IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 58, NO. 6, …aditya/coord-TAC.pdf · 2013. 7. 1. · IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 58, NO. 6, JUNE 2013 1443 On the Value

IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 58, NO. 6, JUNE 2013 1443

On the Value of Coordination and Delayed QueueInformation in Multicellular Scheduling

Aditya Gopalan, Constantine Caramanis, Member, IEEE, and Sanjay Shakkottai, Senior Member, IEEE

Abstract—We study limited-coordination scheduling in a wire-less downlink network with multiple base stations, each servinga collection of users over shared channel resources. When neigh-boring base stations simultaneously schedule users on the samechannel resource, collisions occur due to interference, leadingto loss of throughput. Full coordination to avoid this problemrequires each base station to have complete “instantaneous”channel-state information for all its own users, as well as theability to communicate on the same timescale as channel fluc-tuations with neighboring base stations. As such a scheme isimpractical, if not impossible, to implement, we consider thesetting where each base station has only limited instantaneouschannel-state information for its own users, and can communicatewith other base stations with a significant lag from the channelstate variations to coordinate scheduling decisions.In this setting, we first characterize the throughput capacity

of the system. A key insight is that sharing delayed queue-lengthinformation enables coordination on a slow timescale among thebase stations, and this permits each base station to use limitedand local channel-state along with global delayed queue-state tostabilize its users’ packet queues. Based on this, we develop adistributed, queue-aware scheduling (and information exchange)algorithm that is provably throughput-optimal. Finally, westudy the effect of inter-base-station coordination delay on thesystem packet delay performance under the throughput-optimalalgorithm.

Index Terms—Multicellular scheduling, orthogonal frequencydivision multiple access (OFDMA), resource allocation.

I. INTRODUCTION

N EXT-generation cellular systems like 3 GPP-long termevolution (LTE) [1] are based on the technique of Or-

thogonal Frequency Division Multiple Access (OFDMA), andpromise high-speed packet-based services for a variety of appli-cations. In a typical downlink of such a cellular system, base sta-tions enable multiple mobile users to share available channel re-sources by assigning them different frequency bands or “tones”.These tone-based channels experience temporal fluctuations in

Manuscript received September 20, 2011; revised July 12, 2012; acceptedDecember 01, 2012. Date of publication January 14, 2013; date of currentversion May 20, 2013. This paper was presented in part at the InformationTheory and Applications Workshop, San Diego, CA, February, 2011. Thiswork was supported in part by NSF grants, CNS-0721532, CNS-0964391,CNS-1017549, EFRI-0735905, ECCS-1056028, by the Defence Threat Reduc-tion Agency under grant HDTRA 1-08-0029, and by the DARPA ITMANETprogram. Recommended by Associate Editor S. Mascolo.A. Gopalan is with the Department of Electrical Engineering, Technion-Israel

Institute of Technology, Haifa, Israel (e-mail: [email protected]).C. Caramanis and S. Shakkottai are with the Department of Electrical and

Computer Engineering, The University of Texas at Austin, Austin TX 78713USA (e-mail: [email protected]; [email protected]).Color versions of one or more of the figures in this paper are available online

at http://ieeexplore.ieee.org.Digital Object Identifier 10.1109/TAC.2013.2240098

quality due to fading, and these fluctuations dictate the instan-taneous data rates that can be sustained within a time slot. Basestations obtain channel quality measurements from the mobileusers attached to them, and perform resource allocation, on atimescale of about every 1–2 milliseconds. This helps the basestations opportunistically exploit local channel fluctuations toschedule data transmissions to the users.Data transmission in a multicellular environment with many

base stations is, however, impeded by the following factors:1) Co-channel interference: A system of several base sta-tions is prone to inter-cell interference, where transmis-sions to neighboring mobile users assigned the same fre-quencies in different cells collide, resulting in a loss ofthroughput. The issue of interference management is espe-cially vital in modern LTE-based femtocell networks [2],which are comprised of a heterogeneous deployment ofbase stations, eachwith a small footprint. Due to unplannedor ad-hoc deployments, femtocell base stations suffer fromradio interference from nearby femtocells and macrocells,and this impacts overall throughput. Mitigating inter-cellinterference in femtocell networks has thus been a subjectof much recent research [3], [4].

2) High backhaul latency: Avoiding interference entirelydemands that the base stations coordinate their trans-missions using instantaneous channel state informationacquired from other base stations. Such coordination atthe timescale of channel fluctuations, however, is renderedinfeasible by the relatively high latencies (of the orderof 10–100 s of milliseconds) of backhaul links that con-nect base stations. Femtocell base stations typically usethird-party IP/Ethernet backhauls to coordinate interferingtransmissions, and are constrained to communicate overmuch slower timescales [3]. Communication between basestations is thus limited to sharing information which issignificantly delayed compared to instantaneous channelstate variations.

3) Partial local channel state information: Base stationscannot acquire even the complete instantaneous channelstates for all their own users. This is because OFDMA-based systems like LTE have many sub-channels, and get-ting channel state information for all users on each sub-channel in every time-slot may be prohibitive in terms ofavailable feedback bandwidth.

In such a setting, the challenge is how to effectively use acombination of network state information available at the basestations—(a) partial “local” information, i.e., instantaneouschannel quality estimates, gathered from users at the timescaleof channel state variations, and (b) other “global” information(such as channel statistics, accumulated queue lengths, userinterference patterns etc.) gathered from other base stations

0018-9286/$31.00 © 2013 IEEE

1444 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 58, NO. 6, JUNE 2013

and significantly delayed from the instantaneous channel statevariations—and schedule for maximum throughput.This paper considers a collection of base stations, each

serving an exclusive set of users, in a time-slotted system. Tocapture the fact that certain users may interfere (e.g., users indifferent cells on the same frequencies located close to eachother) while others may not (e.g., users on orthogonal channels,or on common frequencies but far from each other), we modelan arbitrary collection of subsets of interfering users in theuser population. Inter-cell interference is modeled by assumingthat transmissions to interfering users by their respective basestations collide if scheduled simultaneously. At each time slot,each base station can access instantaneous channel states fora subset of its users, exchange delayed information with otherbase stations, and finally schedule users from the chosen subset.With this information structure at the base stations, we first

characterize the network throughput region, i.e., the set of alllong-term joint service rates achievable for all users. A key ob-servation we exploit is that common state information providedby global delayed queues allows coupling of decisions acrossbase stations. We demonstrate the optimal way of using thiscoupled state to coordinate scheduling across multiple base sta-tions, and develop a provably throughput-optimal schedulingalgorithm. In other words, when it is possible to share globaldelayed information among base stations, it is enough to sharedelayed queue lengths to achieve throughput-optimality. To thebest of our knowledge, this is the first throughput-optimalityresult using the information structure of local limited instan-taneous channel state and global delayed information (queuelengths). We also quantify, via analysis and simulations, howthe packet delay performance of our throughput-optimal sched-uling algorithm varies with the amount of delay in the sharedqueue length information.

A. Main Contributions

The main contributions of this paper are as follows:1) We derive the throughput region of a multi-base-stationsystem, with given arbitrary subsets of interfering users,in which the base stations schedule using the informationstructure of (a) local limited channel state information, and(b) globally shared information which is independent ofthe instantaneous channel states. Moreover, we show thatany rate within the throughput region can be obtained bytimesharing, using common randomness, across a simpleclass of static scheduling policies. In a static schedulingpolicy, each base station always picks a fixed subset ofits users, and schedules each user in that subset dependingsolely on its instantaneous channel state and a fixed binaryvector associated with that state.

2) We present a two-tier, distributed and provably throughput-optimal scheduling algorithm that relies on the base sta-tions sharing their users’ queue lengths every (an integerparameter) time slots. Specifically, at every -th time slot,all the base stations communicate their queue lengths toeach other. For the next time slots, each base stationuses this (delayed) queue length information, along withknowledge about channel statistics and interfering users,to locally observe an appropriate subset of its channels’

states and schedule the corresponding users. The param-eter in our algorithm is the maximum “staleness” of ex-changed queue length information, and is also a measureof the inter-base-station coordination time. Moreover, thescheduling algorithm is throughput-optimal for any fixedvalue of , meaning that the coordination time can beeasily adapted to suit the latency of the backhaul betweenbase stations without sacrificing throughput.

3) We provide analytical bounds on the system packet delayperformance as a function of , and carry out simulationsto illustrate the degradation in the packet delay with in-creasing .

B. Related Work

Throughput-optimal scheduling for wireless networks datesback to the pioneering work of Tassiulas et al. [5], [6]. Sincethen, there has been much work on throughput-optimal wire-less scheduling, both with a central scheduler having completenetwork-state information [7]–[9] and distributed implementa-tions [10]–[13]. Further references can be found in [14], [15].Scheduling with partial or limited channel state information hasbeen addressed in [16], where infrequent channel state infor-mation used to schedule, and [17]–[20] where scheduling isstudied with partial or inaccurate observability of the aggregatechannel state. In [21]–[23], the authors develop throughput-op-timal algorithms using delayed channel-state information withchannel state and topology uncertainty in an ad hoc network set-ting, where channels are independent across users. Our resultsdiffer in twoways. First, the authors in [21]–[23] do not considerthe setting as in this paper where only limited channel-state isavailable at base stations—in the ad hoc network setting whereneighborhoods are small, complete local-channel state is avail-able, which is not the case in 4G base stations. In addition tothe challenge of the subset selection problem, the key concep-tual difference and contribution of this paper is that this subsetselection occurs through the base station coordination, as wefurther explain below. Second, our results in this paper allowchannels to be arbitrarily correlated across users. This combi-nation of limited and correlated channel state leads to differenttrade-offs and scheduling algorithms.In the multiple base station setting, two-tiered interference

mitigation through load balancing and base station coordinationhas been studied in [24], but under the assumption that a cen-tral scheduler has instantaneous queue states of all users, andeach base station has complete channel states of its users. Theauthors use the central scheduler to determine (based on statis-tics and instantaneous queue state) which of the base stationsare allowed to transmit (ON base stations) and which are OFFin order to minimize interference, following which each ONbase station schedules users based on its channel state informa-tion. However, the authors do not investigate queue-stability orthroughput-optimality. Further, as we see from our analysis, in adistributed setting where there is no central coordinator, the op-timal scheduler in fact allows collisions between transmissionsfrom multiple base stations. The intuition for this is that due tochannel randomness, it is better to be “optimistic” under somesituations and attempt transmission at a base station with thehope that a contending base station’s channels will be poor, andhence the contending base station will not attempt to transmit.

GOPALAN et al.: ON THE VALUE OF COORDINATION AND DELAYED QUEUE INFORMATION IN MULTICELLULAR SCHEDULING 1445

In [25] the authors propose a gradient power-control algo-rithm to mitigate inter-cell interference and dynamically reusefrequencies, while [26] considers scheduling algorithms to ef-fectively allocate subcarriers or frequencies to users in a mul-ticellular environment to maximize the sum throughput of thesystem. In [27] the authors assume coarse-grained communica-tion among base stations along with a dynamic user model inwhich users randomly enter and exit the network, and presentsimulation results for scheduling strategies with the main metricbeing file transfer delay.The authors in [4], [3] consider the problem of delayed

coordination that results among LTE-Advanced femtocell basestations connected by an IP-based backhaul. They developheuristic scheduling algorithms that account for coordinationlatencies, and carry out extensive numerical studies. None ofthe above works, though, examines the importance of usingglobal information via delayed queue lengths and local in-stantaneous channel state information to stabilize queues andachieve throughput-optimality.Finally, there is work from a physical layer perspective to

maximize sum rate. However, it does not address either delayed/limited information or stability. The reader is referred to [24] fora comprehensive survey.

II. MOTIVATING EXAMPLE: HOW THROUGHPUTDEPENDS ON THE COORDINATION TIMESCALE

In this section, we present an example to illustrate how theextent of coordination in scheduling, i.e. sharing informationacross base stations, affects the throughput/capacity of multi-cellular wireless systems.Let us consider a scenario involving two base stations and

two wireless users: base station serving user and base sta-tion serving user in discrete time slots. Assume that thejoint channel states of the two users are either (1, 2) or (2, 1),each with probability 0.45, or (2, 2) with probability 0.1, in-dependently in each time slot. The channel state denotes howmany packets can be transmitted to the user in the event of asuccessfully scheduled transmission. Corresponding to the sit-uation in an LTE system where the two users are assigned thesame frequencies and are located close to each other at the cellboundary, we assume that transmissions to these two users col-lide if scheduled together. At every time slot, each base stationdecides whether to schedule its respective user or not dependingon the structure of network state information it possesses. Weconsider three possible structures of network state information:1) First, assume that at every time slot, each base stationknows only its own user’s current channel state (i.e., thebase stations have local channel state information with nocoordination). In this case we can show that the throughputregion is enclosed by the solid curved lines connecting thepoints (0, 1.55), (0.9, 0.9) and (1.55, 0) in Fig. 1. Essen-tially, this is equivalent to saying that each base station de-cides independently to schedule its own user with somefixed probability. The first (respectively third) point rep-resents the case when user (respectively ) is alwaysscheduled and the other user is always not scheduled. Thesecond point represents the case when each user is sched-uled if and only if its observed channel state is 2.

Fig. 1. Throughput region for a 2-user system under scheduling with differentinformation structures.

2) Next, assume again that each base station knows only itsown user’s current channel state, but that the base sta-tions can exchange delayed or slowly varying informa-tion—more specifically, any information independent oftheir users’ current channel states. This models the factthat backhaul capabilities between the base stations do notpermit exchange of instantaneous channel state informa-tion occurring on a fast timescale. For instance, the basestations can rely on a source of common randomness tomake their scheduling decisions. This is the situation inwhich the base stations have local channel state informa-tion with “slow” global coordination. We see here that thethroughput region expands to the convex hull of the ear-lier three points (Fig. 1); intuitively, collaboration allowstimesharing.

3) Finally, we assume the base stations can obtain instan-taneous global channel state information, i.e., acquireboth the users’ channel states before making schedulingdecisions. This models the fact where the base stationscan hypothetically exchange information as fast as theinstantaneous channel states vary, and we find that thethroughput region expands further to the convex hull of thepoints (0, 1.55), (0.9, 1.1), (1.1, 0.9) and (1.55, 0) (Fig. 1).This is because the second (respectively third) point canbe achieved by scheduling only user (respectively )when its channel state is 2—the advantage in knowingboth users’ instantaneous channel states comes from thefact that a base station can suitably “back off” when bothchannels have state 2.

The example shows that there can be a significant differencein network throughput depending on the extent of global coor-dination (none/slow/fast) between base stations. As mentionedearlier, Case 3—when the base stations access complete instan-taneous channel state information—is practically infeasible dueto the high latency of inter-base station backhaul links relativeto the timescale of channel quality variations. However, back-hauls in present-day LTE-based systems effectively allow de-layed information on a slower timescale to be shared across basestations. This prompts us to treat Case 2 (where coordinationamong base stations is possible only on a timescale slower thanthat of the channel state) in detail in this paper, and to developthroughput-optimal scheduling algorithms for this setting.

III. SYSTEM MODEL

This section describes the notation and definitions necessaryto develop a formal model for coordinated wireless scheduling,


Fig. 2. Coordinated scheduling with local information.

incorporating the effect of coordination latencies betweenscheduling base stations.

A. Network Model

Consider base stations wishing to send packetdata to users on the wireless downlink. Eachuser is associated with a unique base station from which it canreceive data; we use to denote the set of users associatedto base station , and to denote the base station to whichuser is associated. We denote the set of base stations and theset of users by and , respectively.

B. Arrival and Channel Model

Time is slotted into discrete units. Data packets destined foruser arrive at base station as a stationary nonneg-ative integer-valued random process , . Forsimplicity we will assume that is independent and iden-tically distributed (iid) over time slots with ,

and . Let . Packetsget queued if they are not immediately transmitted. The channelbetween user and its associated base station is time-varying,and we assume that its state stays constant for the duration ofa time slot. We denote the channel state random process by

, , where for any takes values in afinite set . We explicitly assume that consists of the integers

. The aggregate channelstate process is assumed to beindependent and identically distributed (iid) over time slots, butthe channel states can be correlated across users. For a subset

of users, we overload notation and de-note by the channel state of just the subset . Letdenote the probability mass function of the aggregate channelstate , i.e.,

. Here, for each channel , takes valuesin the set of possible channel states . Sucha canonical wireless system is shown in Fig. 2.

C. Queueing Model

Each base station maintains one packet queue for everyuser associated with it, into which data packets destined to

get buffered if they are not immediately transmitted. We de-note the fact that user is successfully scheduled fordata reception at time slot by setting a binary random vari-able . When this happens, up to packets canbe drained from its packet queue. Thus if denotes thequeue-length process for the packet queue of user , then theevolution of can be described as

(1)

Another form of (1) which we use later is

(2)

where . Let represent thevector of queue lengths at time slot .

D. Multiple Base Station Scheduling Model

In the multiple base station network, at every time slot , eachbase station schedules transmissions for a set of its users.Following the motivating example, we model the informationstructure that each base station uses to schedule users as havingtwo important properties:1) Limited, local channel state information at base sta-tions: Each base station accesses instantaneous channelstate information for only a subset of its users prior toscheduling. We let each base station choose a subset

of its users at every time slot , from a fixed arbitrarycollection of subsets of . Following this, the in-stantaneous state of the chosen users’ channelsis available to , and it can use this knowledge to scheduleusers in . We use a binary random variable torepresent whether a user is scheduled at time( denotes that is scheduled). This frameworkformally captures the fact that channel state feedback capa-bilities between base stations and their users are potentiallylimited.

2) Delayed, slower-timescale coordination between basestations: Base stations can share information to help co-ordinate scheduling, but instantaneous channel state infor-mation at each time slot cannot be shared. This capturesthe fact that the backhaul links that allow base stationsto communicate suffer from a high latency relative to thetimescale of channel state variation.To model this formally, we first fix a sufficiently large in-teger which we will call the system history param-eter. At time slot , we assume that each base station canaccess the history of the entire network—queue lengths,channel states, arrivals—for all the previous time slotsup to and not including , denoted by the (random) vector

. This says that the backhaulcoordination links are capable of letting the base stationsshare their past observations with each other; however, anyinstantaneous (at time ) channel state information cannotbe propagated within the same time slot. We remark thatthe restriction of available system history to the previoustime slots is made for technical convenience—our resultshold when, for instance, the entire system history (past


channel states, scheduling decisions and arrivals) is avail-able at the base stations.1

At each time slot , in addition to the system history ,we assume that all the base stations can utilize commonrandomness, represented by a random variablethat is independent from the instantaneous channel state

. The independence of from the instantaneouschannel state is in keeping with the constraint that basestations are incapable of sharing instantaneous channelstate information among themselves. represents anycommon, auxiliary information which can be used by allthe base stations. For instance, can denote measuredchannel/arrival/queue statistics, or just a current time indexthat drives time-dependent scheduling, or the outcomeof a common “coin toss” sequence to timeshare acrossbase stations etc. It models common information that canbe propagated across the inter-base station backhaul inorder to coordinate scheduling (we discuss this in detail inSection IV in the context of time-sharing policies).Thus, the total information available to the base stations ateach time slot , as a result of slow coordination over thebackhaul, is , which we call thesystem state.

In summary, every base station first picks a subset of its usersto observe their instantaneous channel states—this can dependon queue lengths, channel states, arrivals and auxiliary informa-tion in the last slots. After having observed the instantaneouschannel states for that subset, the base station schedules users inthe subset depending on their instantaneous channel states andthe global information it already possesses. We term the collec-tive set of rules applied at each time slot by every base stationto schedule users as the scheduling policy or algorithm used bythe base stations.

E. Interference Model

As introduced earlier, data transmissions to users in differentcells that are close to each other and use the same frequencies areprone to interference. At the same time, transmissions to userson suitably orthogonal channels (e.g. different frequencies ortones in an OFDMA-based system) can occur simultaneouslywithout any interference. For each user , letdenote the set of users that interfere with , meaning that

user cannot receive any data packets in a time slot at which auser in is scheduled. We assume that for all .For instance, means that user does not experienceinterference from any other user. Thus, we have, for all users

(3)

With this, the maximum number of packets that can be drainedfrom the queue for user at time becomes

(4)

1In addition, we note that if queue length information is shared with a latencyof time slots instead of 1, the scheduling algorithm proposed in this workcan be used with a slight modification to yield the same throughput and queuelength performance, so knowing the state of the network at time is not arestrictive modeling assumption. This is because because using delayed queuelength information in scheduling does not affect stability properties as long asthe delay is bounded.

Such a collision interference model together with the“GO/NO-GO” type scheduling model described earlier modelsa rudimentary “binary” power-control scheme for users in thenetwork.2

F. Objective/Performance Metric

For the setup described above, note that under any schedulingpolicy, the system state is a discrete time Markov chain.Let us assume that this Markov chain is irreducible and aperi-odic.3 Following standard terminology, we say that a vector ofarrival rates with , issupported by a scheduling policy if the Markov chain ispositive recurrent under the policy when the packet arrival ratesat the user queues are , (this cor-responds to the intuitive notion that the queues in the networkare drained as fast as they fill up, i.e., they are stable). The goalis then to characterize the stability region, which we define tobe the set of all vectors of arrival rates ( : ) sup-ported by at least one scheduling policy. In addition, we wishto investigate whether there exists a single scheduling policywhich can support any arrival rate vector in the stability re-gion—a property we call throughput optimality of a schedulingpolicy.

IV. STABILITY REGION WITH SLOWGLOBAL COORDINATION

In this section, we explicitly characterize the stability regionof a system of base stations that schedule users with coordina-tion and limited local channel state information. We first intro-duce a class of static scheduling policies, called Static ServiceSplit (SSS) policies, which use only instantaneous local channelstate information at each base station to make scheduling de-cisions. SSS policies can achieve a finite set of rates in the sta-bility region, and by using additional common randomness (e.g.,a common sequence of coin toss outcomes), the base stationscan suitably time-share across SSS policies. We show in The-orem 1 that the stability region is, in fact, the convex hull thatresults from all the possible time-sharing combinations of SSSpolicies. In other words, any given scheduling policy is similar,in the sense of long-term service rates, to a Static Time-sharing(STS) policy, or a time-shared combination of SSS policies.

A. SSS Policies

Let us consider a class of “simple” scheduling policies which,inspired by [7], we will term Static Service Split (SSS) policies.However, unlike the standard SSS policies used in literature forscheduling with complete channel state information [7], [28],

2We remark that this zero-rate interference model is assumed only for ease ofexposition. With minor changes, our results and algorithm can be extended to amore complex model in which interfering users experience reduced rates. Moreprecisely, reduced channel rates under collisions can be captured by modifyingthe definition of in (3), (4) to be a specification of rates for each usercorresponding to which users interfere. This would replace the hard interfer-ence constraint expressed in (4). Thus, the effect of adaptive modulation/codingschemes under interference can also be modeled, but at the expense of signifi-cant notational complications.3As in [18], this can be ensured by imposing appropriate conditions on the

arrival and channel process (e.g. their marginal distributions are supported withpositive probability on the components in ). Weaker conditionsfor different stability definitions are possible, see Section II, [18] and Section III,[9] for additional discussion.


our SSS policies are essentially “two-tiered,” and are specifi-cations of both (i) fixed subsets that base stations must alwayspick and (ii) fixed “binary vectors” for every observed subsetchannel state, that indicate exactly which users must be sched-uled when that channel state is observed.Formally, an SSS policy is defined by a tuple

, where for each , isa permissible subset of users for base station , andis a collection of binary length- vectors, one for eachpossible set of observed channel states of users in . Equiv-alently, we can think of as a map that takes the channelstates observed for subset into a binary vector

. Scheduling using the SSS policy iscarried out as follows. At each time slot ,1) Each base station picks a fixed subset of itsusers in order to observe their instantaneous channel states.

2) All the users for not in are not scheduled,i.e., for such users. A user inis scheduled if and only if , i.e.,

.As an example, suppose that base station picks the subset

of two of its users, where each user’s channelstate can be either 0 or 1. Then, it can use 4 binary vectors

—one for every 2-tuple of observed channel states(there are 4 in all). If, say, , this means that ifboth channel states are observed to be 1, only the second usermust be scheduled to transmit, and so on.Thus, an SSS policy only specifies “local” scheduling rules

per base station: each base station uses channel-state informa-tion from a predefined subset of users and schedules users inthe subset accordingly. Note that scheduling decisions amongdifferent base stations are functions purely of the instantaneouschannel states of their respective users, and are not coupledby any other common/shared information. In what follows, weintroduce a more general class of scheduling policies calledStatic Time-sharing policies, which involve the base stationstime-sharing between SSS policies using a common random-ness sequence .

B. Static Time-Sharing Policies

A Static Time-sharing (STS) policy results when base stationsuse common randomness to time-share between SSS policies.It is parameterized by a finite set of SSS policies , to-gether with a corresponding set of nonnegative weightsthat sum to 1. At each time slot , independent of previous timeslots, all the base stations together decide to schedule accordingto the SSS policy with probability . This can be achieved,if, for instance, the base stations use a common random se-quence where each is the out-come of an independent -sided coin-toss with probability dis-tribution , and indexes the SSS policy for time slot. Thus, in an STS policy, scheduling decisions among differentbase stations are coupled not only through their users’ instanta-neous channel states, but also via the common randomness thatthey use. In the next section, we will see that STS policies canachieve all rates in the convex hull of the rates of SSS policies(i.e., the “corner point rates”), and moreover, that no schedulingpolicy limited by a high-latency backhaul can stabilize rates out-side this convex hull.

C. Characterization of the Stability Region

Towards an explicit characterization of the stability regionwith slow/high-latency backhauls, let us define the rate vectorassociated with an SSS policy . For each user , as in (4),

let

where is simply from (3) but with the superscriptindicating explicit dependence on the scheduling policy .

Next, let , where

(5)

Observe that represents the vector of long-term, ergodicservice rates that the SSS policy delivers to the flows to allthe users in the system.4 In a similar manner, if is an STSpolicy, i.e., is a combination of SSS policieswith weights , then we define the rate vector

associated with by

(6)

Essentially, the rate vector for an STS policy is defined to bethe convex combination of the rate vectors of its component SSSpolicies.For an SSS policy with rate vector , all arrival rate vec-

tors , with , , that aredominated by lie in the system stability region, since thescheduling policy stabilizes them. Furthermore, any arrivalrate vector that is the convex combination of SSS policy ratevectors also belongs to the stability region, since an appropriateSTS policy corresponding to the convex combination stabilizesit. Let

where Int Co refers to the interior of the convex hull ofthe set in standard Euclidean space. Then, the preceding ar-gument indicates that is definitely an inner bound to the sta-bility region (recall that the stability region consists of all thosearrival rate vectors which can be supported by some schedulingpolicy). Theorem 1 below states that in fact, the stability regionis no more than :

4Equation (5) defines the expected offered service rate per time slot thatuser sees under the scheduling policy . For this purpose, the product on theright hand side in (5) is taken over all users that interfere with user (theseusers could be associated to different BSs). Thus, a general term in the productis for a user that interferes with and the (unique) BS towhich is associated. In this regard, the subscript in the sum represents theBS of an interfering user.


TABLE ICHANNEL STATE DISTRIBUTION FOR THREE-USER,

TWO-BASE-STATION SYSTEM EXAMPLE

Theorem 1: The stability region of the system is , i.e., avector of arrival rates with ,

is supported by a scheduling policy if and only if.This result says that any scheduling policy which stabilizes

the system for a certain choice of arrival rates effectively be-haves like an STS policy, i.e., a suitable time-shared combina-tion of SSS scheduling policies, in the sense of the long-termservice rates it delivers. This result is useful later, in Section V,towards showing that a particular scheduling algorithm we de-velop is throughput-optimal. The proof of this theorem is sim-ilar in spirit to the results in [7], [21], [22] used to characterizethe stability region. It uses the fact that a system stable/ergodicunder a policy must have consistent long-term fractions, whichare in turn used to construct STS policies yielding the same ser-vice rates. Refer to Appendix A for the proof.

D. Example: Stability Region for a Three-User,Two-Base-Station System

To illustrate the concepts and result of the previous section,let us derive the stability region for a simple case of two base sta-tions and serving a total of three users . isassociated to whereas and are associated to . Channelstates for all the three users are either 0 or 1 (ON/OFF channels).Consider the case when all the users are in geographic proximityto each other and have been assigned the same frequency bandsby their respective base stations, so that simultaneously sched-uled transmissions to any two users collide. In other words,

, and .Let us assume that base station can pick at most one of its

two users at any time slot to sample, i.e., ,while trivially. For simplicity, we let thejoint channel state distribution of the aggregate channel

take one of four states asshown in Table I.Let us compute the throughput region of the system with the

given channel state statistics, according to Theorem 1. First,consider the case when base station always picks to samplein the first scheduling step. The set of achievable long-termthroughput rates with just users and is the shaded regionshown in Fig. 4(a). In this figure, the extreme points (1/4, 0)and (0, 5/8) are the service rates when users and are al-ways scheduled for service respectively, with the other user ineach case never scheduled. The extreme point (1/8, 1/2) repre-sents the service rates when users and are scheduled ifand only if their respective channel state is 1 (ON). In this case

there is a loss of throughput due to collision when both channelstates are 1.Remark: The dotted line in Fig. 4(a) represents the additional

throughput obtained when both base stations can see the channelstates of both and before scheduling. This helps reducecollisions when both the channels have state 1 and hence in-creases throughput.Similarly, we can compute the set of service rates when base

station always picks . The set of achievable long-termthroughput rates with just users and is the shaded regionshown in Fig. 4(b). Here again, we see three extreme points onthe “northeast” boundary of the region, having similar interpre-tations as in the previous figure. Also, the dotted line representsthroughput gained if both base stations know the joint channelstates of and before scheduling.Theorem 1 now tells us that the stability region of the system

can be found by taking the convex hull of the two “sub”-rateregions we found earlier. This is depicted graphically as theshaded region in Fig. 4(c).

V. THROUGHPUT-OPTIMAL SCHEDULING WITHSLOW GLOBAL COORDINATION

The result of Theorem 1 characterizes the network stabilityregion as the long-term service rates of static timesharing sched-uling policies. However, observe that• Any given static timesharing policy is not throughput-op-timal, since the set of arrival rates stabilizable by the policyis merely the “cube” of points that are dominated in everycoordinate by its long-term service rates, and not the wholestability region.

• For any arrival rate in the stability region, stabilizing thequeueing system with an appropriate static timesharingscheduling policy requires knowledge of the arrival rate.In other words, the static timesharing policies that stabilizethe system for a given arrival rate explicitly depend on thearrival rate. This motivates the need for a throughput-op-timal scheduling algorithm that, for any arrival rate in thestability region, keeps all queues stable.

In this section, we focus on developing a throughput-optimalinformation exchange and scheduling algorithm in which basestations share delayed queue length information among them-selves, and use this information to select local SSS policies.

A. A Throughput-Optimal Scheduling Algorithm With SlowGlobal Coordination

The scheduling algorithm that we describe here uses aninteger parameter , which represents the time interval (inslots) between successive communication exchanges betweenthe base stations. Specifically, the algorithm operates in suc-cessive epochs of time slots, and requires that all the queuelengths in the system be shared globally across the base stationsat the beginning of each epoch. Thus, the timescale at whichthe base stations exchange queue length information for coordi-nating their scheduling actions is once every time slots. Thescheduling policy is as described in Algorithm 1, and essen-tially operates as follows:


Algorithm 1 Scheduling algorithm

Parameter: Integer .

for1) if (mod )• Let the global queue length vector at time slot be

. Each base station thensolves the following (common) optimization problemwith the decision variables being (i) one observablesubset per base station , and (ii) a specificationof users—represented by binary vectors —tobe scheduled for each observable subset and everyjoint channel state of the subset:

(7)

• Let be a choice ofarguments that solves (7).

end if2) Each base station chooses the subset of its users,and schedules every user in that subset for which

.

end for

1) Each base station accesses the global vector of queuelengths every slots.

2) The global vector of queue lengths time slot is usedto choose a “temporally local” SSS policy for the nexttime slots. The subsets and binary decision vectors for thislocal SSS policy are chosen in such a way as to maximizethe sum of “local” service rates delivered to each queueweighted by its corresponding delayed queue length.

Remark: The constraints in the optimization problem (7)mainly deal with the structure of the binary decision vectorsfor BS while observing only a subset of its channel states.The two nontrivial constraints that a vector must satisfy,expressed by the final two constraints, are: (a) no user outsidethe subset can be scheduled by , i.e., any such user’sbinary decision is 0, and (b) for a user within , its schedulingdecision can depend only on channel states observed within thesubset , i.e., the user’s binary decision cannot change if nochannel state in changes.Note that when the history parameter is at least , the

scheduling algorithm formally makes the state processa time-inhomogeneous Markov chain, since

within each scheduling epoch, the queue lengths used as

weights in (7) depend on how far into the epoch the algorithmis operating. To keep the presentation clear, we avoid suchtechnicalities and deal, instead, with the system state sampledat the start of every epoch , which is ahomogeneous Markov chain under the algorithm .Our next result—Theorem 2—establishes two key properties

of the algorithm :1) Throughput-optimality: The scheduling policy isthroughput-optimal for any fixed value of , i.e., it cansupport any arrival rate in the stability region .

2) Packet-delays under are linear in : The averagequeue lengths in the system, under the scheduling algo-rithm , grow at most linearly with the information lag .As a result, by Little’s Law, the average packet delays arelinear in the average queue lengths for fixed arrival rates,and hence also grow at most linearly in . Indeed, the pa-rameter models the lag or delay incurred by the basestations in exchanging queue length information, and withan increasing information lag , queueing delays seen byincoming arrivals grow. Theorem 2 result helps quantifythis intuition precisely by giving an average queuelength bound.

Theorem 2: For a fixed , if the arrival rate , thenunder the scheduling algorithm ,1) is a positive-recurrent Markov chain,2) There exists a constant , not depending on , suchthat under the stationary distribution of

(8)

( denotes the expectation under the stationary distribu-tion of .)

We have seen earlier—in Section IV and Theorem 1—thatscheduling using SSS policies can only achieve finitely manyextreme points of the stability region; additional commonrandomness is required to stabilize the entire throughput re-gion via time-sharing across SSS policies. What Theorem 2shows is that this crucial role of global, common random-ness can be played by delayed/slow-timescale queue-lengthinformation shared among base stations during the operationof the scheduling algorithm . The delayed queue-lengthupdates help to correctly couple the base stations’ local SSSscheduling decisions, so as to achieve the right time-sharingcombination and stabilize any valid arrival rate. This is rem-iniscent of the manner in which queue length information isused in a distributed fashion by Ying and Shakkottai [21] fortransmission contention over interfering collision channels toachieve throughput-optimality. However, their work does notinvestigate a two-time-scale information structure (i.e., slower,delayed backhaul information and faster, instantaneous channelstates restricted to base stations), whereas our model addressesboth (a) delayed inter-base station coordination and (b) theissue of choosing subset-based partial channel state informationat each base station.The proof of Theorem 2 relies on the key observation that

delayed queue length information in the optimization (7) helpsto pick out the “right” set of contending base stations so that,even when collisions occur within each epoch of time slots,


the base stations’ decisions are time-shared to ensure the-slot negative drift of a quadratic Lyapunov function of thequeue lengths. At the same time, since each base station canaccess only partial, subset-based channel state informationfrom its users, we employ techniques from scheduling withsubset-based CSI [18] to show that the “correct” observablesubsets are picked and time-shared by every base station soas to locally achieve the right service rates. The results inthe above work [18] solve the problem of choosing partial,subset-based channel state information at a lone base stationfor throughput-optimality; however, in this paper we face andovercome the novel challenge of simultaneously combining(a) partial channel state information at every base station with(b) global delayed queue-length updates across different basestations, for achieving throughput-optimality in the presence ofinter-cell interference.We show the negative drift of the quadratic Lyapunov func-

tion under by proving that1) The -slot drift under is of the same form as that undera static time-sharing scheduling policy, and

2) Because solves the optimization (7) at every -th timeslot, its local -slot drift is the most negative (and boundedaway from zero) across all static time-sharing policies.

Finally, the queue-length bound in (8) is obtained from the neg-ative -slot Lyapunov drift by using a technique due to Neely[14]. The reader is referred to Appendix B for the completeproof of the theorem.

VI. SIMULATION RESULTS: HOW PACKETDELAYS UNDER VARY WITH

In this section we present simulation results that illustrate theimpact of the coordination delay and system load (i.e., howclose the arrival rate vector is to the boundary of the throughputregion) on the average delay experienced by arriving packets,under the scheduling algorithm . We consider two examplesetups involving multiple base stations and users, interferingchannels and subset-constrained scheduling at base stations andexplore the packet delay performance of the algorithm.

A. 3-User, 2-Base Station Example

The network model we first consider for numerical simula-tion is the one presented and discussed in Section IV-D, with3 users and 2 base stations. With regard to the throughput re-gion of the system, shown in Fig. 4(c), consider the rate vector

which is themidpoint of the edge joiningthe corner points (1/8, 1/8, 0) and (0, 0, 3/8), and on the boundaryof the throughput region. For a scaled version , wesay that represents a “load” of to the system, analogousto the terminology used in describing load in an M/M/1 queue.Arrivals are generated in an iidBernoulli fashion and schedulingis performed using the -slot throughput-optimal policies de-veloped in Section V. We examine the average delay or waitingtime experience by packets that enter the network, in the fol-lowing two cases:1) Effect of Coordination Delay: For five different loads tothe system (0.55 to 0.95 in steps of 0.1), the impact ofvarying the coordination interval from 1 to 100 on thepacket delay is as shown in Fig. 5(a). We observe that thegrowth in average packet delay is linear with which is in

Fig. 3. Stability region example: Three-user, Two-base-station System withslow coordination between the base stations.

accordance with the result of part 2 of Theorem 2, since byLittle’s law the average delay in the network is proportionalto the average queue lengths for a fixed net arrival rate.

2) Effect of Load: For five different values of coordinationinterval ( , , , and), we plot the average packet delay in the system versus

load increasing from 0.5 towards 1. The increase in averagepacket delay is observed to be particularly severe as theload approaches 100%.

B. 10-User, 3-Base Station Example

For the second simulation study, we consider a wireless net-work comprised of 3 base stations , and that serve atotal of 10 users . Users , and are as-sociated with , , and with and the remaining 4users are served by base station . Fig. 6 depicts the user loca-tions within their respective cells and their interference pattern.Users form an interfering or colliding set of users, asdo and . This happens, for instance, wheneach user in the interfering set is close to its cell edge and isallotted the same transmission frequency by its base station (in-dicated in Fig. 3 by a colored ellipse containing the respectiveinterfering users).The channel rates for users 1, 5 and 9 are modeled as

Bernoulli (0.7) independent random variables at each time slot,and the rates for all other users are assumed to be always ON(i.e., 1 packet per time slot at all times). We assume a symmetricarrival rate of 0.05 packets per time slot to all users. Each basestation can observe instantaneous NSI for 3 of its users, whichmeans that must pick a subset of 3 of its 4 users at eachtime to observe channel state. The packet delay performanceof the scheduling algorithm is plotted in Fig. 7, for eachvalue of the coordination delay . This is in agreement withour result that the average packet delay increases linearly withthe coordination latency between base stations.

VII. CONCLUSION

In this work, we considered multi-base-station wirelessdownlink scheduling with slow, global coordination and lim-ited, local channel state information. We characterized thenetwork stability region under this information structure,and developed a throughput-optimal distributed schedulingalgorithm in which it is sufficient for base stations to sharedelayed queue lengths on a slow timescale to pick appropriatesubsets of users, and use the locally observed channel statesof these users to make good scheduling decisions. In this way,coordination between the base stations on a slow timescale—in


Fig. 4. (a) Stability region for users and , (b) stability region for usersand , (c) stability region for all three users.

the form of delayed queue lengths—helps solve the subset-se-lection problem at each base station and, together with theright rules for scheduling users in those subsets, achievesthroughput-optimality. We also investigated the impact of thedelay in shared queue length information on the average packetdelay performance of the system.Future directions of research include: (i) evaluating the

throughput performance of greedy, low-complexity sched-uling strategies, and (ii) refining packet delay estimates usinglarge-deviations/heavy-traffic analysis.

Fig. 5. (a) Average packet delay with lag for various loads, (b) Averagepacket delay with load for various lags .

APPENDIX APROOF OF THEOREM 1

For showing necessity, assume that there exists a sched-uling policy which supports the arrival rate vector

. This means that under , the vectoris a positive recurrent discrete time Markov chain.

Consider this Markov chain in its stationary regime (arbitrarilyclose approximations to the stationary regime will also suffice).We will need the following additional notation for the proof:1) Let be a representation ofthe collection of user subsets that each base station picksto observe at time slot .

2) For user subsets , and in the sup-port of , let

.3) Recall that the scheduling decision for user

is a function of system state , thesubset chosen by its server (since users out-side this subset are not scheduled), and current channelstates . To indicate this, we explicitly write


Fig. 6. Example of a system with 3 base stations and 10 users for numericalsimulation. Interfering user groups, indicated by colored ellipses, are ,

and . Channels are Bernoulli (0.7) for , and andconstant 1 for all other users. Each base station can observe channel states for 3of its users, which means must choose 3 of its 4 users at each time.

Fig. 7. Average packet delay of the scheduling algorithm for various valuesof coordination delay , for the 10-user 3-base station example.

, where for everyand , is a function

that maps into {0, 1}, withwhenever .

Let . To begin, note that at stationarity,, and so

(9)

(10)

Evaluating the expectation gives

(11)

since is independent of . Using (11), (10) finally be-comes

(12)

The fact that

together with the fact that (12) has the same form as that of (6)(relying in turn on (5)) for the long term rates of STS schedulingpolicies, shows that the vector can be dom-inated by a convex combination of rate vectors of SSS sched-uling policies. This finishes the proof of the theorem.

APPENDIX BPROOF OF THEOREM 2

To avoid heavy notation, we prove the theorem assuming thateach base station can pick all its users in the first schedulingstep, viz. . The extension to thegeneral case is straightforward.


First, we bound the amount that any queue in the system cangrow in time slots, using (1):Lemma 1:

(13)Proof: Consider two cases:

1) : In this case, according to (1),both sides of (13) are equal.

2) : For this case, letbe the first time that (if no such

time exists, thenand we are done). We

must then have

which finishes the proof.

Next, for the Markov chain , let us introduce thequadratic Lyapunov function

In what follows, we bound the expected drift in this Lyapunovfunction over an interval of time slots when the system oper-ates under the policy , and show that the expected drift canbe bounded negatively away from zero. Consider

where follows from the fact that if are nonnega-tive real numbers with , then

. Taking conditional expectationsgiven yields

(14)

where the last line follows because by definition, the schedulingchoices of the policy from time upto dependonly on the queue lengths at time and the optimalbinary vectors computed at time slot , and arethus statistically identical from time upto .By hypothesis, . Hence, there

exists and a static time-sharing scheduling policysuch that . Let be a time-sharing(Bernoulli) combination of SSS policies with selec-tion probabilities , respectively, , where

(the superscript indexes theSSS policy). We have, for

(15)

We add and subtract to the right hand side of(14) to get

(16)

The crucial observation here is that the scheduling policyis designed such that the last term above, in round brackets, isalways non-positive:


Lemma 2: For the static time-sharing policy , we have

Proof: With the long term rates of the STS policysatisfying (15), we can write

(17)

We also have, by the definition of our proposed schedulingpolicy via (7), that

where the last line is due to the optimal choice of the ,. Together with (17), this proves the lemma.

Using Lemma 2, (16) implies

(18)

Without loss of generality, . For a fixed, outside the finite set of vectors for which

,we have , so byFoster’s theorem [29], is a positive recurrentMarkov chain. This proves the first part of the theorem.Turning to the second part, we can take expectations of both

sides of (18) and sum over , so that

Rearranging terms and noting that gives

The positive recurrence of , from the previouspart, implies that

for any , and, together with the finiteness of , weget that

for any . This proves thesecond part of the theorem.

REFERENCES

[1] Motorola, “Long term evolution (LTE): Overview of LTEair-interface,” White Paper, 2007 [Online]. Available: http://www.motorola.com/web/Business/Solutions/Industry%20Solu-tions/Service%20Providers/Wireless%20Operators/LTE/_Docu-ment/Static%20Files/6993_MotDoc_New.pdf

[2] D. Lopez-Perez, A. Valcarce, G. de la Roche, and J. Zhang, “OFDMAfemtocells: A roadmap on interference avoidance,” IEEE Commun.Mag., vol. 47, no. 9, pp. 41–48, Sep. 2009.

[3] R. Madan, A. Sampath, N. Bhushan, A. Khandekar, J. Borran, and T.Ji, “Impact of coordination delay on distributed scheduling in LTE-Afemtocell networks,” in Proc. GLOBECOM, 2010, pp. 1–5.


[4] R. Madan, A. Sampath, A. Khandekar, J. Borran, and N. Bhushan,“Distributed interference management and scheduling in LTE-A femtonetworks,” in Proc. GLOBECOM, 2010, pp. 1–5.

[5] L. Tassiulas and A. Ephremides, “Stability properties of constrainedqueueing systems and scheduling policies for maximum throughputin multihop radio networks,” IEEE Trans. Autom. Control, vol. 4, pp.1936–1948, Dec. 1992.

[6] L. Tassiulas and A. Ephremides, “Dynamic server allocation to par-allel queues with randomly varying connectivity,” IEEE Trans. Inform.Theory, vol. 39, pp. 466–478, Mar. 1993.

[7] M. Andrews, K. Kumaran, K. Ramanan, A. L. Stolyar, R. Vi-jayakumar, and P. Whiting, “Scheduling in a queueing system withasynchronously varying service rates,” Prob. Eng. Informational Sci.,vol. 14, pp. 191–217, 2004.

[8] S. Shakkottai and A. Stolyar, Scheduling For Multiple Flows Sharinga Time-Varying Channel: The Exponential Rule. Providence, RI:American Mathematical Society, 2002, vol. 7, pp. 185–201.

[9] A. Eryilmaz, R. Srikant, and J. R. Perkins, “Stable scheduling policiesfor fading wireless channels,” IEEE/ACM Trans. Networking, vol. 13,pp. 411–424, 2005.

[10] X. Lin and S. B. Rasool, “Constant-time distributed scheduling policiesfor ad hoc wireless networks,” in Proc. Conf. Decision Control, 2006,pp. 1258–1263.

[11] X. Wu and R. Srikant, “Scheduling efficiency of distributed greedyscheduling algorithms in wireless networks,” in Proc. IEEE IN-FOCOM, 2006, pp. 1–12.

[12] A. Eryilmaz, A. Ozdaglar, and E. Modiano, “Polynomial complexityalgorithms for full utilization of multi-hop wireless networks,” in Proc.IEEE INFOCOM, 2007, pp. 499–507.

[13] S. Sanghavi, L. Bui, and R. Srikant, “Distributed link scheduling withconstant overhead,” in Proc. ACM SIGMETRICS, 2007, pp. 313–324.

[14] M. J. Neely, “Dynamic Power Allocation and Routing for Satellite andWireless Networks With Time Varying Channels,” Ph.D. dissertation,Massachusetts Institute of Technology, Cambridge, MA, November2004.

[15] L. Georgiadis, M. J. Neely, and L. Tassiulas, “Resource allocation andcross-layer control in wireless networks,” Foundations and Trends inNetworking, vol. 1, pp. 1–144, 2006.

[16] K. Kar, X. Luo, and S. Sarkar, “Throughput-optimal scheduling inmultichannel access point networks under infrequent channel measure-ments,” IEEE Trans. Wireless Commun., vol. 7, no. 7, pp. 2619–2629,Jul. 2008.

[17] N. Chang and M. Liu, “Optimal channel probing and transmissionscheduling for opportunistic spectrum access,” in Proc. ACM Int. Conf.Mobile Comp. Netw. (MobiCom), Montreal, QC, Canada, Sep. 2007,pp. 27–38.

[18] A. Gopalan, C. Caramanis, and S. Shakkottai, “On wireless schedulingwith partial channel state information,” IEEE Trans. Inform. Theory,vol. 58, no. 1, pp. 403–420, Jan. 2012.

[19] A. Pantelidou, A. Ephremides, and A. Tits, “Joint scheduling androuting for ad-hoc networks under channel state uncertainty,” inProc. Int. Symp. Modeling Optim. Mobile, Ad-Hoc Wireless Networks(WiOpt), Apr. 2007, pp. 1–8.

[20] M. Ouyang and L. Ying, “On scheduling in multi-channel wirelessdownlink networks with limited feedback,” in Proc. Annu. AllertonConf. Commun., Control Comp., Piscataway, NJ, USA, 2009, pp.455–461, IEEE Press.

[21] L. Ying and S. Shakkottai, “On throughput optimality with delayednetwork-state information,” IEEE Trans. Inform. Theory, vol. 57, no.8, pp. 5116–5132, Aug. 2011.

[22] L. Ying and S. Shakkottai, “Scheduling in mobile ad hoc networks withtopology and channel-state uncertainty,” IEEE Trans. Autom. Control,to be published.

[23] A. Reddy, S. Banerjee, A. Gopalan, S. Shakkottai, and L. Ying, “Onwireless scheduling with heterogeneously delayed network-state infor-mation,” Queueing Syst., vol. 72, pp. 193–218, Dec. 2012.

[24] S. Das, H. Viswanathan, and G. Rittenhouse, “Dynamic load balancingthrough coordinated scheduling in packet data systems,” in Proc. IEEEINFOCOM, 1, Mar./Apr. 2003, vol. 1, pp. 786–796.

[25] A. Stolyar and H. Viswanathan, “Self-organizing dynamic fractionalfrequency reuse for best-effort traffic through distributed inter-cell co-ordination,” in Proc. IEEE INFOCOM, Apr. 2009, pp. 1287–1295.

[26] G. Fodor, M. Telek, and C. Koutsimanis, “Performance analysisof scheduling and interference coordination policies for OFDMAnetworks,” Comput. Netw., vol. 52, no. 6, pp. 1252–1271, Jun. 2008.

[27] B. Rengarajan and G. de Veciana, “Architecture and abstractions forenvironment and traffic aware system-level coordination of wirelessnetworks: The downlink case,” in Proc. IEEE INFOCOM, Apr. 2008,pp. 502–510.

[28] S. Shakkottai, R. Srikant, and A. L. Stolyar, “Pathwise optimality of theexponential scheduling rule for wireless channels,” Adv. Appl. Prob.,vol. 36, no. 4, pp. 1021–1045, 2004.

[29] P. Brémaud, Markov Chains: Gibbs Fields, Monte Carlo Simulation,and Queues, ser. Texts in applied mathematics. New York: Springer,1999.

Aditya Gopalan received the Ph.D. degree from the Electrical and ComputerEngineering Department, University of Texas at Austin in 2011.He is a Postdoctoral Fellow with the Department of Electrical Engineering,

Technion—Israel Institute of Technology, Haifa. His current research interestsinclude network algorithms, stochastic control and machine learning.

Constantine Caramanis (M’06) received the Ph.D. degree in electrical engi-neering and computer science from the Massachusetts Institute of Technology,Cambridge, in 2006.Since then, he has been on the faculty in Electrical and Computer Engi-

neering, TheUniversity of Texas at Austin. His current research interests includerobust and adaptable optimization, machine learning and high-dimensional sta-tistics, with applications to large scale networks.Dr. Caramanis received the NSF CAREER award in 2011.

Sanjay Shakkottai (M’02–SM’11) received the Ph.D. degree from the Elec-trical and Computer Engineering Department, University of Illinois at Urbana-Champaign in 2002.He is with TheUniversity of Texas at Austin, where he is currently a Professor

and the Engineering Foundation Centennial Teaching Fellow in the Departmentof Electrical and Computer Engineering. His current research interests includenetwork architectures, algorithms and performance analysis for wireless andsensor networks.Dr. Shakkottai received the NSF CAREER award in 2004.

Documents

IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 58, NO. 6, …aditya/coord-TAC.pdf · 2013. 7. 1. · IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 58, NO. 6, JUNE 2013 1443 On the Value