IEEE INFOCOM 2001 1

Opportunistic Scheduling with QoS Requirementsin Time-varying Wireless Channels

Xin Liu, Edwin K. P. Chong, and Ness B. Shroff

Abstract—In this paper, we present a framework for “op-portunistic scheduling” that exploits the variation of thewireless channel conditions to improve spectrum efficiency.The objective of the scheduling schemes in this paper is tomaximize the system performance in an opportunistic fash-ion, while satisfying various QoS requirements. We studythree types of QoS requirements: the first is fairness in re-source sharing, the second is a performance-based fairnessrequirement, and the third is that each user has a minimum-performance requirement. We provide optimal solutions,and discuss the advantages and disadvantages of the differ-ent scheduling schemes. We also demonstrate via simulationthat the schemes result in significant performance improve-ment compared with scheduling schemes that are not oppor-tunistic.

Keywords— Scheduling, wireless, resource allocation,time-varying channel, time-slotted system.

I. INTRODUCTION AND MOTIVATIONWIRELESS scheduling algorithms should be tailoredto fit the unique characteristics of wireless net-

works. On one hand, good scheduling schemes in wire-less networks should be able to schedule users “oppor-tunistically” to achieve higher utilization of wireless re-sources. (Here, the term “opportunistic” denotes the abil-ity to schedule users based on favorable channel condi-tions.) On the other hand, the potential to exploit higherdata throughput in an opportunistic way, when channelconditions permit, introduces the tradeoff between wire-less resource efficiency and level of satisfaction amongdifferent users. For example, allowing only users closeto the base station to transmit at high transmission powermay result in very high throughput, but sacrifices the trans-missions of other users. Such a scheme compromises theappealing feature of wireless networks—“anywhere, any-time” information access. In the emerging high-rate datawireless networks, there is an increasing demand for qual-

Xin Liu and Ness B. Shroff are with the School of Electrical andComputer Engineering, Purdue University, West Lafayette, IN 47907-1285. Email:fxinliu, shroffg@ecn.purdue.edu.

Edwin K. P. Chong is with the Dept. of Electrical and Computer Engi-neering, Colorado State University, Fort Collins, CO 80523; Tel: 970-491-7858, Fax: 970-491-2249, Email: echong@engr.colostate.edu.

This research is supported in part by the National Science Foundationthrough grants ECS-9501652, NCR-9624525, and ANI-9805441.

ity of service (QoS) provisioning. To address this prob-lem, we present here a framework for scheduling users inan opportunistic way. The objective is to improve wire-less resource efficiency by exploiting time-varying chan-nel conditions while at the same time control the level ofQoS among users. Under this framework, we study oppor-tunistic scheduling policies with different QoS constraints.The first two QoS requirements that we study are in factfairness requirements. The third QoS metric considered inthis paper is the minimum “performance” (e.g., data rate)a user receives. We also present optimal solutions for thestudied scheduling problems.

The authors of [1], [2] and [3] have studied wireless fairscheduling policies. They extend the scheduling policiesof wireline networks to wireless networks, and the newschemes provide various degrees of performance guaran-tees, including short-term and long-term fairness, as wellas short-term and long-term throughput bounds. A com-prehensive survey of these algorithms can be found in [4].However, these works model a channel as either “good”or “bad,” which may be too simple to characterize realisticwireless channels, especially for data services.

In [5], [6], the authors present a scheduling scheme forthe Qualcomm/HDR system. The scheduling scheme ex-ploits time-varying channel conditions while maintainingproportional fairness, as defined in [6], [7]. Their mea-sure of fairness is different from ours, as is their objectivefunction.

In [8], [9], the authors study scheduling algorithms forthe transmission of data to multiple users. Both delay andchannel conditions are taken into account (see Section IV-C).

The paper is organized as follows. In Section II, we in-troduce the system model. We consider a time-slotted sys-tem in which time is the resource to be shared among theusers. In Section III, we present three scheduling problemswith different QoS requirements. We provide an optimalsolution for each scheme in Section IV. We also discussthe advantages and disadvantages of the different schedul-ing formulations. In Section V, we provide simulation re-sults to illustrate the performance of the scheduling poli-cies. The conclusions are presented in Section VI.

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II. SYSTEM MODEL

We introduce our system model in this section. We con-sider a time-slotted system; i.e., time is the resource to beshared among all users. The system can have more thanone channel (frequency band), but at any given time, onlyone user can occupy a given channel within a cell. Here,we focus on the scheduling problem for any given channel.

As explained previously, channel conditions in wirelessnetworks are time-varying, and thus users experience time-varying performance. We use a stochastic model to capturethe time-varyingandchannel-condition-dependentperfor-mance of each user. Specifically, letfUki g be a stochasticprocess associated with useri, whereUki is the level of per-formance that would be experienced by useri if it is sched-uled to transmit at timek. The value ofUki measures the“worth” or “performance value” of time-slotk to the useri,and is in general a function of its channel condition. Usu-ally, the better the channel condition of useri, the largerthe value ofUki . We assume thatUki is nonnegative andbounded. Let ~Uk = (Uk1 ; � � � ; UkN ) be theperformancevectorat time-slotk, whereUki is the performance valueachieved by useri if time-slot k is assigned to useri. As-sume from now on thatf~Ukg is stationary. We use the no-tation ~U = (U1; � � � ; UN ), whereUi is a random variablerepresenting the performance value of useri at a generictime-slot. Note that the stationarity assumption does notpreclude correlations across users or across time. Exam-ples of the performance measure could be throughput orvalue of throughput minus the cost of power consumption.We assume throughout the paper that performance valuesfor different users arecomparable and additive.

Each of our scheduling scheme could be implemented inTDMA systems as well as in time-slotted CDMA systems(an example of the latter is the Qualcomm/HDR system).We consider both the uplink and the downlink of a wire-less network. In both cases, the base station serves as thescheduling agent. The scheduling scheme does the follow-ing: at the beginning of a time-slot, the scheduler (i.e., thebase station) decides which user should be assigned thetime-slot based on the performance values of the users atthat time-slot. For the uplink case, if a user is assigned atime-slot, the user will transmit in that time-slot. For thedownlink case, if a user is assigned a time-slot, the basestation will transmit to the user in that time-slot.

III. SCHEDULING PROBLEMS

A. Basic Framework

We present a framework for opportunistic schedulingthat maximizes the average system performance under cer-tain QoS requirements. Under this framework, we study

three different scheduling problems. In the first two prob-lems, the QoS requirements are fairness constraints. Inthe third problem the QoS requirement is a minimum-performance constraint. We present solutions for thesescheduling problems in Section IV.

Recall that~U is the performance vector of users, whichis a random vector representing the performance values ofusers at a generic time-slot. The generic scheduling prob-lem is: given~U , determine which user should be scheduledto transmit (in the given time-slot and under certain con-straints). We define apolicyQ to be a mapping from theperformance-vector space to the index setf1; 2; � � � ; Ng.Given ~U , the policyQ determines the user to be sched-uled: if Q(~U) = i, then useri should use the time-slot,and the system receives a performance “reward” ofUQ(~U)(i.e.,Ui). Hence,E(UQ(~U)) is the average system perfor-mance value associated with policyQ. Note that the policyQ is potentially “opportunistic” in the sense that it can useinformation on the performance vector~U to decide whichuser to schedule.

We are interested only in policies that satisfy specificQoS requirements. We say that a policyQ is feasible ifit satisfies the QoS constraints for all users. Our goal is tofind a feasible policyQ that maximizes the average systemperformanceE(UQ(~U)).B. Resource-Based Fairness Constraints

First, we describe the scheduling problem withresource-based fairness constraints. Because time is theresource shared among users, a natural fairness criterion isthat each user be given at least a certain share of the entireresource; i.e., time.

Let ri denote the minimum time-fraction that should beassigned to useri, whereri � 0,

PNi=1 ri � 1, andN isthe number of users in the cell. Here, we assume that theri’s are predetermined and serve as prespecified fairnessconstraints; i.e., on average, at least a fractionri of the en-tire time should be scheduled to useri. The time-fractionassignment scheme then dictates the minimum fraction oftime that a user should transmit on the channel, which canbe determined by the user’s class, the price the user iswilling to pay for the wireless service, or the user’s cur-rent channel conditions [10]. The scheduling algorithmthen decides which time-slot should be assigned to whichuser, given the minimum time-fraction requirement. Ourgoal is to find a scheduling policyQ that exploits the time-varying channel conditions to maximize the total expectedsystem performance while satisfying the resource-sharingconstraint. The problem can be stated formally as follows:maximizeQ2� E �UQ(~U)�

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subject to PfQ(~U ) = ig � ri; i = 1; � � � ; N; (1)

where� is the set of all scheduling policies,ri � 0 for alli, andPNi=1 ri � 1.

In a previous paper [10], we studied a special caseof the resource-based fairness scheduling scheme, wherePNi=1 ri = 1, which means that the inequality constraintsin Eq. (1) are actually equality constraints:PfQ(~U ) =ig = ri for all i. The current problem is more general,and allows more flexibility in resource sharing—it allowsfor a minimum amount of fairness in the system. Note that� := Pi ri � 1 is a tuning parameter such that the smallerthe value of�, the less restrictive the fairness constraint,and the greater the opportunity to improve the system per-formance.

C. Performance-Based Fairness Constraints

We now describe a performance-based fairness schedul-ing problem. Similar to the problem defined in Eq. (1),we want to maximize the total expected system perfor-mance under a fairness constraint. However, the fairnessconstraint in this problem is that each user gets at least acertain share of the total expected performance. Formally,the problem is stated as:maximizeQ2� E �UQ(~U)�

subject to E �Ui1fQ(~U)=ig� � aiE �UQ(~U)� ;i = 1; 2; � � � ; N; (2)

where� is the set of all policies,ai is the minimum frac-tion of the overall average performance required by useri,ai � 0 for all i,Pi ai � 1, andE(Ui1fQ(~U)=ig) is the av-erage performance value of useri given policyQ. Theai’sare predetermined fairness parameters, and�0 = Pi ai isa tuning parameter—the smaller its value, the greater theflexibility in making scheduling decisions.

Any policyQ that satisfies the above fairness constrainthas the property thatai1� �0 + aj � E �Ui1fQ(~U)=ig�E �Uj1fQ(~U)=jg� � 1� �0 + aiaj ;for i; j = 1; 2; � � � ; N: In other words, the performance-based fairness constraint controls the maximum discrep-ancy of performance values among users.

The constraint given in Eq. (2) ensures that a user isgiven at least a certain share of the total performance, andis hence potentially fairer than the resource-based fairnessconstraint given by Eq. (1). However, there is also a disad-vantage of such a scheduling scheme: a user experiencing

poor channel conditions could have a detrimental impacton the overall system performance. By observing the con-straint in Eq. (2), we haveE �UQ(~U)� � E �Ui1fQ(~U)=ig�ai � E(Ui)ai :Hence, if a user is experiencing poor channel conditions(a very small value ofE(Ui)) and has a large value ofai,then it could compromise the overall system performancesignificantly because a substantial portion of the total time-slots may have to be allocated to this user in order to meetits fairness requirement. In contrast, in the resource-basedfairness scheme, a user in a poor condition would at mostsacrificeri fraction of the overall resources.

To alleviate this potential problem, we could adjust thevalues of theai’s dynamically. Let�i = E(1Q(~U)=i); i.e.,�i is the portion of resource (time-slots) allocated to userigiven policyQ. LetRe be the efficiency ratio, whereRe = �iE �UQ(~U)�E �Ui1fQ(~U)=ig� :If the value ofRe is very large, then useri is not an “effi-cient” user of the scarce spectrum. In other words, it usesa large portion�i of the resource to get a small portionof the overall performanceE(Ui1fQ(~U)=ig)=E(UQ(~U )). Astraightforward remedy is to set up a threshold�. IfRe � �;then we decrease the values ofai; i.e., the minimum por-tion of the system performance promised to the user.

This scheme provides fairness in terms of performancevalues, which, to some extent, parallels the concept ofweighted fair queueing used in wireline networks. The dif-ference is that the overall capacity here (i.e.,E(UQ(~U))) isnot fixed; it depends on channel conditions, theai values,and the scheduling policy. In practice, to convince a userthat it is getting itsfair share of the total performanceisnot as easy as in the case where fairness is based on actualresources (e.g., fraction of time-slots).

The two scheduling problems discussed thus far provideusers with different fairness guarantees. However, whilethey satisfy a relative measure of performance (i.e., fair-ness), they do not consider any absolute measure of per-formance. This motivates our study of the next problem.

D. Minimum-Performance Guarantees

In this section, we describe a scheduling problem wherethe average system performance is maximized subject to

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meeting each user’s minimum-performance requirement;i.e., QoS is defined as the minimum-performance con-straint. The problem is formulated as:maximizeQ E �UQ(~U)�

subject to E �Ui1fQ(~U)=ig� � Ci;i = 1; � � � ; N; (3)

where E(Ui1fQ(~U)=ig) is the average performance ofuseri, andCi � 0 is the minimum performance require-ment of useri. We call the vector~C = fC1; C2; � � � ; CNgtherequirement vector.

The problem defined in Eq. (3) has two aspects:� Is ~C a feasible requirement vector; i.e., does there exista policyQ such thatE(Ui1fQ(~U)=ig) � Ci for all i?� If ~C is a feasible requirement vector, which policy max-imizes the overall performance under the given QoS con-straint, and how does one implement it?

Unlike our previous two problems, the QoS constraintin this problem is defined as a minimum-performance re-quirement instead of a fairness constraint. Hence, the for-mulation here offers users a more “direct” service guar-antee. For example, if the performance measure is de-fined as the data-rate, then each user is guaranteed a min-imum data-rate, which may be more important to usersthan knowing that a minimum amount of resource willbe assigned to it. While appealing to users, providingminimum-performance guarantees also raises the feasibil-ity issue—can the system satisfy the performance require-ments for all users? Note that feasibility is not a concernin the fairness-based constraints. In the resource-basedfairness scheduling problem defined in Eq. (1), as long asPi ri � 1, the system is feasible. In the performance-based fairness scheduling problem defined in Eq. (2),Pi ai � 1 is the feasibility constraint. Both of them areeasy to verify. However, there is no easy way in general todetermine whether a given~C is feasible or not.

On the other hand, there are some natural instances of~Cwhere feasibility is not a problem. For example, letCi =�iE(Ui), where�i � 0 for all i and

P �i � 1. Such asetting can be satisfied by a round-robin scheduling policy,and hence is feasible for opportunistic scheduling policies.

IV. OPTIMAL SCHEDULING POLICIES

In Section III, we have discussed three scheduling prob-lems. The objectives are to maximize the overall sys-tem performance under certain QoS requirements for eachuser. In this section, we present opportunistic schedul-ing policies that solve the problems defined in Section III.The scheduling policies are optimal in the sense that they

maximize the overall system performance under the cor-responding QoS constraints. We shall mention that thereare some parameters involved in each policy. In general,these parameters depend on the distribution function of~U ,which is typically not knowna priori. In Section IV-D,we give a simple description on how to estimate these pa-rameters, and refer readers to [11] for a detailed discus-sion. Furthermore, due to page limitations, we have omit-ted the proofs here, and refer readers to [11] for proofs ofall propositions in this section.

A. Resource-Based Fairness Constraints

Recall that the scheduling problem defined in Eq. (1)is to maximize the expected system performanceE(UQ(~U)=i) under the fairness constraint. We define a pol-icy Q� as follows:Q�(~U ) = argmaxi (Ui + v�i ); (4)

where thev�i ’s are chosen such that:1. mini(v�i ) = 02. PfQ�(~U ) = ig � ri for all i3. For alli; if PfQ�(~U) = ig > ri; thenv�i = 0:Proposition 1 The policyQ� is a solution to the problemdefined in Eq. (1); i.e., it maximizes the average systemperformance under the resource-based fairness constraint.

We can think of the parameter~v� in Eq. (4) as an “off-set” used to satisfy the fairness constraint. Consider thecase where we want to maximize the overall performancewithout any QoS constraints. It is straightforward to showthat we should always choose the “best” user (i.e., theuser with the maximum performance value) to transmitin a generic time-slot; i.e.,Q(~U) = argmaxUi. How-ever, such a scheme may be unfair to certain users. Hence,to satisfy the fairness constraint, the scheduling policyschedules the “relatively-best” user to transmit. Useri is“relatively-best” ifUi + v�i � Uj + v�j for all j. If v�i > 0,then useri is an “unfortunate” user; i.e., the channel con-ditions it receives is quite poor. Hence, it has to take ad-vantage of some other users (i.e., users withv�j = 0) to sat-isfy its fairness constraint. Thus, to maximize the overallsystem performance, we can only give the “unfortunate”users the amount of resource equivalent to their minimumrequirements. WhenPfQ�(~U ) = jg > rj for userj, theuser gets more than its minimum requirement—this usercannot take advantage of other users; i.e.,v�j = 0.

The policyQ� maximizes the average system perfor-mance even if the users’ performance values are arbitrarilycorrelated, both in time and across users. The following

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proposition establishes, under a more restrictive assump-tion, that each user can be guaranteed a minimum perfor-mance level given the distribution of~U .

Proposition 2 If the performance valuesUi, i =1; � � � ; N , are independent, then for alli,E �Ui1fQ�(~U)=ig� � PfQ�(~U) = igE(Ui) � riE(Ui):Note thatE(Ui1fQ�(~U)=ig) is the average performance

value of useri using our opportunistic scheduling policy,andPfQ�(~U) = igE(Ui) is the average performance ofuseri when using a non-opportunistic scheduling schemegiven thatPfQ�(~U ) = ig portion of the resource (i.e.,time) is assigned to useri. (Recall that a non-opportunisticscheduling policy is one that does not use information on~U to decide which user to transmit.)

The above proposition says that: if the users’ per-formance values are independent, then the average per-formance ofeach userin our opportunistic schedulingscheme will be at least equal to the performance of thecorresponding user of any non-opportunistic schedulingscheme (i.e.,PfQ(~U ) = igE(Ui)). Furthermore, becausePfQ(~U ) = ig � ri, each user gets a guaranteed min-imum performance ofriE(Ui). This result is intuitivelyappealing. When a user is experiencing good channel con-ditions, it has a higher chance to have a maximum value ofUi + v�i among all users, and thus be chosen to transmit.When a user is experiencing bad channel conditions, it hasless opportunity to be the relatively-best user and then tobe scheduled. Hence, a user tends to transmit more oftenunder favorable conditions, resulting in performance im-provement for each user. In this sense, the opportunisticscheduling policy gives all the users the chance to improvetheir expected performance. Of course, different users mayexperience different amounts of improvement. In general,the larger the variability of a user’s performance value, thegreater the improvement.

B. Performance-Based Fairness Constraints

Recall that the scheduling problem defined in Eq. (2) isto maximize system performance under the performance-based fairness constraints. We define a policyQ� takingthe form:Q�(~U ) = argmaxi ((�+ ��i )Ui) ; (5)

where� = 1�PNi=1 ai��i , and the��i ’s are chosen so that:

1. mini(��i ) = 02. E �Ui1fQ�(~U)=ig� � aiE �UQ�(~U)� for all i

3. For all i, if E �Ui1fQ�(~U)=ig� > aiE �UQ�(~U)�, then��i = 0 .

Proposition 3 The policyQ� defined in Eq. (5) is a so-lution to the problem defined in Eq. (2); i.e., it maximizesthe average system performance under the performance-based fairness constraint.

Similar to ~v� in the last section, the parameter~�� inEq. (5) can be considered an “offset” used to satisfythe performance-based fairness constraint. The optimalscheduling policy always chooses the relatively-best userto transmit. In this case, the useri is relatively-best if(�+ ��i )Ui = maxj ��+ ��j �Uj :Note that� is constant for all users. As before, if��i > 0,then useri is an “unfortunate” user, and its average per-formance value equals the minimum requirement; i.e.,E(Ui1fQ�(~U)=ig) = aiE(UQ�(~U)).C. Minimum-Performance Guarantees

Recall that the scheduling problem defined in Eq. (3) isto maximize the system performance given that each userobtains a minimum average performance value. The prob-lem has two aspects: (1) whether the requirement vectoris feasible; and (2) how to find the optimal policy whenthe requirement vector is feasible. In this section, we firstpresent the optimal scheduling policy assuming feasibility,and we then discuss the feasibility problem.

Suppose ~C = fC1; C2; � � � ; CNg is feasible; i.e.,there exists some scheduling policyQ such thatE(Ui1fQ(~U)=ig) � Ci for all i. We then define a policyQ� by: Q�(~U ) = argmaxi (��iUi); (6)

where the��i ’s are chosen so that:

1. mini(��i ) = 12. E �Ui1fQ�(~U)=ig� � Ci for all i3. For alli, if E �Ui1fQ�(~U)=ig� > Ci, then��i = 1:Proposition 4 The policyQ� defined in Eq. (6) is a solu-tion to the problem defined in Eq. (3); i.e., it maximizesthe average system performance under the minimum-performance constraint.

Note that the parameter~� “scales” the performance val-ues of users, and the scheduling policy schedules therelatively-best user, where useri is relatively-best if��iUi = maxj ��jUj :

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If the scale factor for a user is larger than 1, then the useris an “unfortunate” user, and it is only granted an averageperformance value that equals its minimum performancerequirement.

In [8], [9], the authors study scheduling algorithmswhere both delay and channel conditions are takeninto account. Roughly speaking, the algorithm is:argmax biWiti, whereWi is the head-of-the-line packetdelay for queuei, ti is the channel capacity, andbi is someconstant. Furthermore, they have the following result (as-suming there is a finite set of channel states): to maximizethe system throughput with minimum-throughput require-ments, there exists some constant i, such that one shouldchoose a user with the maximum value of iti. However,there is no discussion on how to obtain i’s, how to breakties, or how feasibility plays a role. We will address themhere.

Our opportunistic scheduling policy dominates non-opportunistic policies in the following sense: Considera non-opportunistic scheduling policy (one that does notuse information on~U ) in which useri shares a portion�i of the resource (time-slots), where

Pi �i = 1, anduser i gets an average performance value�iE(Ui). LetCi = �iE(Ui) for all i. Then~C is feasible, and the oppor-tunistic scheduling policy always provides “no-loss” per-formance values for each user relative to that of the non-opportunistic scheduling policy, assuming that the signal-ing cost is negligible.

The ability to provide a specific performance guaranteeis an advantage of this scheme. However, to satisfy sucha constraint introduces the question of feasibility. In thefollowing, we discuss the feasibility problem and how todetermine the feasibility region of our scheduling policy.The feasibility region of a policyQ is the set of require-ment vectors that are feasible under the policyQ.

Proposition 5 The feasible region of our opportunisticscheduling policy isconvexand contains the feasibility re-gions of all policies.

The feasibility region of our scheduling policy is deter-mined by the distribution of~U . In general, there is noclosed-form expression for the feasibility region even ifthe distribution function of~U is known. The distributionof ~U depends on the user’s channel condition, its mobility,and the form of the performance value function. It is prac-tically impossible to know the distribution of~U a priori inthe system. Hence, we estimate the feasibility region usingsample paths; i.e., the realization offUki g. Convexity isan important feature in determining whether a requirementvector is feasible. Due to page limitations, we refer readersto [11] for further discussion on how to estimate feasibil-

ity regions and the impact of this opportunistic schedulingscheme.

D. Discussions on the Scheduling Schemes

In this paper, we have described three different schedul-ing problems and provided their solutions. The objectivesof the scheduling problems are similar: to improve spec-trum efficiency while maintaining certain levels of QoS foreach user using opportunistic scheduling algorithms. Wehave studied two scheduling problems with fairness con-straints and one with minimum-performance constraint.The solutions to these scheduling problems have certainsimilarities—all the schemes choose the “relatively-best”user to transmit. Although in different scheduling policies,“relatively-best” has different meanings, the approach is touse offset/scaling to satisfy the QoS constraint for users.When a user is “unfortunate,” i.e., it has to take advan-tage of other users to satisfy its QoS constraint (in termsof fairness or performance value), the user does not getmore than its minimum requirement. This is done in orderto increase the system performance.

In general, the larger the number of users sharing thesame channel, or the larger the variance of~U , the largerthe “opportunistic” scheduling gain compared with non-opportunistic scheduling policies. Furthermore, the morerestrictive the QoS constraints, the less flexibility we havein decision making.

In this paper, we consider fairness in two ways:resource-based fairness and performance-based fairness.In wireline networks, when a certain amount of resourceis assigned to a user, it is equivalent to granting the usera certain amount of throughput/performance value. How-ever, the situation is different in wireless networks, wherethe amount of resource and the performance value are notdeterministically related (though correlated). This is thereason we consider two types of fairness. Both schemeshave flexible structures that allow the system to decide thelevel of fairness among users. The disadvantage of theperformance-based fairness is that we have to be carefulwith the requirement of each user to avoid the situationwhere one unfortunate user takes up most the resource inorder to satisfy its QoS requirements.

Max-min fairness can be considered as a special caseof the fairness constraint presented here. Roughly speak-ing, max-min fairness means that any user is entitled to asmuch performance/resource as any other user. Max-minfairness can be applied in two different ways. First, if weapply the max-min fairness criterion to the performance-based fairness scheduling scheme, then the system shouldmaximize the minimum performance of users. This isequivalent to settingai = 1=N for all i; i.e., each user

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obtains the same performance value. However, it is worthnoting that this max-min fairness in terms of performancevalues may not be a good fairness criterion in wireless net-works due to the fact that a user experiencing poor channelconditions can substantially compromise the overall sys-tem performance by consuming a significant amount ofresources. Second, if max-min fairness is applied to theresource-based fairness scheduling scheme, then we haveri = 1=N for all i (N be the number of users sharingthe same channel); i.e., each user is entitled to the sameamount of resource.

Feasibility is not a concern in the scheduling problemswith fairness constraints. The feasibility issue arises in theproblem with minimum-performance constraints, whichprovides a more direct service performance guarantee tousers.

Note that the objectives and constraints in all the prob-lem formulations are expressed in terms ofexpectation;i.e., a “long-term” performance measure. There is no guar-antee of “short-term” performance. In [10], we discussan extension to improve short-term performance—a sim-ilar procedure can be used here with all the schedulingschemes. The basic idea is as follows: if a user is be-hind its share, we increase its probability of transmission.In general, when users’ performance values have strongcorrelation across time, the short-term performance is bad.The stricter the short-term performance requirement, theless the opportunity to exploit time-varying channel con-ditions, and the less the performance improvements. Wealso refer interested readers to [8], [9], where queueingdelays are considered.

Thus far, we have studied three scheduling schemes anddiscussed the attributes of each scheme. Different schemesmay be suitable for different scenarios. For example, if theservice provider wants to build a simple wireless networkwith pricing schemes, the resource-based fairness schedul-ing scheme might be a good choice. The resource-basedfairness scheduling scheme is simple and flexible. There isno feasibility problem. The amount of resource consumedby a user implies the minimum performance the user gets(with technical assumptions). The resource consumed bya user can be directly connected with the price the usershould pay.

The minimum-performance guarantee scheme givesusers a direct performance guarantee, but involves the fea-sibility problem. If the service provider wants to builda network that provides data-rate guarantees, then thisscheme is a good choice. Note that providing service per-formance guarantee is a challenging problem in both wire-less and wireline networks.

In each of the opportunistic scheduling policies, there

are some parameters involved. The values of these pa-rameters are determined by the distribution of~U and theQoS constraints. In practice, the distribution of~U is un-known, and hence we need to estimate the parameters.This can be done using stochastic approximation algo-rithms, as in [10], [11]. Here, we use the resource-basedfairness scheduling scheme to give a simple descriptionof our parameter estimation procedure. Recall that ouropportunistic scheduling policy is given byQ�(~U) =argmaxi(Ui+v�i ), wherev�i s should satisfy the conditionsdiscussed in Section IV-A. We can rewrite these conditionsas:1. PfQ~v(~U) = ig � ri for all i2. (PfQ~v�(~U ) = ig � ri)(vi �minj vj) = 0, for all i.We need to estimate the parametersv�i , i = 1; � � � ; N � 1.Let us consider a scheduling algorithmQ~v:Q~v(~U) = argmaxi Ui + vi:If vi = 0 for all i, then, clearly, the scheduling policymaximizes the system performance. If this schedulingpolicy meets the above constraints, we can set~v� = ~0.In general,~v = ~0 does not meet the constraints. Fur-thermore, if we increase the value ofvi, we increase thevalue ofPfQ~v(~U) = ig while decreasing the values ofPfQ~v(~U ) = jg for j 6= i. Intuitively, if PfQ~v(~U) =ig < ri, we increase the value ofvi, and ifPfQ~v(~U) =ig > ri and vi � minj vj , we decrease the value ofvi. By making such adjustments, we hope to find the~v�eventually such that the above two conditions are satisfied.Stochastic approximation algorithms for each scheme isdiscussed in [11].

V. SIMULATION RESULTS

In this section, we present simulation results of ourscheduling schemes. Each policy dynamically decideswhich user should be scheduled to transmit in a time-slotbased on users’ current performance values and the QoSconstraint.

For comparison, we also simulate two other schedul-ing policies. The first is round-robin, a non-opportunisticscheduling policy, which schedules (active) users follow-ing a predetermined order. This scheduling scheme servesas a benchmark of the system performance. As a compar-ison, we measure how much gain the system can obtainusing opportunistic scheduling policies. The second is agreedy scheduling policy, which always selects the userwith the maximum performance at a generic time-slot totransmit. This policy achieves the maximum performancethe system can obtain without any constraints, thereby pro-viding an upper bound on the achievable performance. In

IEEE INFOCOM 2001 8

the following, we first describe the simulation model ofa cellular cell, then show the simulation results for eachscheduling policy using the cellular model.

A. Cellular Model

We consider a multi-cell system consisting of a centerhexagonal cell surrounded by hexagonal cells of the samesize. The base station is at the center of each cell, andsimple omni-directional antennas are used by mobiles andbase stations. We focus on the downlink of the center cell.The frequency reuse factor is 3, and the co-channel inter-ference from the six first-ring neighboring cells are takeninto account. We focus on one frequency band, which isshared by 10 users in the central cell. The scheduling pol-icy decides which user should transmit in this frequencyband at each time-slot. The users have exponentially dis-tributed “on” and “off” periods.

Users move with random speed and direction in the cell.They perceive time-varying and location-dependent chan-nel gains. The channel gains of the users are mutuallyindependent random processes determined by the sum oftwo terms: one due to path (distance) loss and the other toshadowing. We adopt the path-loss model (Lee’s model)and the slow log-normal shadowing model in [12]. To beconservative, we ignore the effects of fast multi-path fad-ing in the simulation. If fast fading could be tracked accu-rately, our scheme would provide even higher performanceimprovements than shown here. The mobility model, thepropagation model, and the parameters of the simulationare discussed in detail in [11].

Figure 1 shows the forms of the performance valuesused by different users (there are 10 users in the system).The performance values of users1, 5, and 8 are step-functions of SINR. The performance values of users2, 6,and 9 are linear functions of SINR (in dB). Users3, 4,7, and10 have performance values that are S-shape func-tions of SINR. Furthermore, the 10 users are divided intothree “distance” groups; i.e., when a user becomes active,its distance from the base station is fixed, depending onwhich group it belongs to. Users1–4 belong to the “far”group. Specifically, when the user becomes active, its dis-tance from the base station is0:9R, whereR is the ra-dius of the cell. Users5–7 belong to the “middle” group;their starting distance from the base station is0:5R. Users8–10 belong to the “near” group with a starting distance0:2R. Note that when the user is active, it moves aroundin the cell freely and randomly. However, a user in a “near”group has a much larger chance to be close to the base sta-tion than a user in a “far” group. Hence, we can studyhow the distance from the base station effects users’ per-formance in different scheduling schemes.

0 5 10 15 20 25 30 35 400

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Fig. 1. Users’ performance as a function of SINR.

The system performs the same procedure at every time-slot. Basically, the users measure their channel conditions,calculate their performance values, and then send the in-formation to the base station. The base station decides theuser to be scheduled using the corresponding opportunisticscheduling policy, based on the information of the users’performance values. The base station then updates the pa-rameters used in the scheduling policy. More details of thesimulation procedure are explored in [11].

B. Resource-Based Fairness Constraints

In this scheduling scheme, each user requires a min-imum portion of the resource. For the simulations, weset all users to have the same minimum resource require-ment. Specifically,N is the number of active users shar-ing the channel in the central cell, then each active userhas a resource requirementri = 1=(N + 3). In this case,Pi2A ri = N=(N + 3) < 1, whereA is the set of activeusers. Hence, the system has the freedom to assign a por-tion of the resource (3=(N +3)) to some “fortunate” users(beyond their minimum requirements) to further improvethe system performance, as discussed in Section IV-A.

Figures 2 and 3 show the results of our simulation ex-periments. In both figures, the x-axis represents the users’IDs. In Figure 2, the y-axis represents the portion of re-source each user gets in the different scheduling policies.The first bar is the result of round-robin, where the re-source is equally shared by all active users. Note that theamounts of resource consumed by different users are notequivalent because different users are active at differenttimes. For example, if there are 2 users active during atime period, then each user gets50% of the time-slots ofthis time period. However, if there are 10 users active,then each user only gets10%. The second bar shows theportion of resource used by users in our resource-basedfairness scheduling policy, while the third bar shows the

IEEE INFOCOM 2001 9

minimum requirements of users. Note that the second baris higher than the third bar for all the users. Hence, ourscheduling policy satisfies the fairness requirements of allusers. Because users 9 and 10 are most likely to be closeto the base station and have large performance values, theyare the “fortunate” users in the system. It is not surpris-ing that they get a lot more resource than their minimum-requirements. The rightmost bars represent the greedyscheduling scheme that always chooses the user with thelargest performance value to transmit. Note that in thisscheme, users 1, 2, 3, 4, 5, and 8 get very little or almostzero resource while users 9 and 10 have a very large share.As expected, the greedy algorithm is biased very heavilytowards the users close to the base station with higher per-formance values.

Figure 3 shows the average performance obtained byusers in different scheduling policies. The first bar repre-sents round-robin, the second bar represents our schedul-ing policy, and the third bar is the greedy algorithm. Notethat in the opportunistic scheduling scheme, all users ex-cept users5 and 8 obtain higher average performance val-ues than in the round-robin scheme. Moreover, amongthese users with better performance, users1, 2, 3, 4 and6actually consumes less resource compared to round-robin.Because in the opportunistic scheduling scheme, users aremore likely to be chosen to transmit while experiencinggood channel conditions, they (e.g., users1, 2, 3, 4 and6 in this simulation) may consume less resource but stillget better average performance values compared to that ofround-robin. To explain why users 5 and 8 do not have ahigher performance value, recall that both users 5 and 8are non-elastic users, whose performance values are step-functions of the SINR. Furthermore, user 5 belongs to the“middle” distance group, and user 8 is in the “near” group.Most of the time, they experience SINR values that arehigher than the threshold in the step-functions. Hence,there is little chance to improve their performance op-portunistically. Compared with round-robin, because theyconsume less resource, these two users get smaller perfor-mance values.

Last, the overall system performance is improved by58% in our scheduling scheme while the greedy algorithmhas a112% performance improvement compared withround-robin. Of course, the greedy algorithm achieves thisperformance improvements by violating the QoS require-ments of some of the users. In summary, the simulationresults show that the opportunistic scheduling policy cansatisfy the fairness constraint with significant system per-formance gains.

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Fig. 2. Portion of resource shared by users in the resource-basedfairness scheduling simulation.

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Fig. 3. Average performance value in the resource-based fair-ness scheduling simulation.

C. Performance-Based Fairness Constraints

In this section, we show results for the performance-based fairness scheduling scheme. In the simulation, weset the performance requirement of users as:~a = [0:02 0:08 0:05 0:05 0:02 0:1 0:1 0:02 0:1 0:1℄;where

Pi ai = 64%. Note that in this setting, users 1,5, and 8 (i.e., the non-elastic users) have relatively lowpercentage requirements. Because these users have lowand non-elastic performance values, to achieve the sameamount of performance as other users, they require a largerportion of the resource. Hence, we assign these users lowrequirements to prevent them from using up too much re-source in the system. Simulation results show that the per-formance values achieved by the users are comparable tothat of the round-robin scheme although we do not haveguarantees that this scheme outperforms that of round-robin for each user.

IEEE INFOCOM 2001 10

Figure 4 shows the average performance values of theusers in different scheduling schemes. The x-axis repre-sents the users’ IDs, and the y-axis is the average per-formance value. In the figure, the first bar indicates theaverage performance values of users via the round-robinscheduling scheme. The second bar is the minimum per-formance value the user should get, which is calculated as:aiPTk=1 UkQ(~Uk)1fuseri is activegPTk=1 1fuseri is activeg ;where T is the total number of time-slots simulated,T =1,000,000. The denominator is the amount of timethe user is active, whereas the numerator is the amount ofsystem performance when the user is active; i.e., a userrequires a share of the system performance only when itis active. The third bar indicates the performance valuea user obtained from our scheduling policy. We can seein Figure 4 that all users obtain performance values largerthan their minimum requirements. The rightmost bar is theresult of the greedy algorithm. Compared with the round-robin scheme, our scheduling policy improves the overallsystem performance by50% while the greedy schedulinghas an improvement of110%.

Figure 5 shows the average amount of resource con-sumed by different users. The first bar represents round-robin scheduling policy, the second bar represents ourscheduling policy, and the third bar is the greedy schedul-ing policy.

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Fig. 4. Average performance value in the performance-basedfairness scheduling simulation.

D. Minimum-Performance Guarantees

In the following, we show simulation results forthe opportunistic scheduling scheme with minimum-performance guarantees. First, we run the simulation for

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Fig. 5. Portion of resource shared by users in the performance-based fairness scheduling simulation.

1,000,000 time-slots using the round-robin scheduling pol-icy, where the resource is equally distributed among allusers, and active users are scheduled in a predeterminedorder. Then we get an average performance value for eachuser. We use it as the minimum-performance requirementfor each user. Then we run a simulation using the oppor-tunistic scheduling policy, the round-robin policy, and thegreedy scheduling policy.

Figure 6 shows the average performance values of usersresulting from the different scheduling policies. Thefirst bar indicates the average performance values usingthe round-robin scheduling policy, the second bar is theminimum-performance requirement of a user, the third barindicates the result from our scheduling policy, and therightmost bar is that of the greedy algorithm. Note that ourscheduling policy outperforms the round-robin policy uni-formly, which illustrates the “no-loss” situation discussedearlier in Section IV-C. Compared with the round-robin,our scheduling policy improves the overall system perfor-mance by51% while the greedy scheduling has an im-provement of109%. Similar to the previous simulationresults, the greedy algorithm results in the highest over-all performance value at the cost of the extreme unfairnessamong users.

Figure 7 shows the amount of resource consumed byeach user in different scheduling policies. The first bar rep-resents round-robin, the second bar represents our schedul-ing policy, and the third bar is the greedy schedulingscheme.

In summary, the simulations show that using ourscheduling policies, the system can achieve significantperformance gains while satisfying the QoS constraints.In the simulations with fairness constraints, there is noguarantee that every user performs better than using the

IEEE INFOCOM 2001 11

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Fig. 6. Average performance value in the minimum-performance guarantee scheduling simulation.

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Fig. 7. Portion of resource shared by users in the minimum-performance guarantee scheduling simulation.

round-robin policy. In the simulations with the minimum-performance requirement, we set the requirement to bethe performance value obtained from round-robin; conse-quently, all users perform better using our policy than thatin round-robin. In all the simulations, the greedy scheme,as expected, has the best performance at the cost of ex-treme unfairness among users.

VI. CONCLUSIONS

Opportunistic scheduling is a way to improve spectrumefficiency by exploiting time-varying channel conditions.In this paper, we present a framework for opportunisticscheduling—to maximize the average system performancevalue by exploiting variations of the channel conditionswhile satisfying certain QoS constraints. The frameworkprovides the flexibility to study a variety of opportunisticscheduling problems. We discuss three different schedul-ing problems: to maximize the system performance with

a resource-based fairness constraint, a performance-basedfairness constraint, and a minimum-performance require-ment for each user. We discuss the properties of eachscheduling problem, and provide optimal solutions to theseproblems. Different scheduling schemes may be suitablefor different application scenarios. Furthermore, althoughnot discussed in this paper, the framework for opportunis-tic scheduling can also cover cases where there are differ-ent constraints from different users. For example, someusers may have resource requirements while other userscan have a minimum-performance requirements. In suchscenarios, similar optimal solutions can be provided.

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[10] X. Liu, E. K. P. Chong, and N. B. Shroff, “Transmission schedul-ing for efficient wireless network utilization,” inIEEE Infocom2001, (Alaska), April 2001. An extended version of this paper isin press in IEEE Journal on Selected Areas in Communications.http://shay.ecn.purdue.edu/�xinliu/papers.html.

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