tnet-zhang-2064175

IEEE/ACM TRANSACTIONS ON NETWORKING, VOL. 19, NO. 2, APRIL 2011 447

Scheduling Algorithms for Multicarrier WirelessData Systems

Matthew Andrews and Lisa Zhang

Abstract—We consider the problem of scheduling multicarrierwireless data in systems such as IEEE 802.16 (WiMAX). Eachscheduling decision involves assigning carriers to users for eachtime slot, subject to the constraint that each carrier is assigned toat most one user, but multiple carriers can potentially be assignedto the same user. One important aspect of our problem is that ascheduler knows the channel rates across all users and all carrierswhenever a scheduling decision is made. This “global” informationmay give a potential for enhancing performance via an optimizedallocation of carriers to users. We analyze this problem in asituation where finite queues are fed by a data arrival process. Thewell-known MaxWeight algorithm for the single-carrier settingmaximizes the product of queue size and service rate. We focus onhow to adapt MaxWeight to the multicarrier setting. If the sameobjective is pursued, more service than needed may be assigned todrain a queue, thereby creating wastage. While a simple variantin the objective forbids this wastage, it turns an easy-to-computeold objective into an intractable new objective. We state thehardness of the new optimization problems and propose severalextremely simple algorithms with provable performance bounds.We conclude with supporting simulation examples.

Index Terms—Communication systems, communications tech-nology, max weight, multicarrier, scheduling, stability, wirelesscommunication, wireless networks, wireless systems.

I. INTRODUCTION

T HE ADVENT of wireless data systems has led to renewedinterest in scheduling data in multiuser systems. In recent

years, a large body of work has looked at the problem of sched-uling over time-varying user-dependent channels in a cellularwireless system. (See Fig. 1.) This work examines a number ofdifferent models. For example, in the finite-queue model (e.g.,[29] and [30]), the aim is to keep the system stable, assumingthe queues are fed by an exogenous arrival process. Alterna-tively, in the infinitely backlogged model (e.g., [19], [28], and[31]), the aim is to maximize the system utility, assuming thequeues are permanently backlogged. Other work examines thedifference between models where the channel rates are gov-erned by some stationary stochastic process and models where aworst-case adversarial channel process is assumed, e.g., [2] and[7]. However, most of the previous work looks at a situation

Manuscript received November 12, 2009; revised June 04, 2010; acceptedJuly 27, 2010; approved by IEEE/ACM TRANSACTIONS ON NETWORKING EditorT. Bonald. Date of publication September 16, 2010; date of current version April15, 2011. An earlier version of this paper was published in the Proceedingsof the ACM 13th Annual International Conference on Mobile Computing andNetworking (MobiCom), Montreal, QC, Canada, September 9–14, 2007.

The authors are with Bell Labs, Murray Hill, NJ 07974 USA (e-mail:[email protected]; [email protected]).

Digital Object Identifier 10.1109/TNET.2010.2064175

Fig. 1. Cellular wireless system.

with a single wireless carrier in which we can make a sched-uling decision on a time slot by time slot basis. Some wirelesssystems, however, have multiple carriers in which we can as-sign different carriers to different users. Examples include mul-ticarrier CDMA systems and also systems such as IEEE 802.16(WiMAX), EV-DO Revision C, and the Long-Term Evolution(LTE) of UMTS that use an orthogonal frequency-division mul-tiple access (OFDMA) physical layer in which different “tones”can be assigned to different users at each time.

In this paper, we study the problem of scheduling in a time-slotted multicarrier system. A straightforward approach for mul-tiple carriers is to schedule carriers one by one independently byusing an existing scheduling algorithm for each carrier in turn.Under such a simple adaptation, it is unclear a priori if the per-formance of a scheduling algorithm can be directly translatedfrom a single-carrier system to a multicarrier system. For ex-ample, all carriers could favor the same user, which could leadto an excessive amount of service to one user. A main goal ofthis paper is to examine how to adapt the popular algorithmknown as MaxWeight to the case of multiple carriers. We presenta number of natural analogs of MaxWeight in the multicarriersetting and prove their performance bounds against different ob-jective functions. We show the tradeoff between the complexityof the variants and their performance.

Our approach is based on the natural assumption that a multi-carrier scheduler knows the channel rates across all users and allcarriers whenever a scheduling decision is made. This “global”information may give a potential for enhancing performance viaan optimized allocation of carriers to users. Another purposeof this paper is to investigate the benefits of jointly allocatingmultiple carriers versus the isolated local optimization of eachcarrier.

A. Model, Problems, and Results

We consider a single base station transmitting data to a setof wireless users on a set of carriers. Due to the wireless

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448 IEEE/ACM TRANSACTIONS ON NETWORKING, VOL. 19, NO. 2, APRIL 2011

Fig. 2. Schedule during a single time slot �. Shaded squares indicate�� , and unshaded ones indicate �� .

nature of the channel, the channel rates are user-dependent, car-rier-dependent, and time-varying. In particular, we useto denote the channel rate at which carrier can serve user attime slot . Knowing the global rate information, a scheduler de-cides the carrier-to-user assignment during each time slot, sub-ject to the constraint that multiple carriers may be assigned tothe same user, but each carrier however can be assigned to atmost one user. In particular, we use indicator variableto indicate whether or not carrier is assigned to user at time. If , then data of size can be transmitted

to user at time on carrier (see Fig. 2). Our goal is to choosethe values in the most advantageous way.

We consider a finite-queue model with an external ar-rival process. In the single-carrier situation, the well-studiedMaxWeight algorithm always serves the user that maximizes

at each time slot . Here, denotes the queuesize of user at the beginning of time slot . The MaxWeight al-gorithm is known to have desirable stability properties, namelyit keeps queues bounded whenever possible. The proof relieson showing that if the queue sizes are large, then MaxWeightcreates a negative drift in the Lyapunov function .

The focus of this paper is how to emulate the MaxWeight al-gorithm in multicarrier systems. Let

be the amount of service user receives at time . Weconsider three objective functions when scheduling time slot

(1)

(2)

(3)

Objective (1) is the simplest analog of MaxWeight thatmaximizes for the single-carrier case.However, optimizing (1) has the potential shortcoming of as-signing more service to a user than it can actually use. Note thatthis problem exists for the single-carrier setting. However, it isparticularly acute for multicarrier systems where the amount ofservice that can be assigned in a single time slot is relativelylarge. Objective (2) offers a natural fix that forbids excessiveassignment by replacing with inthe objective function. Objective (3) explicitly maximizesthe negative drift of the Lyapunov function, where and

denote the queue size of user at the beginning and atthe end of time slot . Both objectives (2) and (3) are more

sensitive to maintaining small queues than objective (1). Wepropose four variations of the MaxWeight algorithm and referto them as MaxWeight-Alg1 through MaxWeight-Alg4 (or Alg1through Alg4 in short). In this paper, we prove a collection ofresults regarding these four algorithms in relation to the threeobjectives.

1) We describe MaxWeight-Alg1, which assigns each carrierto the user that maximizes . This carrier-by-carrier algorithm optimizes objective (1).

2) Somewhat surprisingly, both objectives (2) and (3) areNP-hard to optimize. Furthermore, they cannot be approx-imated to within a factor of for some constant .1

3) Since we cannot hope for optimum solutions to objectives(2) and (3), we focus on developing approximation algo-rithms. We present MaxWeight-Alg2 and MaxWeight-Alg3,which are variations of MaxWeight-Alg1. They provide a

-approximation and a -approximation for objectives (2)and (3), respectively.In addition, there are scenarios for which MaxWeight-Alg2and Alg 3 achieve at most a fraction of their respectiveoptimal objective values.

4) We present a more complex algorithm MaxWeight-Alg4that is based on an algorithm for the Generalized Assign-ment Problem [14]. It improves the approximation ratio forobjective (2) to for any . Although webelieve that Alg4 is not simple enough to be practical forwireless systems, we feel that it does provide theoreticalinsight into the multicarrier scheduling problem.

5) We show that the stability properties of the single-carrierMaxWeight algorithm also apply to the multicarrier algo-rithms MaxWeight-Alg1 through Alg4.

6) We present simulation results to show that althoughMaxWeight-Alg2 and Alg3 may not optimize objec-tives (2) and (3), they still significantly outperformMaxWeight-Alg1 due to the fact that they are trying tooptimize better objectives. The reason for the improvedperformance is that MaxWeight-Alg1 often tries to assignmore service to a user than it can actually use. This be-havior does not occur for algorithms MaxWeight-Alg2 andAlg3.

B. Related Work

The MaxWeight algorithm was first shown to perform wellin wireless networks by Tassiulas and Ephremides [29], [30].Other papers that study MaxWeight include [4], [3], and [24].Two algorithms that are similar to MaxWeight are MaxDelay[4], [3] and Exp [25], [26]. MaxDelay allocates service to user

, where denotes the head-of-linedelay for user at time . Exp is a more complex algorithm thatprovides more control over the relative delays that the usersexperience.

These algorithms were designed for the case in which the fi-nite queues are fed by an arrival process. For the case in which

1If �� is the optimal value of a maximization problem, we say an algorithmis an �-approximation algorithm if it always returns a solution whose value isat least ��. If for every algorithm, there are instances for which the algorithmcannot guarantee an �-approximation, then we say that the problem cannot beapproximated to within a factor �.

ANDREWS AND ZHANG: SCHEDULING ALGORITHMS FOR MULTICARRIER WIRELESS DATA SYSTEMS 449

the queues are infinitely backlogged and we wish to maximize asystem utility function, the Proportional Fair algorithm was in-troduced by [17] and [31] and studied in [1], [19], and [28]. Itwas shown in [2] that Proportional Fair does not work so wellwhen the queues are fed by an arrival process. In particular, itcan cause the queues to be unstable. Algorithms for optimizingutility functions subject to fairness requirements and constraintson minimum/maximum throughput have been studied by [5],[21], and [22]. Algorithms that combine the goals of systemutility maximization and queue stability have recently been pre-sented by [12], [23], and [27]. In [6], it was shown that un-less these problems are studied jointly, system oscillations canoccur. We remark that all of this previous work on wirelessscheduling has looked at a single carrier in isolation.

II. CANDIDATE ALGORITHMS

In this section, we define a number of algorithms that aim toemulate the MaxWeight algorithm in the multicarrier scenario.We focus on constructing a schedule for a single time slot. Forconvenience, the dependence on is omitted

queue size for user at the beginning of time

which includes the arrival for time

queue size for user at the end of time

rate for user carrier during time

total service to user during time

queue size for user after carrier is assigned.

Two equations that relate the above quantities are

Recall the objectives (1)–(3) defined in Section I. We analyze thefollowing four algorithms with respect to these three objectives.The first three algorithms go through the carriers in order. Attime , carrier serves user defined below.

• MaxWeight-Alg1: , where argmaxmeans is the index that maximizes .

• MaxWeight-Alg2: .• MaxWeight-Alg3: .• MaxWeight-Alg4 begins by approximately solving a relax-

ation of an integer linear program for objective (2) followedby rounding the fractional approximate solution. We deferthe detailed description to Section V.

We conclude this section with two simple observations.The following theorem follows directly from the definition ofMaxWeight-Alg1.

Theorem 1: MaxWeight-Alg1 optimizes objective (1). Oursecond result shows that objectives (2) and (3) are related.

Lemma 2: Any -approximation algorithm for objective (2)provides an -approximation for objective (3).

Proof: Since , we have

In addition, . Therefore,objectives (2) and (3) are always within a factor of 2 of eachother.

III. HARDNESS OF OBJECTIVES (2) AND (3)

As we discussed in the Introduction, optimizing objective (1)is not ideal since it could lead to more service being allocatedto a user than it is able to use, and hence the queue sizes maybecome larger than necessary. Hence, it would be preferable touse objectives (2) and (3). In this section, we show that, unfortu-nately, we cannot hope for an efficient algorithm that optimizesobjectives (2) and (3) in general.

Theorem 3: For some , there is no -approxima-tion algorithm for objectives (2) and (3) unless .

Proof: We use a reduction from the three-bounded three-dimensional matching problem. In this problem, we are givena set , where . Athree-dimensional matching is a subset such thatno elements in agree in any coordinate. In a three-boundedinstance, each element in appears at most three timesin . The goal is to find a matching of maximum cardinality.Kann [18] showed that there exists an such that it is NP-hardto decide whether the maximum size matching equals or is atmost . Specifically, given any 3SAT instance, we canconstruct a matching instance such that if the 3SAT instance issatisfiable, then the matching instance has a solution of size ,and the matching size is at most otherwise. Without lossof generality, we assume for the constructed matchinginstance since trivially implies the unsatisfiability of3SAT.

We now convert this matching instance (which is a resultof a reduction from 3SAT) into an instance of our schedulingproblem. We use a reduction similar to that of [11] for a problemknown as the Generalized Assignment Problem. For each hy-peredge , we are given a user . For each element

, we have a carrier . We call these carriers reg-ular carriers. We set the channel rate if is acomponent of , and otherwise. We have anotherset of dummy carriers for which forall users . Let for all users .

Given a scheduling solution, we partition the users into threesets , and . Each user in is assigned three regular car-riers only. Note that the users in correspond to a three-dimen-sional matching, and hence . Each user in is assignedone dummy carrier and possibly one regular carrier; each userin is assigned one or two regular carriers only. To see that

, and form a partition of the users that receive service,we observe that there is no benefit to assigning one dummy car-rier and regular carriers to a user since . There isalso no benefit to assigning two dummy carriers to a user sincethere is more benefit to reassigning one dummy carrier to a userwith regular carriers. Such a user always exists since thenumber of users in is at least , the number ofdummy carriers. Therefore, and .

Consider the regular carriers not assigned to users in. With respect to objectives (2) and (3), there is more benefit to

assigning them to users in than assigning them to users in .However, we can assign at most two regular carriers to each user


in . Hence, at least regular carriersare assigned to users in . Therefore, objectives (2) and (3) canbe upper-bounded as follows. In this proof, we use OBJ2 andOBJ3 to denote objectives (2) and (3). We also use the subscript

to denote the case in which the maximum size matchinghas size and the subscript to denote the casein which the maximum size matching has size at most

(4)

and

(5)

We now consider two cases. If the size of the maximum three-dimensional matching is indeed , then and .In this case, the upper bounds on objectives (2) and (3) that wehave just derived are actually tight, i.e., equals (4)and equals (5). If the maximum three-dimensionalmatching has size at most , then . For bothobjectives, the drop in value is at least , i.e.,

and.

We note that since the matching instance is three-bounded. Therefore, both objectives (2) and (3) are at most .This means that the relative difference in the objective valuesbetween the two cases is at least . By setting ,we obtain our result.

IV. APPROXIMATION RATIOS OF ALG2 AND ALG3

In this section, we show that algorithms MaxWeight-Alg2 andAlg3 are constant-factor approximations for objectives (2) and(3). The hardness results of Section III imply that for these ob-jectives, constant-factor approximation algorithms are the bestthat we can hope for. Moreover, in Section IV we present sim-ulation results to show that although these algorithms may notoptimize objectives (2) and (3), they still significantly outper-form MaxWeight-Alg1 due to the fact that MaxWeight-Alg1 willoften try to assign more service to a user than it can actually use.

Theorem 4: MaxWeight-Alg2 is a -approximation algorithmfor objective (2). By Lemma 2, this immediately implies that itis a -approximation algorithm for objective (3).

Proof: We show thatMaxWeight-Alg2 is a special case ofthe greedy algorithm for maximizing a nondecreasing submod-ular function over a matroid. In order to clarify this relationship,we first define the following terms.

• Consider a ground set , and let be a set of subsets of .The set is a matroid if .— If and , then .

— If and , then there exists an elementsuch that .

• A special case of a matroid is a partition matroid. We saythat a matroid is a partition matroid if there is a partition of

into components such that if and onlyif , for all .

• Let be a function on sets in .— It is a submodular function on if for all such

that and

— It is a nondecreasing submodular function if in additionand for all such that

• The Greedy algorithm for maximizing a nondecreasingsubmodular function over a matroid works as follows.— Initially let .— Repeat the following procedure for as long as possible.

Let . Set.

For partition matroids, the algorithm can be simplified. Atstep , instead of considering all elements in , we onlyneed to find .

• Fisher et al. [13] proved the following property of theGreedy algorithm.

Lemma 5: The Greedy Algorithm gives a - approximationto the problem of maximizing a nondecreasing submodularfunction over a matroid.

We now show that MaxWeight-Alg2 is special case of theGreedy Algorithm for partition matroids. The ground set

. A subset if and only ifthere is at most one element in for each carrier. In other words,

defines a valid schedule. Let . Thisclearly defines a partition matroid. The function is definedby

where . Note that if, then . From this, it is easy to see that for

any element where forms a valid schedule,, i.e., the func-

tion is submodular. Moreover, it is clear that functioncorresponds directly to objective (2). Hence, when we try to op-timize objective (2), we are trying to find an assignment (whichcorresponds to an element of a partition matroid) that maxi-mizes a submodular function. Recall that MaxWeight-Alg2 goesthrough each carrier in turn and assigns it to the user that max-imizes the increase in the objective. Hence, MaxWeight-Alg2corresponds to the Greedy algorithm, and so, by the result ofFisher et al., it is a -approximation algorithm for objective (2).

We now provide an analysis of algorithm MaxWeight-Alg3.Theorem 6: MaxWeight-Alg3 is a -approximation for

objective (3).


Proof: Recall that is the queue size for user after al-gorithm MaxWeight-Alg3 has assigned carrier . Let OPT bean optimal assignment, and let be the analogous queue sizeafter the first carriers are assigned according to OPT. Con-sider any carrier . Suppose that OPT assigns carrier to user ,and MaxWeight-Alg3 assigns it to user . We first show that thegain obtained by algorithm MaxWeight-Alg3 due to carrier isat least one half of the gain obtained by OPT due to carrier thatis never obtained by MaxWeight-Alg3. More precisely, we show

(6)

Suppose that . We have

by definition of

since change in queue size bounded by

by assumption that

Inequality (6) also holds true for the case that sincethen there is no gain obtained by OPT that is not obtained byMaxWeight-Alg3. Algebraically, the inequality holds triviallyin this case since the right-hand side is zero. Note that the in-equality also holds when .

Now that we have verified inequality (6), we proceed to provethe lemma. For clarity of the rest of the proof, we let be theuser that carrier serves under Alg3, and let be the user thatserves under OPT. Note that can be the same as . We knowthat for , and for . Wetherefore have

by telescoping on

since for

by Inequality

since for

This immediately implies

which completes the proof.We conclude this section by showing that our analysis

of MaxWeight-Alg2 is essentially tight and our analysis ofMaxWeight-Alg3 cannot be significantly improved.

Theorem 7: For any constant , there exists an instanceon which MaxWeight-Alg2 and Alg3 achieve at most afraction of the optimal value of objectives (2) and (3).

Proof: The example is as follows. There are two users,each with . The channel rates are given byfor , and . The optimalalgorithm assigns carrier 1 to user 2 and carrier 2 to user 1.Hence, the optimal values of objectives (2) and (3) are and

, respectively. On the other hand, algorithms MaxWeight-Alg2 and MaxWeight-Alg3 both assign carrier 1 to user 1 since

.

V. MAXWEIGHT-ALG4: IMPROVED APPROXIMATION

FOR OBJECTIVE (2)

In this section, we show that for any , it is actuallypossible to obtain a randomized -approximation forobjective (2). (Note that .) The algorithm,which we call MaxWeight-Alg4, is based on a recent algorithmfor the Generalized Assignment Problem (GAP) due to Fleis-cher et al. [14]. (In the GAP problem, we are given a set of binsof different sizes. Each item has a bin-dependent profit and abin-dependent size. The goal is to pack the items into bins soas to maximize the profit in such a way that no bin size is vi-olated.) MaxWeight-Alg4 is somewhat complex, and so we feelthat it is impractical for scheduling wireless systems. However,we include it here since we feel that it is of theoretical interestto understand what are the limits regarding the approximabilityof objective (2).

Let be the set of all possible subsets of carriersthat could be assigned to user . For , let

. For convenience, we also calcu-late a new set of rates such thatand . This can clearly be done inlinear time. The variable is used to indicate whether or notsubset is assigned to user . We could optimize objective (2)


by solving the following integer program:

Note that since we showed in Theorem 3 that optimizing objec-tive (2) is NP-hard, we cannot hope to solve the above integerprogram exactly. Algorithm MaxWeight-Alg4 finds an approx-imate solution as follows. It first finds a solution to the linearrelaxation of the above integer program in which the constraint

is replaced by . Note that we cannotdirectly use a standard linear programming algorithm for this re-laxation since there are exponentially many variables. However,following [14], we can apply standard iterative Lagrangian LPalgorithms (e.g., [15]) to obtain a -approximation to thelinear relaxation of Alg4-IP in time polynomial in , and

.MaxWeight-Alg4 then rounds the solution to this linear relax-

ation by choosing a single set to assign to user . In particular,we set with probability . We still do not have a validassignment since a carrier may be assigned to two differentusers. In this case, we pick the user that gives the maximumvalue of .

Theorem 8: The value of the solution obtained byMaxWeight-Alg4 is at least a fraction of theoptimal value of objective (2).

Proof: For each carrier , we set to be theuser-set pair that has the highest value of . We set

to be the user-set pair that has the next highest valueof , etc. The definition of our rounding algorithmmeans that carrier is assigned to user as part of withprobability at least .

The contribution of carrier in the solution to the relaxationof Alg4-IP is . By the above argument, theexpected contribution in the rounded solution is at least

Fleischer et al. [14] show, using the arithmetic/geometric meaninequality, that expressions of this form are at least

Hence, we have obtained an assignment whose value with re-spect to objective (2) is at least a fraction of the solutionto the fractional relaxation. Since this is in turn at least a

fraction of the optimal solution to objective (2), our final ap-proximation ratio is at least .

VI. STABILITY OF FOUR VARIANTS OF MAXWEIGHT

In this section, we consider the stability of the four variantsof MaxWeight that we have proposed. Informally, an algorithmis said to be stable if it keeps the queue sizes bounded. However,bounded queue is impossible if traffic arrival is unrestricted. Wetherefore define admissible traffic for which some carrier as-signment can keep queues bounded. In particular, we use thenotion of -admissible, where is a window size and

measures how close the incoming traffic is to “fullload.” Let be the amount of data injected for user in timeslot . For given and , a system is -admissible if thereis a schedule such that in any window ofsize , we have

Obviously, more injections are admissible to a system with alarger value or a smaller value. We say that an algorithmis stable if it keeps the queues bounded as long as the traffic is

-admissible.The single-carrier MaxWeight algorithm is known to be stable

as long as the channel rates for a user cannot be zero for ar-bitrarily long periods. (This condition holds for example whenthe rates are governed by a stationary stochastic process withnonzero mean.) The following theorem states that this propertyalso holds for the four multicarrier MaxWeight algorithms pre-sented in this paper. The proof uses a standard technique (e.g.,[29] and [30]) of showing that the Lyapunov functionhas a negative drift when the queues are large.

Theorem 9: If any fixed and , algorithmsMaxWeight-Alg1 through Alg4 are stable for -admissibletraffic as long as for each the rates cannot be zerofor arbitrarily long periods.

Proof: We assume that the channel rates are bounded.Let be the supremum of these rates. For simplicity, wealso assume that the channel rates are bounded away fromzero. Let be the infimum of these rates. By looking overlarger time windows, it is straightforward to adapt the argu-ment to the case where channel rates can be zero, but onlyfor bounded periods of time. Consider the potential function

. We first show that if the queues aresufficiently large, then this potential function has negativedrift for MaxWeight-Alg1. Recall thatindicates whether or not carrier serves user at time . If

, then


and otherwise

In both cases this implies

Since the arrival process is -admissible, the total arrivalsper user is upper-bounded by a function of , and .In particular

Similarly, the second term can be upper-bounded by a functionof , and . Let denote an upper bound of the firsttwo terms. To bound the third term, we note that for

Hence

Again, the second term of the above expression can be upper-bounded by a function of and . Let be this con-stant. To bound the third term, letfor given time slot and carrier . The definition of max weightimplies and for all . Therefore

Given and , let . In other words,. Therefore, given and , we have

This implies

Hence, when the queues are sufficiently large, i.e., larger than, the potential function decreases.

This implies stability for MaxWeight-Alg1.For algorithms MaxWeight-Alg2 and Alg4, stability follows

from a similar argument to the above and the fact that if, then each of these algorithms assigns each carrier to

user . Hence, in all cases

where is a function of , and .Similarly, if , then algorithm MaxWeight-Alg3

assigns carrier to user for some. Therefore, we once again have

VII. SIMULATIONS

We now present simulation work. Our focus is on the firstthree MaxWeight algorithms, as their simplicity is more likelyto lead to practical consequences, whereas we feel the signifi-cance of Alg4 is more theoretical. The question we address iswhether we are better off using an optimal algorithm, namelyMaxWeight-Alg1, for a not-so-good objective function oran approximate algorithm, for example MaxWeight-Alg2 orMaxWeight-Alg3, for a more sensible objective function. Ourfinding chooses the latter.

The algorithms are implemented in simple homegrown pro-grams written in Python. We use a field trace that representsmeasured channel conditions from a third-generation wirelesssystem as well as synthetic traces in which the channel ratesfluctuate around a mean value according to 3 km/h Rayleighfading. We assume a constant rate arrival model. The number ofcarriers is 40, and the number of users varies between 30 and 40.In all cases, MaxWeight-Alg2 and Alg3 have extremely similarperformance, and so we combine them onto a single plot. Bothof these algorithms significantly outperform MaxWeight-Alg1.This is due to the fact that MaxWeight-Alg1 is wasteful and often


Fig. 3. Simulated trace: total queue size. (Left) MaxWeight-Alg1. Mean:119440; 95%: 141900; Median: 118904. (Right) Alg2 and Alg3. Mean: 4275;95%: 8888; Median: 3736.

Fig. 4. Simulated trace: total queue size over all users per time step.(Left) MaxWeight-Alg1. Mean: 30247; 95%: 41328; Median: 29488. (Right)Alg2 and Alg3. Mean: 31; 95%: 176; Median: 0.

assigns more service to a user than it can actually use. See Figs. 3and 4 for total queue size plots under the simulated traces, andFig. 5 for plots under the field trace. The figure captions offersummary statistics for the mean, 95th percentile, and medianqueue sizes.

VIII. CONCLUSION

In this paper, we studied a variety of scheduling algorithmsfor time-slotted multicarrier wireless data systems. We pre-sented a set of algorithms that aim to emulate the MaxWeightalgorithm for the single-carrier case.

A number of immediate open problems remain. First, wewould like to know if it is possible to improve on the -approx-imation for objective (3). In particular, we would like to knowif algorithm MaxWeight-Alg3 has a better approximation ratiothan since, in the worst example that we can construct, theperformance of MaxWeight-Alg3 only differs from the optimumby a factor of . We would also like to know if there is a simple

Fig. 5. Field trace: total queue size over all users per time step.(Left) MaxWeight-Alg1. Mean: 140746; 95%: 167828; Median: 131448.(Right) Alg2 and Alg3. Mean: 226; 95%: 260; Median: 22.

Fig. 6. Simulated trace: delay averaged over packets that complete transmis-sion at each time step. (Left) MaxWeight-Alg1. (Right) Alg2 and Alg3.

algorithm that improves on the -approximation for objective(2). We feel that MaxWeight-Alg4 is probably too complex to beimplemented in practice.

There are also other longer term open problems such as de-riving delay bounds for scheduling problems in the multiusermulticarrier setting. The relationship between small queues andsmall delays is however intuitive, illustrated by Fig. 6, whosequeue-size counterpart is plotted in Fig. 3.

Since the preliminary version of this work appeared in ACMMobiCom in 2007, more work has been done on multicarrierscheduling, e.g., [10] presents an iterative longest-queue-firstalgorithm that aims to minimize buffer overflow probabilities.A number of new research problems stemming from multicar-rier wireless scheduling has been defined. One line of researchfocuses on accommodating a contiguity constraint, which is im-posed by the standards for the uplink of LTE. This constraintrequires that if multiple carriers are assigned to the same userduring a time slot, then these carriers have to be consecutive.Approximation algorithms for this problem are proposed in [8]and [20].


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Matthew Andrews received the B.A. degree (first class honors) in mathematicsfrom Oxford University, Oxford, U.K., in 1993, and the Ph.D. degree in theo-retical computer science from the Massachusetts Institute of Technology, Cam-bridge, in 1997.

He is a Distinguished Member of Technical Staff with the Mathematics ofNetworks and Systems Department, Bell Labs, Murray Hill, NJ. He holds arange of patents in the area of telecommunications. His research interests in-clude wireless resource allocation, packet scheduling, and approximation algo-rithms. His recent work includes algorithms for joint scheduling and congestioncontrol in ad hoc networks and complexity theoretic results on the hardness ofnetwork design.

Lisa Zhang received the B.A. degree (summa cum laude) in mathematics fromWellesley College, Wellesley, MA, in 1993, and the Ph.D. degree in theory ofcomputing from the Massachusetts Institute of Technology, Cambridge, in 1997.

She is a Member of Technical Staff with the Algorithms Research Group,Bell Labs, Murray Hill, NJ. She holds a range of patents in the area of telecom-munications. Her research area is algorithm design and analysis. Her researchbroadly concerns algorithmic and complexity issues of networking, with a focuson design and optimization, routing and scheduling protocols, and stability andquality-of-service analyses.

Dr. Zhang twice won the Bell Labs President’s Gold Award and the LucentChairman’s Award.

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