novel Lyapunov-drift-based argument, inspired by the approach in [3]. Wefurther show that, by properly scaling the design parameterin our policy, we can guarantee a degree of service regularitywithin a factor of our fundamental lower bound. This factor isa function of the system statistics and design parameters andcan be as low as two under symmetric arrival rates in somespecial networks.• We support our analytical results with extensive numericalinvestigations (cf. Section V), which show significant per-formance gains in the service regularity over the traditionalthroughput-optimal policies. Furthermore, the numerical inves-tigations indicate that the service regularity performance of ourpolicy actually approaches the lower bounds as the weight ofthe time-since-last-service increases in some special networks.

This work significantly extends our earlier work [14] inseveral key aspects: (1) we conduct novel analyses that extendboth throughput optimality and service regularity guaranteeresults to general multi-hop fading networks; (2) we showthe existence of all moments of the system state under ourproposed algorithm, which establishes the foundation to utilizethe Lyapunov-drift based analysis of the steady-state behaviorof stochastic processes; (3) we conduct simulations to compareour policy with traditional throughput-optimal scheduling algo-rithms in more general setups, including switch topologies andfading scenarios.

II. SYSTEM MODEL

We consider a wireless network with L links, where a linkrepresents a pair of a transmitter and a receiver that are within

the transmission range of each other. We assume that the systemoperates in slotted time with normalized slots t ∈ {1, 2, ...}.Due to the interference-limited nature of wireless transmissions,the success or failure of a transmission over a link depends onwhether an interfering link is also active in the same slot, whichis called the link-based conflict model. We call a set of linksthat can be active simultaneously as a feasible schedule anddenote it as S[t] = (Sl[t])

Ll=1, where Sl[t] = 1 if the link l

is scheduled in slot t and Sl[t] = 0, otherwise. We use S todenote the set of all feasible schedules.

We capture the channel fading over link l via a non-negative-integer-valued random variable Cl[t], with Cl[t] ≤ Cmax, ∀l, t,for some Cmax < ∞, which measures the maximum amountof service available in slot t, if the link l is scheduled. Weassume that C[t] = (Cl[t])

Ll=1, ∀t ≥ 0, are independently and

identically distributed (i.i.d.) over time. We assume that cmin ,minlE[Cl[t]] > 0. Let S(c) , {Sc : S ∈ S} denote the setof feasible rate vectors when the channel is in state c, whereab = (albl)

Ll=1 denotes the component-wise product of two

vectors a and b. Then, the capacity region is defined as

R ,∑

c

Pr{C[t] = c} · CH{S(c)}, (1)

where CH{A} denotes a convex hull of the set A, and thesummation is a Minkowski addition of sets.

We assume a per-link traffic model1, where Al[t] denotesthe number of packets arriving at link l in slot t that areindependently distributed over links, and i.i.d. over time withfinite mean λl > 0, and Al[t] ≤ Amax, ∀l, t, for someAmax < ∞. Accordingly, a queue is maintained for each linkl with Ql[t] denoting its queue length at the beginning of timeslot t. Then, the evolution of queue l is described as follows:

Ql[t+ 1] = (Ql[t] +Al[t]− Cl[t]Sl[t])+,∀l, (2)

where (x)+ = max{x, 0}. We say that the queue l is stronglystable if it satisfies

lim supT→∞

1

T

T∑

t=1

E[Ql[t]] <∞. (3)

We call system stable if all queues are strongly stable. In thispaper, we consider the policies under which the system evolvesas a Markov Chain. We call an algorithm throughput-optimalif it makes all queues strongly stable for any arrival rate vectorλ = (λl)

Ll=1 that lies strictly within the capacity region.

In this work, we are interested in providing regular servicefor each link, which relates to the statistics of the inter-service time. We use Il[m] to denote the time between the(m − 1)th and the mth service for link l. If the system isstable, the steady-state distribution of the underlying MarkovChain exists (see [19]) and thus we use Q = (Ql)

Ll=1,

S = (Sl)Ll=1 and I = (I l)

Ll=1 to denote the random vector

with the same steady-state distribution of the queue-length,service processes and inter-service time, respectively. We use

1We note that our algorithm can be extended to serve multi-hop traffic, butthe notion of service regularity is clearer in the per-link context.

3

the normalized second moment of the inter-service time underthe steady-state distribution, i.e., E[I

2

l ]/(E[I l])2, as a measure

of the “regularity” of the service that link l receives. Notingthat E[I

2

l ]/(E[I l])2 = Var(I l)/(E[I l])

2 + 1, the normalizedsecond moment of the inter-service time reflects its normalizedvariance. Hence, the smaller the normalized second moment ofthe inter-service time, the smaller its normalized variance andthus the received service is more regular.

We would like to develop throughput-optimal policiesthat achieve low values of a linear increasing function of(E[I

2

l ]/(E[I l])2)Ll=1 in steady-state, implying more regular ser-

vice. However, unlike queue-lengths with Markovian evolution,the dynamics of inter-service times do not lend themselves tocommonly used Markovian analysis methods. To overcome thisobstacle, we introduce the following related quantity, namelythe time-since-last-service, which has much more tractable formof evolution, and whose mean has a close relationship tothe normalized second moment of the inter-service time (cf.Lemma 1).

For each link l, we introduce a counter Tl, namely Time-Since-Last-Service (TSLS), to keep track of the time since itwas lastly served, i.e., it was scheduled and the channel wasavailable. Let

τl[t] , maxτ={1,...,t−1}

{Sl[τ ]Cl[τ ] > 0, Sl[τ + 1]Cl[τ + 1] =· · · = Sl[t− 1]Cl[t− 1] = 0

},

be the last time when link l was served before time slot t, thenTl[t] = t − τl[t] − 1. By definition, each counter Tl increasesby 1 in each time slot when link l has zero transmission rate,either because it is not scheduled, or because its channel isunavailable, i.e., Cl[t] = 0, and drops to 0, otherwise. Moreprecisely, the evolution of the counter Tl can be written as

Tl[t+ 1] =

{0 if Sl[t]Cl[t] > 0;Tl[t] + 1 if Sl[t]Cl[t] = 0.

(4)

It can be seen from (4) that the evolution of Tl[t] differssignificantly from that of a traditional queue (also see Fig.2). In particular, unlike the slowly evolving nature of queue-lengths, the Tl[t] is incremented until link l receives service atwhich time it drops to zero. In our design, we will considerpolicies that not only use queue-lengths to achieve throughput-optimality, but also include TSLS to improve service regularity.

The evolution of Tl is tightly related to the inter-service timeIl, where Il is the time between two consecutive instances whenTl hits zero, as shown in Fig. 2. In fact, we have the followinglemma relating the two in steady-state.

Fig. 2: A sample trajectory of Il[m] and Tl[t], where the curveshows the evolution of Tl[t].

Lemma 1: For any policy under which the steady-state dis-tribution of the underlying Markov Chain exists, we have

E[T l] =1

2

(1

E[I l]E[I

2

l ]− 1

), (5)

where T l and I l denote the steady-state TSLS and inter-servicetime at link l, respectively.

Proof: The detailed proof is provided in Appendix A.Lemma 1 reveals the connection between the second moment

of the inter-service time I l and the mean of TSLS T l insteady-state. This can be intuitively seen in Fig. 2, where thearea of each “triangle” under the trajectory of Tl[t] is roughly12I

2l . In this work, we are interested in designing throughput-

optimal algorithms that reduce the total weighted-sum2 ofthe normalized second moment of the inter-service time, i.e.,∑Ll=1 βlρlE[I

2

l ]/(E[I l]

)2, where µl , 1/E[I l], ρl , λl/µl,

and βl ≥ 0 is some parameter related to the link. We can setβl > 0 if link l prefers regular service and βl = 0 otherwise.

According to Lemma 1, we haveL∑

l=1

βlρlE[I

2

l ](E[I l]

)2 = 2

L∑

l=1

βlλlE[T l] +

L∑

l=1

βlλl. (6)

Since∑Ll=1 βlλl only depends on the system parameters,

we will use∑Ll=1 βlλlE[T l] as our measure for the service

regularity. In this work, we aim to design a scheduling policythat is not only throughput-optimal, but also yields provablegood characteristics in the service regularity.

We achieve this dual objective by developing a parametricclass of throughput-optimal schedulers (cf. Section III-B) thatutilize a combination of queue-lengths and TSLS in its deci-sions. Our policy is shown to guarantee a ratio (as a functionof the system statistics) in its service regularity with respect toa fundamental lower bound (cf. Section IV-A).

III. ALGORITHM DESIGN FOR REGULAR SERVICE

In this section, we first discuss the inefficiency of thewell-known throughput-optimal Maximum Weight Scheduling(MWS) Algorithm in terms of service regularity. We then pro-pose Regular Service Guarantee policy which can be shown thatnot only achieves the throughput optimality but also possessesgood service regularity performance.

A. Inefficiency of the MWS Algorithm

In this subsection, we describe a well-known schedulingpolicy, namely the Maximum Weight Scheduling (MWS) Algo-rithm and discuss its inefficiency in terms of service regularityperformance. We first give the definition of the MWS Algorithmfor completeness.

2The weighting parameter ρl is the arrival intensity at link l, and indicatesthat the link with higher load prefers more regular service. Despite this, ρl isincluded in the objective function primarily for technical reasons. Noting thatthe link preference parameter βl can be any non-negative real number, andthus this weighted form is still general enough.

4

Algorithm 1 (Maximum Weight Scheduling (MWS) Algorithm):Under our model, the MWS Algorithm selects a scheduleS(MWS)[t] with the largest total sum of the product of queue-length and the maximum channel available rate within thatschedule, i.e., it chooses a schedule S(MWS)[t] such that

S(MWS)[t] ∈ arg maxS∈S

L∑

l=1

Ql[t]Cl[t]Sl[t]. (7)

The MWS Algorithm is known to be throughput-optimal(e.g., [24], [16], [21], [2]), i.e., it stabilizes the network forany arrival rate vector λ that strictly lies within the capacityregion R. In our setup, the MWS Algorithm can be expectedto have close-to-lower-bound average delay performance (see[5]). It has also been shown to be heavy-traffic optimal (see[22], [3]), i.e., it minimizes the mean steady-state queue-lengthunder heavy-traffic conditions, where the arrival rate vectorapproaches the boundary of the capacity region from below.

However, despite its throughput optimality and a numberof favorable properties on the delay performance, the MWSAlgorithm may result in poor performance in terms of serviceregularity. This can be observed in Fig. 3 when the MWSAlgorithm serves a set of links with heterogeneous arrivalstatistics in a non-fading fully-connected network with 8 links,where each link can transfer one packet in one time slot ifscheduled and at most one link can be scheduled in each slot.

1 2 3 4 5 6 7 810

0

101

102

103

104

105

106

Link #

Var[Il]

Asymmetric Arrival Rates

Symmetric Arrival Rates

Fig. 3: The variance of the inter-service time under the MWSAlgorithm for links with different arrival processes in fully-connected networks with 8 links. The links with smaller ratesor more bursty arrivals suffer from high variance of the inter-service time.

In Fig. 3, the blue line shows a scenario where the lth linkhas a Bernoulli arrival with rate 2−l for l ∈ {1, . . . , 8}. Inthis case, we observe that the variance of the inter-service timeincreases exponentially as the arrival rate of the link reduces.The red curve illustrates a different scenario where all 8 linkshave the same mean arrival rate, but increasing variances (i.e.,burstiness) in their arrivals, where we observe that the link withmore bursty arrivals suffers from higher variance in its inter-service time.

B. The Regular Service Guarantee Policy

As discussed above, the MWS Algorithm is throughput-optimal but inefficient in providing regular services. Note thatthe introduced TSLS counter has a direct impact on serviceregularity: the smaller the mean TSLS value, the more regularthe service. This interesting connection motivates the followingparametrized policy which is later revealed to possess thecharacteristics of throughput optimality and service regularity.

Algorithm 2 (Regular Service Guarantee (RSG) Algorithm):In each time slot t, select a schedule S∗[t] such that

S∗[t] ∈ arg maxS∈S

L∑

l=1

(αlQl[t] + γβlTl[t])Cl[t]Sl, (8)

where αl > 0 and γ ≥ 0 are fixed control parameters.

We note that there are two sets of control parameters inthe RSG Algorithm3 and they affect different behaviors ofthe algorithm. Yet, it will be revealed later that none ofthem affects its throughput optimality. The parameters αl areweighing factors for the queue-lengths, where a larger αl willresult in a smaller average queue-length. The parameter γis a common weighing factor of TSLS for all links. It willbe revealed in Section IV-B that the design parameter γ canimprove the service regularity as it increases. Also note thatwhen γ = 0, our policy coincides with the MWS Algorithm.When γ > 0, with the addition of Tl[t] terms in the weightof each link, our algorithm operates completely different fromthe MWS and its approximate algorithms, which, to the bestof our knowledge, are the only known policies possessingthe throughput-optimality characteristic in general multi-hopnetwork topologies. Despite of this, we can still show that ouralgorithm is throughput-optimal.

Proposition 1: The RSG Algorithm with any αl > 0 andγ ≥ 0, is throughput-optimal, i.e., for any arrival rate vectorλ ∈ Int(R), the RSG Algorithm stabilizes the system, with

lim supK→∞

1

K

K−1∑

t=0

L∑

l=1

αlE[Ql[t]] ≤B(α,β, γ)

2ε, (9)

where Int(A) denotes the interior points of the region A,B(α,β, γ) , 4γCmax

∑Ll=1 βl +

∑Ll=1 αlE

[A2l [t] + C2

l [t]],

ε is some positive constant satisfying λ + ε1 ∈ R, and 1 is avector of ones.

Proof: Consider the Lyapunov function

W (Q[t],T[t]) ,L∑

l=1

αlQ2l [t] + 4γCmax

L∑

l=1

βlTl[t]. (10)

3The RSG Algorithm inherits the same complexity issue as the well-knownMWS Algorithm. The low complexity or the distributed implementations ofthe RSG Algorithm are always attractive in practical networks and are left forfuture research.

5

It is shown in Appendix B that there exists a positive constantε > 0 such that

∆W ,E [W (Q[t+ 1],T[t+ 1])−W (Q[t],T[t])|Q[t],T[t]]

≤− 2ε

L∑

l=1

αlQl[t] +B(α,β, γ). (11)

Taking the expectation on the both sides of (11) and summingover t = 0, 1, ...,K − 1, we have the desired result.

Proposition 1 establishes the throughput optimality of theRSG Algorithm, thus Ql[t] and Tl[t] will converge in distribu-tion to Q

∗l and T

∗l , which attain the steady-state distribution

under our policy. Proposition 1 also gives an upper bound forthe expected total queue-length under the steady-state, whichincreases linearly with the design parameter γ. It will berevealed later that γ controls the tradeoff between the averagetotal queue-length, and the service regularity performance,especially in the heterogenous networks.

Next, we will show that all moments of steady-state systemvariables, such as queue-lengths and TSLS, are bounded underthe RSG Algorithm, which enables us to analyze the serviceregularity performance by using the Lyapunov-type approachdeveloped in [3]. In [6], the sufficient condition for all momentsof state variables of a Markov Chain to exist in steady state isgiven as finding a Lyapunov function that satisfies: (1) it hasa negative Lyapunov drift when the system variable is largeenough; (2) the absolute value of the Lyapunov drift is boundedor has an exponential tail. From equation (11), we know that thefirst condition holds. Yet, the second condition is hard to holdfor the Lyapunov function W (Q,T) or its variant

√W (Q,T)

due to the unique evolution of TSLS counters (see equation (4)),which have bounded increment but unbounded decrement. Wetackle this challenge by properly partitioning the system space.

Proposition 2: For any arrival rate vector λ ∈ Int(R), allmoments of steady-state queue length and TSLS exist underthe RSG Algorithm with any αl > 0 and γ ≥ 0.

Proof: We show the boundedness of E[eη‖Y[t]‖2

]for

some η > 0 by intelligently partitioning the system space,where Y[t] ,

(√αQ[t],

√4γCmaxβT[t]

),√x denotes the

component-wise square root of the vector x, and xy denotesthe component-wise product of the vectors x and y. Please seeour technical report [13] for details.

Having established the throughput optimality and the mo-ment existence of the system state variables of the RSGAlgorithm, we are ready to analyze the service regularityperformance, i.e.,

∑Ll=1 βlλlE[T l].

IV. SERVICE REGULARITY PERFORMANCE ANALYSIS

In this section, we study the service regularity performanceof our proposed RSG Algorithm analytically. We first establisha fundamental lower bound on the service regularity for anyfeasible scheduling algorithm. Then, we derive an upper boundon the service regularity under the RSG Algorithm. Theseinvestigations reveal that the service regularity performance ofthe RSG Algorithm can be guaranteed to remain within a factor

of the lower bound, which is expressed as a function of thesystem statistics and the design parameters, and can be as lowas 2 in some special networks. We assume the parameter γ > 0throughout this section.

A. Lower Bound Analysis

In this subsection, we derive a lower bound based on aLyapunov drift argument inspired by the technique used in[3]. To study the lower bound of the service regularity bythe Lyapunov drift argument, we consider a class of policies,called P , that not only stabilize the system but also yield thebounded second moment of the steady-state TSLS4. Note thatour proposed algorithm, as well as the MWS algorithm, fallsinto this class by Propositions 1 and 2.

Let T(p)

l and S(p)

l be the steady-state TSLS and schedulingvariable for link l under policy p ∈ P , respectively. Thefollowing lemma gives key identities for the first and secondmoment of the steady-state TSLS, which are useful in derivinga lower bound on the service regularity.

Lemma 2: For any policy p ∈ P , we have

E

∑

l∈H(p)

βlλlT(p)

l

=

L∑

l=1

βlλl − E

∑

l∈H(p)

βlλl

, (12)

2

L∑

l=1

βlλlE[T

(p)

l

]=

L∑

l=1

βlλl − E

∑

l∈H(p)

βlλl

+ E

∑

l∈H(p)

βlλl

(T

(p)

l

)2 , (13)

where H(p)

, {l : ClS(p)

l > 0}, and C = (Cl)Ll=1 has the

same probability distribution as C[t] = (Cl[t])Ll=1.

Proof: See Appendix C for the proof.We are ready to give a lower bound on the service regularity

for any feasible policy p ∈ P .Proposition 3: For any policy p ∈ P , we haveL∑

l=1

βlλlE[T

(p)

l

]≥ 1

2

( ∑Ll=1 βlλl

maxS∈S∑l∈S βlλl

− 1

)L∑

l=1

βlλl.

Proof: In the rest of proof, we will omit superscript pfor conciseness. For any sample path, by Cauchy-Schwarzinequality, we have

∑

l∈H

βlλlT l

2

=

∑

l∈H

√βlλl ·

√βlλlT l

2

≤

∑

l∈H

βlλl

∑

l∈H

βlλlT2

l , (14)

4We conjecture that the second moment of the steady-state TSLS is boundedas long as the system is stable.

6

where we recall that H , {l : ClSl > 0}. This implies

∑

l∈H

βlλlT2

l ≥(∑

l∈H βlλlT l)2

∑l∈H βlλl

. (15)

Hence, we have

E

∑

l∈H

βlλlT2

l

≥E

[(∑l∈H βlλlT l

)2∑l∈H βlλl

]

(a)

≥(E[∑

l∈H βlλlT l])2

E[∑

l∈H βlλl]

(b)=

(∑Ll=1 βlλl − E

[∑l∈H βlλl

])2

E[∑

l∈H βlλl] , (16)

where the step (a) uses the fact that f(x, y) = x2

y is convexand Jensen’s inequality for a multi-variable function; step (b)follows from (12). By substituting (16) into (13), we have

L∑

l=1

βlλlE[T l]≥ 1

2

( ∑Ll=1 βlλl

E[∑

l∈H βlλl] − 1

)L∑

l=1

βlλl. (17)

Note that

E

∑

l∈H

βlλl

= E

[L∑

l=1

βlλl1{ClSl>0}

]

=

L∑

l=1

βlλl Pr{ClSl > 0}

≤L∑

l=1

βlλl Pr{Sl = 1} ≤ maxS∈S

∑

l∈S

βlλl. (18)

By substituting (18) into (17), we have the desired result.

Consider a fully-connected non-fading network, where onlyone link is scheduled in each time slot. Let βl = β and λl = λfor each link l. Then, the lower bound becomes

L∑

l=1

E[T

(p)

l

]≥ 1

2L(L− 1). (19)

This lower bound can be achieved by the Round-Robin(RR) policy, which serves each link periodically. Indeed, inthe steady-state, the TSLS vector under the RR policy is apermutation of {0, 1, 2, ..., L− 1} and thus

∑Ll=1E

[T

(RR)l

]=

12L(L− 1).

Yet, we would like to point out that the RR policy is notthroughput-optimal. Thus, for an arrival rate vector λ thatcannot be supported by the RR policy, we do not expect athroughput-optimal policy to approach the above lower boundwhen serving it. However, for the arrival rate vectors that canbe supported by the RR policy, we shall see in our numericalresults that the performance of our policy can approach thislower bound when we increase the scaling parameter γ.

B. Upper Bound Analysis

In this subsection, we obtain an upper bound on the serviceregularity under the RSG Algorithm. Let Q

∗l , S

∗l and T

∗l be

the steady-state queue-length, scheduling variable and TSLSfor link l under the RSG Algorithm, respectively.

Proposition 4: For the RSG Algorithm, we have

L∑

l=1

βlλlE[T∗l

]≤Cmax

1 + ε

L∑

l=1

βl − E

∑

l∈H∗

βl

+1

2γ(1 + ε)

L∑

l=1

αlE[A

2

l + C2

l

], (20)

where ε > 0 satisfies λ(1+ε) ∈ R, H∗, {l : ClS

∗l > 0}, and

A = (Al)Ll=1 has the same distribution as A[t] = (Al[t])

Ll=1.

Proof: See Appendix D for the details.Note that the second term of the right hand side of (20)

captures various random effects in the network: the burstinessof the arrival processes and the channel variations. Under ourpolicy these effects diminish as the scaling factor γ goesto infinity. Hence, together with Proposition 1, Proposition 4reveals a tradeoff: when increasing γ, the upper bound onthe total queue-length increases linearly with γ, but the upperbound for the service regularity decreases.

Consider the fully-connected non-fading network as in Sec-tion IV-A. Let βl = β and λl = λ = 1

L(1+ε) for each link l.Then, as γ goes to infinity, (20) becomes

L∑

l=1

E[T∗l

]≤ L(L− 1), (21)

which is always within twice the value of the lower bound(19). In the more general case, the upper bound convergesto a constant that is determined by the system statistics anddesign parameters as γ goes to infinity. Moreover, we shallsee in the numerical results presented in Section V-B thatas γ increases, the service regularity performance under theRSG Algorithm actually converges to the lower bound (19)in the fully-connected non-fading network with the symmetricparameters.

V. NUMERICAL RESULTS

In this section, we provide simulation results for our pro-posed RSG Algorithm and compare its performance to theMWS Algorithm and the Oldest Cell First (OCF) Algorithm5

(see [17]). In particular, we investigate the throughput (cf. Sec-tion V-A) and service regularity (cf. Section V-B) performancesof our policy in both fully-connected network with L = 4links and 3 × 3 switch. We also look at the behavior of theRSG Algorithm as well as the potential benefit of the serviceregularity in our technical report [13]. In all of our simulations,we assume Bernoulli arrivals to each link and αl = βl = 1 foreach link l.

5The OCF Algorithm prioritizes activation of links with the greatest Head-of-Line (HOL) delay.

7

0.248 0.2485 0.249 0.2495 0.250

200

400

600

800

1000

1200

1400

1600

1800

2000

Arrival Rate

∑L l=

1E[Q

l]

MWS AlgorithmRSG Algorithm with γ=10RSG Algorithm with γ=100

(a) Fully-connected non-fading network

0.248 0.2485 0.249 0.24950

100

200

300

400

500

600

700

800

900

1000

Arrival Rate

∑L l=

1E[Q

l]

MWS AlgorithmRSG Algorithm with γ=10RSG Algorithm with γ=100

(b) Fully-connected fading network

0.331 0.3315 0.332 0.3325 0.333 0.33350

500

1000

1500

2000

2500

3000

3500

4000

4500

5000

Arrival Rate

∑L l=

1E[Q

l]

MWS AlgorithmRSG Algorithm with γ=10RSG Algorithm with γ=100

(c) 3× 3 switch

Fig. 4: The throughput performance of the RSG Algorithm

101

102

103

0.5

1

1.5

2

2.5

3

Total Queue Length

∑L l=

1λlE

[Tl]

Symmetric setup

Asymmetric setup

Lower Bound in symmetric setup

Lower bound in asymmetric setup

MWS Algorithm (γ=0)

OCF Algorithm

γ=2−7

γ=128

(a) Fully-connected non-fading network

101

102

103

0.5

1

1.5

2

2.5

3

Total Queue Length

∑L l=

1λlE

[Tl]

Symmetric setup

Asymmetric setup

Lower Bound in symmetric setup

Lower bound in asymmetric setup

γ=128

γ=2−7

OCF Algorithm

MWS Algorithm (γ=0)

(b) Fully-connected fading network

102

103

1

1.5

2

2.5

3

3.5

4

Total Queue Length

∑L l=

1λlE

[Tl]

Symmetric setup

Asymmetric setup

Lower Bound in symmetric setup

Lower bound in asymmetric setup

γ=128

OCF Algorithm

MWS Algorithm (γ=0)

γ=2−7

(c) 3× 3 switch

Fig. 5: Trade-off between mean queue length and the service regularity

A. Throughput Performance

In this subsection, we illustrate the throughput performanceof the RSG Algorithm in three different network setups withsymmetric arrivals: (i) fully-connected non-fading network,(ii) fully-connected network with symmetric ON-OFF fadingchannels with probability q = 0.8 that the channel is available,and (iii) 3 × 3 switch. The achievable rate regions for thesethree networks, respectively, are

Λ1 ,

{λ = (λl)

4l=1 : λ1 = λ2 = ... = λ4 <

1

4

},

Λ2 ,

{λ = (λl)

4l=1 : λ1 = λ2 = ... = λ4 <

1− (1− q)4

4

},

Λ3 ,

{λ = (λl)

9l=1 : λ1 = λ2 = ... = λ9 <

1

3

}.

In Fig. 4, we compare the total mean queue-length under theMWS Algorithm, as well as the RSG Algorithm with differentγ values. It can be observed in Fig. 4 that the RSG Algorithmcan stabilize the system in the above network setups. It also canbe seen that the total mean queue-length of the RSG Algorithmincreases with the parameter γ. This is expected since as γincreases, it becomes more likely for the RSG Algorithm tochoose a queue with less packet to serve, potentially wastingsome service while improving the service regularity, as we shallsee next.

B. Service Regularity Performance

In this subsection, we investigate the service regularityperformance of our RSG Algorithm, as well as illustrate thetradeoff between the total mean queue-length and the serviceregularity. We present our results in three different networks:fully-connected non-fading network, fully-connected fadingnetwork and 3 × 3 switch. In both fully-connected nonfad-ing and fading networks, we consider the symmetric setupwith the arrival rate vector λ , [0.225, 0.225, 0.225, 0.225],and the asymmetric setup with the arrival rate vector λ ,[0.4, 0.3, 0.15, 0.05]. For a fully-connected ON-OFF fadingnetwork, the probability vectors that the channels are avail-able are q = [0.8, 0.8, 0.8, 0.8] in symmetric setup andq = [0.6, 0.5, 0.4, 0.3] in asymmetric setup. For a 3 × 3switch, we consider the symmetric setup with the arrivalrate vector λ , [0.3, 0.3, 0.3; 0.3, 0.3, 0.3; 0.3, 0.3, 0.3] andthe asymmetric setup with the arrival rate vector λ ,[0.5, 0.3, 0.1; 0.2, 0.4, 0.3; 0.1, 0.2, 0.5]. In all simulations, wechoose the scaling parameter γ to be the powers of 2, rangingfrom 2−7 to 27.

Fig. 5 shows the relationship between the total mean queue-length and the service regularity in different network setups.The tradeoff between the service regularity and the total meanqueue-length can be clearly seen: as γ increases, the serviceregularity improves while the total mean queue-length alsoincreases. Also, we can observe that the OCF Algorithm does

8

not improve the service regularity performance significantlycompared to the MWS Algorithm. The reason lies in thatboth MWS and OCF Algorithms perform similarly amongthe links that have continuous arrivals, which can be seenfrom the Little’s Law. In addition, it has been shown that theMWS Algorithm is delay-optimal in fully-connected networkswith symmetric Bernoulli arrivals and ON-OFF fading channels(see [25]). From Fig. 5(b), we can see that the delay-optimalalgorithm may fail to meet the certain service regularity re-quirement. Furthermore, it can be observed that the simulatedvalues converge to the fundamental lower bound in non-fadingnetworks with symmetric setup (Figs 5(a) and (c)), while theystay away from the lower bound in asymmetric setups. Thismotivates us to refine the lower bound analysis in asymmetricsetups, which is left for future investigation. Here, it is worthmentioning that even with very small γ values (e.g., 2−5), ourRSG Algorithm significantly improves the service regularity,while introducing negligible increase in the total mean queue-length.

VI. CONCLUSION

In this work, we investigated the problem of designing ascheduling policy that is both throughput-optimal and possessesfavorable service regularity characteristics. We introduced anew parameter of time-since-last-service, and proposed a novelscheduling policy that combines this parameter with the queue-lengths in its weight. After establishing the throughput optimal-ity of our policy, we showed that it also has provable serviceregularity performance. In particular, the service regularity ofour policy can be guaranteed to remain within a factor distanceof a fundamental lower bound for any feasible schedulingpolicy. We explicitly expressed this factor as a function ofthe system statistics and the design parameters. We performedextensive numerical studies to illustrate the significant gainsachieved by our policy over the traditional queue-length orhead-of-line-delay based policies. Our results show the signif-icance of utilizing the time-since-last-service in improving theservice regularity performance of throughput-optimal policies.

APPENDIX APROOF OF LEMMA 1

Without loss of generality, assume that link l is served attime 0. For any positive integer M , there exists an m such that

Il[1] + · · ·+ Il[m] ≤ M,

Il[1] + · · ·+ Il[m] + Il[m+ 1] > M.

We can write

1

M

M∑

t=1

Tl[t] =1

M

(Il[1]∑

t=1

Tl[t] +

Il[1]+Il[2]∑

t=Il[1]+1

Tl[t] + . . .

+

M∑

t=Il[1]+···+Il[m]+1

Tl[t]

). (22)

We observe the following fact: assume link l receives its(m− 1)th and mth service at time slot t1 and t2, respectively,where t2 > t1. Then, by definition, Il[m] = t2−t1, Tl[t1+1] =0 and Tl[t2] = t2 − t1 − 1 = Il[m] − 1. Using this fact, weknow the kth summation on the right hand side of (22) gives12 (Il[k](Il[k]− 1)), except for the last one. Thus we have:

1

M

M∑

t=1

Tl[t] ≥1

M

m∑

k=1

Il[k](Il[k]− 1)

2, (23)

1

M

M∑

t=1

Tl[t] ≤1

M

m+1∑

k=1

Il[k](Il[k]− 1)

2. (24)

By the definition of m and the fact that E[I l] < ∞ wheneach link is served with strictly positive probability, we knowthat m → ∞ when M → ∞. Since the policy considered inthis paper is Markovian, we have

limM→∞

1

M

M∑

t=1

Tl[t] = E[T l],

limm→∞

1

m

m∑

k=1

Il[k] = E[I l],

limm→∞

1

m

m∑

k=1

I2l [k] = E[I2

l ].

Note that 1m

∑mk=1 Il[k] ≤ M

m ≤m+1m

1m+1

∑m+1k=1 Il[k], which

implies limM→∞Mm = E[I l].

By taking limit on (23) and (24) as M →∞, we have

limM→∞

m

M

1

m

m∑

k=1

Il[k](Il[k]− 1)

2≤ limM→∞

1

M

M∑

t=1

Tl[t]

≤ limM→∞

m+ 1

M

1

m+ 1

m+1∑

k=1

Il[k](Il[k]− 1)

2,

and thus we have the desired result.

APPENDIX BPROOF OF INEQUALITY (11)

∆W ,E [W (Q[t+ 1],T[t+ 1])−W (Q[t],T[t])|Q[t],T[t]]

=E

[L∑

l=1

αlQ2l [t+ 1] + 4γCmax

L∑

l=1

βlTl[t+ 1]

−L∑

l=1

αlQ2l [t]− 4γCmax

L∑

l=1

βlTl[t]

∣∣∣∣∣Q[t],T[t]

]

≤L∑

l=1

αlE[(Ql[t] +Al[t]− Cl[t]S∗l [t])2 −Q2

l [t]∣∣Q[t],T[t]

]

+ 4γCmaxE

[L∑

l=1

βlTl[t+ 1]−L∑

l=1

βlTl[t]

∣∣∣∣∣Q[t],T[t]

],

(25)

9

where the last step follows from the evolution of each queue,and (max{x, 0})2 ≤ x2.

Let H∗ , {l : S∗l [t]Cl[t] > 0}. According to the definitionof the TSLS counter, we have

L∑

l=1

βlTl[t+ 1]

=∑

l/∈H∗

βl (Tl[t] + 1)

=

L∑

l=1

βlTl[t]−∑

l∈H∗

βlTl[t] +

L∑

l=1

βl −∑

l∈H∗

βl (26)

≤L∑

l=1

βlTl[t]−∑

l∈H∗

βlTl[t] +

L∑

l=1

βl. (27)

By substituting inequality (27) into (25), we have

∆W ≤L∑

l=1

αlE[(Ql[t] +Al[t]− Cl[t]S∗l [t])2 −Q2

l [t]∣∣Q[t],T[t]

]

+ 4γCmaxE

[L∑

l=1

βl −∑

l∈H∗

βlTl[t]

∣∣∣∣∣Q[t],T[t]

]

≤L∑

l=1

αlE [2Ql[t](Al[t]− Cl[t]S∗l [t])|Q[t],T[t]]

+

L∑

l=1

αlE[(Al[t]− Cl[t]S∗l [t])2

∣∣Q[t],T[t]]

+ 4γCmax

L∑

l=1

βl − 4γCmaxE

[∑

l∈H∗

βlTl[t]

∣∣∣∣∣Q[t],T[t]

]

≤2

L∑

l=1

αlλlQl[t]− 2E

[L∑

l=1

αlQl[t]Cl[t]S∗l [t]

∣∣∣∣∣Q[t],T[t]

]

− 4γCmaxE

[∑

l∈H∗

βlTl[t]

∣∣∣∣∣Q[t],T[t]

]+B(α,β, γ), (28)

where B(α,β, γ) is defined in Proposition 1.

Let S(MWS)[t] ∈ arg maxS∈S

L∑

l=1

αlQl[t]Cl[t]Sl. Then, by the

definition of the RSG Algorithm, we haveL∑

l=1

(αlQl[t] + γβlTl[t])Cl[t]S∗l [t]

≥L∑

l=1

(αlQl[t] + γβlTl[t])Cl[t]S(MWS)l [t]

≥L∑

l=1

αlQl[t]Cl[t]S(MWS)l [t],

which impliesL∑

l=1

αlQl[t]Cl[t]S∗l [t]

≥L∑

l=1

αlQl[t]Cl[t]S(MWS)l [t]− γ

L∑

l=1

βlTl[t]Cl[t]S∗l [t]. (29)

By substituting (29) into (28), we have

∆W ≤ 2

L∑

l=1

αlλlQl[t]− 2E

[L∑

l=1

αlQl[t]Cl[t]S(MWS)l [t]

∣∣∣∣∣Q,T]

+ 2γE

[L∑

l=1

βlTl[t]Cl[t]S∗l [t]

∣∣∣∣∣Q[t],T[t]

]

− 4γCmaxE

[∑

l∈H∗

βlTl[t]

∣∣∣∣∣Q[t],T[t]

]+B(α,β, γ). (30)

Given Q[t] and T[t], we have

CmaxE

[∑

l∈H∗

βlTl[t]

∣∣∣∣∣Q,T]≥ E

[L∑

l=1

βlTl[t]Cl[t]S∗l [t]

∣∣∣∣∣Q,T],

(31)

where we recall that H∗ = {l : S∗l [t]Cl[t] > 0}. By substituting(31) into (30), we have

∆W ≤ 2

L∑

l=1

αlλlQl[t]− 2E

[L∑

l=1

αlQl[t]Cl[t]S(MWS)l [t]

∣∣∣∣∣Q,T]

− 2γE

[L∑

l=1

βlTl[t]Cl[t]S∗l [t]

∣∣∣∣∣Q[t],T[t]

]+B(α,β, γ). (32)

Note that the capacity region R (see [23]) is also equivalent toa set of arrival rate vectors λ such that there exist non-negativenumbers θ(c; s) satisfying

λl ≤∑

c

Pr{C[t] = c}∑

s∈Sθ(c; s)clsl,∀l, (33)

where s = (sl)Ll=1 and

∑s∈S θ(c; s) = 1,∀c. For any λ ∈

Int(R), there exists an ε > 0 such that

λl ≤∑

c

Pr{C[t] = c}∑

s∈Sθ(c; s)clsl − ε,∀l. (34)

Hence, we haveL∑

l=1

αlλlQl[t] + ε

L∑

l=1

αlQl[t]

≤∑

c

Pr{C[t] = c}∑

s∈Sθ(c; s)

L∑

l=1

αlQl[t]clsl

(a)

≤∑

c

Pr{C[t] = c}∑

s∈Sθ(c; s)

L∑

l=1

αlQl[t]clS(MWS)l [t]

=E

[L∑

l=1

αlQl[t]Cl[t]S(MWS)l [t]

∣∣∣∣∣Q[t],T[t]

]. (35)

10

where the step (a) follows from the definition of S(MWS). Bysubstituting (35) into (32), we have

∆W ≤− 2ε

L∑

l=1

αlQl[t] +B(α,β, γ)

−2γE

[L∑

l=1

βlTl[t]Cl[t]S∗l [t]

∣∣∣∣∣Q[t],T[t]

](36)

≤− 2ε

L∑

l=1

αlQl[t] +B(α,β, γ). (37)

APPENDIX CPROOF OF LEMMA 2

In the rest of proof, we will omit the superscript p for brevity.

Proof of identity (12):L∑

l=1

βlλlTl[t+ 1] =∑

l/∈H

βlλl (Tl[t] + 1)

=

L∑

l=1

βlλlTl[t]−∑

l∈H

βlλlTl[t] +

L∑

l=1

βlλl −∑

l∈H

βlλl, (38)

where H , {l : Sl[t]Cl[t] > 0}. Taking expectation on bothsides with respect to the steady state distribution of (Q,T) andrearranging terms, we have the desired result.

Proof of identity (13):L∑

l=1

βlλlT2l [t+ 1] =

∑

l/∈H

βlλl (Tl[t] + 1)2

=∑

l/∈H

βlλlT2l [t] + 2

∑

l/∈H

βlλlTl[t] +∑

l/∈H

βlλl

=

L∑

l=1

βlλlT2l [t]−

∑

l∈H

βlλlT2l [t] + 2

L∑

l=1

βlλlTl[t]

− 2∑

l∈H

βlλlTl[t] +

L∑

l=1

βlλl −∑

l∈H

βlλl. (39)

Taking expectation on both sides with respect to the steady statedistribution of (Q,T) and rearranging terms, we have

2

L∑

l=1

βlλlE[T l]

=2E

∑

l∈H

βlλlT l

+ E

∑

l∈H

βlλlT2

l

−

L∑

l=1

βlλl − E

∑

l∈H

βlλl

. (40)

Using Identity (12), we have the desired result.

APPENDIX DPROOF OF PROPOSITION 4

Consider the quadratic Lyapunov function WQ(Q,T) ,12

∑Ll=1 αlQ

2l . We have

∆WQ(Q,T)

=E [WQ(Q[t+ 1],T[t+ 1])−WQ(Q[t],T[t])|Q[t],T[t]]

=E

[1

2

L∑

l=1

αlQ2l [t+ 1]− 1

2

L∑

l=1

αlQ2l [t]

∣∣∣∣∣Q[t],T[t]

]

≤1

2

L∑

l=1

E[αl (Ql[t] +Al[t]− Cl[t]S∗l [t])

2 − αlQ2l [t]∣∣∣Q[t],T[t]

]

≤L∑

l=1

αlE [Ql[t] (Al[t]− Cl[t]S∗l [t])|Q[t],T[t]]

+1

2

L∑

l=1

αlE[A2l [t] + C2

l [t]]. (41)

Taking expectation on both sides with respect to thesteady state distribution of (Q,T), and using the fact thatE[∆WQ(Q,T)] = 0 followed from E[Q

2

l ] <∞ for all l ∈ L,we have

0 ≤L∑

l=1

αlλlE[Q∗l

]−

L∑

l=1

αlE[Q∗lClS

∗l

]

+1

2

L∑

l=1

αlE[A

2

l + C2

l

], (42)

which impliesL∑

l=1

αlE[Q∗lClS

∗l

]≤

L∑

l=1

αlλlE[Q∗l

]+

1

2

L∑

l=1

αlE[A

2

l + C2

l

].

Hence, we haveL∑

l=1

E[(αlQ

∗l + γβlT

∗l

)ClS

∗l

]

≤L∑

l=1

αlλlE[Q∗l

]+ γ

L∑

l=1

βlE[T∗l S∗lCl

]+

1

2

L∑

l=1

αlE[A

2

l + C2

l

].

(43)

Recall that given Q[t] = Q, T[t] = T and the channel stateC[t], we have

L∑

l=1

(αlQl[t] + γβlTl[t])Cl[t]S∗l [t]

= maxS∈S

L∑

l=1

(αlQl[t] + γβlTl[t])Cl[t]Sl. (44)

11

According to the definition of the capacity region R, we canshow

L∑

l=1

E [(αlQl[t] + γβlTl[t])Cl[t]S∗l [t]|Q[t] = Q,T[t] = T]

= maxr∈R

L∑

l=1

(αlQl[t] + γβlTl[t]) rl, (45)

The proof is available in our technical report [13].Since λ ∈ Int(R), there exists an ε > 0 such that λ(1+ ε) ∈

R. Hence, we haveL∑

l=1

E [(αlQl[t] + γβlTl[t])Cl[t]S∗l [t]|Q[t] = Q,T[t] = T]

≥L∑

l=1

λl(1 + ε)E [αlQl[t] + γβlTl[t]|Q[t] = Q,T[t] = T] .

Taking expectation on both sides with respect to the steady statedistribution of (Q,T), we have

L∑

l=1

E[(αlQ

∗l + γβlTl

∗)ClS

∗l

]

≥L∑

l=1

λl(1 + ε)E[αlQ

∗l + γβlT

∗l

].

By substituting above inequality into (43) and canceling thecommon term in both sides, we have

L∑

l=1

βlλlE[T∗l

]

≤ 1

1 + ε

L∑

l=1

βlE[T∗l S∗lCl

]+

1

2γ(1 + ε)

L∑

l=1

αlE[A

2

l + C2

l

]

≤Cmax

1 + εE

∑

l∈H∗

βlT∗l

+

1

2γ(1 + ε)

L∑

l=1

αlE[A

2

l + C2

l

]

=Cmax

1 + ε

L∑

l=1

βl − E

∑

l∈H∗

βl

+1

2γ(1 + ε)

L∑

l=1

αlE[A

2

l + C2

l

],

where the last step uses identity (12).

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Bin Li (S ’12) received his B.S. degree in Electronicand Information Engineering in 2005, M.S. degreein Communication and Information Engineering in2008, both from Xiamen University, China, and Ph.D.degree in Electrical and Computer Engineering fromThe Ohio State University in May 2014. He is cur-rently a Postdoctoral Researcher in the CoordinatedScience Laboratory at the University of Illinois atUrbana-Champaign.

His research interests include wireless communi-cation and networks, cloud computing, smart grids,

resource allocation and management, distributed algorithm design, queueingtheory, and optimization theory. He received the Presidential Fellowship fromThe Ohio State University and Chinese Government Award for OutstandingPh.D. Students Abroad in 2013.

Ruogu Li (S ’10) received his B.S. degree in Elec-tronic Engineering from Tsinghua University, Beijing,in 2007. He received his PhD degree in Electricaland Computer Engineering from Ohio State Univer-sity. His research interests include optimal networkcontrol, wireless communication networks, low-delayscheduling scheme design and cross-layer algorithmdesign. His PhD dissertation discussed the principlesand methods of adaptive network algorithm designunder various quality-of-service requirements. He isnow a senior system engineer at ASSIA Inc. in

Redwood City, California.

Atilla Eryilmaz (S ’00-M ’06) received his B.S.degree in Electrical and Electronics Engineering fromBogazici University, Istanbul, in 1999, and the M.S.and Ph.D. degrees in Electrical and Computer En-gineering from the University of Illinois at Urbana-Champaign in 2001 and 2005, respectively. Between2005 and 2007, he worked as a Postdoctoral Associateat the Laboratory for Information and Decision Sys-tems at the Massachusetts Institute of Technology. Heis currently an Associate Professor of Electrical andComputer Engineering at The Ohio State University.

Dr. Eryilmaz’s research interests include design and analysis for communi-cation networks, optimal control of stochastic networks, optimization theory,distributed algorithms, pricing in networked systems, and information theory.He received the NSF-CAREER and the Lumley Research Awards in 2010.