Combinatorial Reverse Auction based Scheduling in Multi-rate Wireless Systems

Combinatorial Reverse Auction-BasedScheduling in Multirate Wireless Systems

Sourav Pal, Student Member, IEEE, Sumantra R. Kundu, Student Member, IEEE,

Mainak Chatterjee, Member, IEEE, and Sajal K. Das, Member, IEEE

Abstract—Opportunistic scheduling algorithms are effective in exploiting channel variations and maximizing system throughput in

multirate wireless networks. However, most scheduling algorithms ignore the per-user quality-of-service (QoS) requirements and try to

allocate resources (for example, the time slots) among multiple users. This leads to a phenomenon commonly referred to as the

exposure problem, wherein the algorithms fail to satisfy the minimum slot requirements of the users due to substitutability and

complementarity requirements of user slots. To eliminate this exposure problem, we propose a novel scheduling algorithm based on

two-phase combinatorial reverse auction, with the primary objective of maximizing the number of satisfied users in the system. We also

consider maximizing the system throughput as a secondary objective. In the proposed scheme, multiple users bid for the required

number of time slots and the allocations are done to satisfy the two objectives in a sequential manner. We provide an approximate

solution to the proposed scheduling problem, which is NP-complete. The proposed algorithm has an approximation ratio of ð1þ logmÞwith respect to the optimal solution, where m is the number of slots in a schedule cycle. Simulation results are provided to compare the

proposed scheduling algorithm with other competitive schemes.

Index Terms—Scheduling, multirate wireless system, reverse auction, performance optimization.

Ç

1 INTRODUCTION

THE concept of opportunistic scheduling in wireless net-works was first introduced in [23]. The basic idea is to

continuously monitor the uncertainty of the underlyingwireless channel and take decisions opportunistically so asto optimize the objective functions under consideration.Extensive research has been conducted with varyingobjectives such as maximizing the system throughput [24],maintaining both long and short-term fairness among users[26], [27], and maximizing the user utility [23]. In general,the goal has been to maximize a concave utility functionrepresenting the specified objective function. Unfortunately,such concave functions fail to capture the importance of thetimelineness of decision making in user scheduling.

On the other hand, the next-generation multirate wire-

less data networks, such as Evolution-Data Optimized

(1xEV-DO) [1], High Data Rate (HDR) [2], and Enhanced

Data Rates for Global Evolution (EDGE) [3], promise to

provide data services and applications with strict timing

constraints. Examples of such applications include stream-

ing multimedia, voice over Internet Protocol (VoIP), instant

messaging (IM), and real-time videoconferencing, all of

which demand that packets be delivered within certaindelay bounds so as to comply with the application-level

quality of service (QoS). We justify that time constraintscheduling is a necessity for delay-sensitive applications byexplaining the timing requirements of VoIP applications.According to the International Telecommunication Union(ITU-T) G.114 specifications [10], for good and pleasing

voice quality, the end-to-end delay for both the forward andreverse paths should not be more than 150 ms. This delay iscontributed by various sources:

1. the voice coder, with a processing delay of 10 ms,2. the bit compression module, with a delay of up to

7.5 ms,3. the packetization scheme, which introduces a delay

between 20 and 60 ms,4. serialization, with varying delay between 0.20 and

15 ms,5. a queuing/buffering and network switching delay of

around 65 ms, and6. a dejitter buffer, with a worst-case delay figure of

40 ms.

Summing up these figures, it is easy to observe that thedelay budget is already exceeds the acceptable ITU-G.114requirements. That, too, is without taking into account thelast-hop wireless link, where additional delay may occur

due to the uncertainty associated with the underlyingwireless channel. Thus, to keep the end-to-end delay withinacceptable limits, the wireless delivery system mustschedule user data delivery within a strict timing constraint.

Therefore, the objective of scheduling is not only to

improve the throughput of the system and enforce fairnessamong participating users but also to meet the minimum

data requirements of users at each scheduling time slot. It is

IEEE TRANSACTIONS ON COMPUTERS, VOL. 56, NO. 10, OCTOBER 2007 1329

. S. Pal, S.R. Kundu, and S.K. Das are with the Center for Research inWireless Mobility and Networking (CReWMaN), Department of Compu-ter Science and Engineering, The University of Texas at Arlington,416 Yates Street Nederman Hall, Room 300, Arlington, TX 76019.E-mail: {spal, kundu, das}@cse.uta.edu.

. M. Chatterjee is with the School of Electrical and Computer Science,University of Central Florida, Orlando, FL 32816.E-mail: [email protected].

Manuscript received 1 May 2006; revised 23 Jan. 2007; accepted 30 Jan. 2007;published online 22 May 2007.Recommended for acceptance by A. Zomaya.For information on obtaining reprints of this article, please send e-mail to:[email protected], and reference IEEECS Log Number TC-0168-0506.Digital Object Identifier no. 10.1109/TC.2007.1082.

0018-9340/07/$25.00 � 2007 IEEE Published by the IEEE Computer Society

not possible to provide such delay-sensitive scheduling withthe help of existing scheduling techniques. It is worthpointing out that the challenges associated with delay-sensitive scheduling have been extensively studied in thecontext of wired networks (see [20] and references within).However, the solutions applicable to wired networks cannotbe directly ported to wireless networks because of thefundamental differences in transmission behavior, whichstem from the physical-layer transmission characteristics.Moreover, the wireless data systems support incrementalerror-correction mechanisms, medium access control(MAC) layer retransmission of lost packets, and multiratetransmission capabilities, all of which significantly impactthe dynamics of the underlying wireless channel. Beforeproceeding further, let us review the related work onmultirate wireless systems for multiple users.

1.1 Related Work

Most of the existing opportunistic scheduling schemessuffer from a syndrome, popularly referred to as theexposure problem [5] in auction theory. This refers to thephenomenon where a bidder who bids straightforwardlyaccording to his demand schedule is exposed to thepossibility that he may end up winning a collection of slotsthat he does not want at the prices that he bid because thecomplementary slots have become too expensive. Such asituation arises when the minimum data requirement of theusers is not met. Since opportunistic scheduling algorithmsmake their decisions on a slot-by-slot basis, they fail toprovide the users with the minimum amount of requesteddata until the very end of the schedule cycle. Suchlimitations in scheduling decisions negatively impact theperformance of delay-sensitive applications. The schedulingalgorithm is an important component that determines theperformance of multirate wireless systems supporting real-time data streams. The scheduler needs to be aware notonly of the wireless channel conditions but also of the QoSrequirements of the users.

In the literature, significant research has focused onvaried issues such as user fairness [21], [26], throughputmaximization [4], [14], and efficiency [27]. Existing oppor-tunistic scheduling algorithms exploiting time-varyingchannel conditions concentrate mainly on throughputmaximization while satisfying other QoS requirements.For example, it has been shown in [4], [14] how the systemthroughput in code division multiple access (CDMA)-basedHDR systems can be maximized while maintaining “pro-portional fairness” among users. Similarly, in [27], it hasbeen shown how we can formulate the opportunisticproblem for a multichannel scenario with resource con-straints, along with a scheduling scheme that aims toprovide fairness among users. The work reported in [23]considered techniques that exploit the wireless channelconditions while guaranteeing each user a predeterminedtime share in a schedule cycle. In [16], a bandwidth pricingmechanism was proposed which solves congestion-relatedproblems in wireless networks. Based on the second-priceauction, this scheme shows how the allocation of resourcesmaximizes social welfare. This work was subsequentlyextended in [15] for designing a pricing mechanism for thedownlink transmission power in a CDMA-based wireless

system. In [12], an auction-based algorithm was proposed,which allowed users to compete for time slots in a fadingwireless channel. Using the second-price auction mechan-ism, the users in the system were allocated channel slotsand the existence of a Nash equilibrium for such a strategywas proven. Later, in [13], the Nash equilibrium strategywas found when the channels for two users are uniformlydistributed.

To summarize, existing opportunistic scheduling algo-rithms aim at maximizing the overall system throughputand do not focus on the delay-sensitive requirements of theapplications.

1.2 Contributions of This Paper

In this paper, we take a fresh approach to the delay-sensitive scheduling problem by borrowing techniquesfrom the auction theory [22]. We consider a cellular networkwith one base station and multiple users. The resourcesavailable to the base station (for example, time slots,frequency bands, and codes) form the goods, which aresold to the users in a marketlike environment. The usersvalue these goods distinctively and express the values interms of a common transaction unit called money.1 We alsoconsider a time-slotted wireless packet data system wherethe duration of an individual time slot is smaller than theaverage fading duration of the received signal. Thus, duringsymbol transmission, we can assume that the underlyingwireless channel exhibits time-invariant properties.

Each user demands a certain number of slots (called abundle and denoted as S) in order to satisfy the minimumdata requirement within a specific schedule cycle. Thenumber of such slots depends on the condition of theunderlying wireless channel. Since the market has multipleindivisible goods and each user’s individual valuation ofthe goods depends on the bundle of goods received, weformulate the scheduling problem as a specific case ofcombinatorial auction. This is due to the fact that a single itemtransaction of the goods does not suffice since the user ismore interested in the sum total of the data received. Thisunderlying condition is exactly the reason that the single-slotallocation approach is not appropriate for delay-sensitiveapplications in multirate wireless systems. Consequently,schedulers based on the principles of opportunisticscheduling are unable to satisfy the minimum data rateconstraints demanded by the users.

In contrast, scheduling based on combinatorial auctiondeals with multislot allocation. Our proposed scheme canbe used to satisfy the minimum data rate constraint ofindividual users. To model our system, we use both formsof combinatorial auction: forward and reverse. In the forwardauction, there exists a single seller who wants to sellmultiple distinct goods to multiple buyers, whereas, in thereverse auction, there exists a single buyer who wants toprocure goods from multiple sellers. In the former case, theintention of the seller is to maximize the total moneyreceived, whereas, in the latter, the buyer tries to choosefrom sellers who quote the minimum price.

1330 IEEE TRANSACTIONS ON COMPUTERS, VOL. 56, NO. 10, OCTOBER 2007

1. We use the concept of “money” as a tool for defining the resourceallocation problem and, as such, this has no significance in real life.

In our study, we establish that existing opportunisticscheduling algorithms are at best equivalent to ourproposed scheme. We first formulate a combinatorialforward-auction-based multiple-slot scheduling schemethat guarantees the minimum data requirements of theusers. However, such an approach is shown to be NP-complete [22] and, hence, computationally intractable. Todesign a tractable solution, we therefore reformulate theproblem based on the reverse auction and propose anapproximation algorithm.

The main contributions of this paper are summarized asfollows:

. We demonstrate that most of the existing schedulingalgorithms suffer from the exposure problem and,hence, fail to guarantee the minimum data require-ments of the admitted users.

. We use combinatorial reverse auction to formulate thescheduling problem with two different objectives:1) toguarantee the minimum data rate of the usersand 2) to maximize the overall system throughput.

. By mathematical analysis, we show that the pro-posed scheme is capable of supporting more userswith hard real-time requirements than the existingschedulers. Our approach also leads to a significantgain in the system throughput.

. We prove that the worst-case performance of theproposed approximate algorithm is bounded by amultiplicative factor ð1þ logmÞ corresponding tothe optimal solution, where m denotes the numberof slots in a schedule cycle. We have also derived thetime complexity of the algorithm.

. We conduct simulation experiments to evaluate theperformance of our proposed algorithm with respectto two extreme scheduling disciplines: round-robinand throughput maximization. It is observed thatour approach can schedule more users whoseminimum QoS requirements are met than existingschemes.

. Finally, we propose a design parameter � thatdetermines the trade-off between guaranteeing theuser utility level (a measure for user satisfaction) andsystem throughput. The variation of system capacitywith the number of satisfied users for differentscheduling algorithms is also shown.

The rest of the paper is organized as follows: In Section 2,we formulate the scheduling problem for a multiratemultiuser time-slotted system. Section 3 models the userutility, identifies the exposure problem, and studies itsimplications on the user utility. The mapping of combina-torial auction to multirate scheduling is presented inSection 4. A scheduling scheme based on reverse combina-torial auction is presented in Section 5. We analyze theproposed algorithm and compare its performance withother existing schemes in Section 6. Simulation results arepresented in Section 7, followed by conclusions in Section 8.

2 PROBLEM FORMULATION

In this section, we describe the system model underconsideration and qualitatively formulate the scheduling

problem. We also define the objective functions for optimalscheduling. For the sake of completeness, we start by brieflydescribing the basics of auction theory, which forms thebasis of our proposed scheduling scheme.

2.1 Preliminaries on Auction Theory

An auction is the process of buying and selling goods byoffering them up for bid (that is, an offered price), acceptingbids, and then selling the item to the highest bidder [22]. Ineconomics, an auction is a method to determine the value ofa commodity that has an undetermined or variable price. Insome cases, there is a minimum or reserve price and, if thebidding does not reach the minimum price, then notransaction between buyers and sellers is executed. Mostof the auctions are primarily forward auctions which involvea single seller and multiple buyers. The buyers competeamong themselves in order to procure the goods of theirchoice by placing an initial bid that they feel is anappropriate price for the item under consideration. How-ever, in reverse auctions, the role of the buyers and the sellerare reversed. A buyer places a request to purchase aparticular item and multiple sellers bid to sell the requesteditem. The winner of a reverse auction is the seller who offersthe lowest price. Sometimes, the bidders are interested inbidding for multiple items at the same time. In such acombinatorial bid, the bidder offers a price for the collectionof goods according to the choice of the bidder rather than byplacing a bid on each individual items separately. Thisresults in combinatorial auction, where the auctioneer selectsa set of combinatorial bids that provides the maximumreturn in revenue without assigning any item to more thanone bidder.

2.2 Scheduling in Wireless Networks: A QualitativeFormulation

Wireless users derive utility from the services received fromthe wireless service providers. The utility perceived is afunction of the amount of data received in a specific timeepoch. In our study, we define a nonzero minimum utility,Umin, that must be met for user satisfaction. Correspondingto Umin, there exists a certain minimum amount of data Dmin

that must be made available to each user within a specificdeadline. Failure to transmit the “entire” Dmin to the userwithin the deadline or the schedule epoch results in atwofold penalty that not only leaves the user dissatisfied,but also penalizes the system throughput since partialtransmission of the data ð< DminÞ does not contributetoward increasing the user utility. A representative exampleis the scenario of streaming multimedia (MPEG-4 video),where a delayed transmission of packets associated withany I-frame results in the frame being discarded [7]. Thus,in real-time scheduling systems such as multirate wirelesspacket networks, each user is assumed to require at leastDmin bytes of data every schedule cycle. Hence, instead ofsolely maximizing the system throughput, our proposedscheduler aims at maximizing the number of users whoseminimum utility is guaranteed.

If the minimum utility of a user cannot be satisfiedwithin the schedule cycle, then the scheduler does not grantany slots to the user to avoid in the twofold penalty, asdiscussed above. However, once the number of allocated

PAL ET AL.: COMBINATORIAL REVERSE AUCTION-BASED SCHEDULING IN MULTIRATE WIRELESS SYSTEMS 1331

users for the particular schedule cycle has been decided, theschedule then endeavors to maximize the utility amongthose users. In general, we argue that the throughputmaximization assuming the “pay-per-byte” philosophy isdetrimental to maximizing the revenue of the serviceprovider since it does not maximize the number of satisfiedusers whose minimum data rate ðDminÞ is guaranteed. It isthus rational to assume that the generated revenue isproportional to the number of users who are satisfied in thelong run if the service provider wants to keep the churn rate(measure of the user attrition rate) under control [8], [9].

2.3 Wireless System Model

We consider a single-cell multirate time-division multipleaccess (TDMA) wireless data system supporting n users.2

Downlink scheduling of the wireless frames is realized bythe base station in a time-division manner, whereby, in eachtime slot, the data is transmitted to only one user, as inHDR-based systems [14]. The schedule cycle, the ratesupported by each user, and the slot allocation for themultirate wireless system are illustrated in Fig. 1. Table 1lists the various parameters for system description and isused by our proposed algorithm in Section 5. Throughchannel-state prediction and feedback mechanisms, thebase station is made aware of the channel quality and thecorresponding data rate experienced by each user for aspecific time window corresponding to the schedule cycle.For each user, the slots in the schedule cycle comprise theschedule vector.

Among the admitted users, let rij denote the possibletransmission bit rates for slot i and experienced by user j.Consequently, ri;j 2 f0; r1; . . . ; rRg, where R denotes thetotal number of feasible transmission bit rates and 0signifies that the user is not allocated any slot in theschedule vector. Scheduling decisions are periodic and madeevery m slots (the actual value of m is implementationspecific). We also denote the length of each slot as ts ms.Thus, if a schedule decision is performed at time instanceTd ¼ a, then subsequent decisions are made at times Td ¼aþ i�m� ts for i � 1. Associated with every schedulingdecision is the schedule matrix Sv ¼ ½wi;j� , where 1 � i �m and 1 � j � n. If user j is granted slot i, then wi;j ¼ 1;

otherwise, wi;j ¼ 0. We also introduce a value function V ðÞthat maps the data received in a particular slot to thecorresponding satisfaction or utility UðÞ of the userreceiving in that time slot.

2.4 Optimal Scheduling of Wireless Users

Let N be the set of satisfied users whose minimum datarequirements, Dmin, have been met at the end of a schedulecycle. The primary objective is to maximize the size jNj in each

cycle. In addition, the secondary objective is to maximize theutility of those jN j users and, hence, the system throughput.After the scheduler has allocated time slots to the users,there might exist residual slots which are insufficient tosatisfy the minimum data requirement of any additionalunallocated user. In order to maximize the systemthroughput, these slots are distributed among the allocatedusers. Thus, the Optimal Scheduling Policy can be constructedas follows:

maximize jN j ð1Þ

such that

Pnj¼1 wi;j ¼ m

Uj � Umin for 1 � j � nwi;j � 0:

8<: ð2Þ

In order to achieve the optimal schedule, we first formulatethe problem in terms of linear programming (LP). In thenext section, we show that the LP is equivalent to theoptimal scheduling policy.

2.5 LP Formulation

Considering that wi;j determines the schedule matrix andri;j determines the data rate of user j for slot i, the systemthroughput ðTpÞ for a schedule cycle is given by

Tp ¼Xi

Xj

wi;jri;j: ð3Þ

However, since the optimal schedule does not maximize thethroughput for each slot, the system suffers throughput lossgoverned by a penalty function �P . The penalty functionmeasuring the system throughput loss for every schedulecycle is given by

�P ¼Xi

Xj

�ðmax ri;jÞ � wi;jri;j

�: ð4Þ

Consequently, the total utility �U derived by the users isgiven by

�U ¼XNj¼1

Uj; ð5Þ

where Uj denotes the utility of user j as a function of thedata received. Since the effective objective function is toobtain the joint performance measure of all of the userutilities as well as the system throughput loss, we employ avalue function V to map both the penalty and the user utilityto a common unit (for example, money metric) so that thejoint optimization can be achieved. We define the valuepenalty ðV P Þ as a function of �p and the value usersatisfaction ðV UÞ as a function of �U . Thus,


2. In this study, we assume that the n users have already been admittedby the session admission control algorithm, the specific details of which arebeyond the scope of this paper.

Fig. 1. Illustration of the schedule cycle for a multirate wireless system.

V P ¼ fð�P Þ; ð6Þ

V U ¼ gð�UÞ: ð7Þ

For the time being, let us ignore the specific nature of fð�Þand gð�Þ. Consequently, the overall objective function of thesystem, the optimization of which would provide thesolution to (1), can be written as

System Objective : Maximize ðV U � V P Þ; ð8Þ

subject to the conditions stated in (2). Note that, in theprocess, we have mapped (1) to an alternative formulationgiven by (8). Next, we model the user utility functions.

3 MODELING USER UTILITY

The methodology of quantifying the user satisfactionderived from the services received using the concept ofutility functions has been established in [8], [9], and [18]. Weassume that the utility is an increasing function of the datareceived ðDrÞ. However, the utility remains zero unless anduntil a minimum amount of data ðDminÞ is received; that is,even for nonzero Dr, the utility is zero if Dr < Dmin. Thiscan be justified by the fact that most applications require aminimum amount of data, below which the applications failto execute. For example, for streaming multimedia, themedia player needs to wait for a certain number of packetsbefore the media frame can be successfully constructed.Formally, we define the utility function for user j receivingDr amount of data as follows:

UjðDrÞ ¼0 0 < Dr < Dmin

UjðDrÞ Dmin � Dr < Dmax

Umax Dr � Dmax:

8<: ð9Þ

We consider a generic utility function, as shown in Fig. 2.

Clearly, the change in utility is more prominent between

Dmin and Dmax. This kind of utility function is very intuitive

and can be better illustrated by the following example:

Consider that a video demands a bandwidth of somewhere

between 1 and 4 Mbps. This means that, with an effective

bandwidth of less than 1 Mbps, the quality of the video is


TABLE 1Notations Used in Auction-Based Scheduling

Fig. 2. Utility curve illustrating the marginal utility of the user as a

function of the data received.

too poor to be perceivable. On the other hand, with aneffective bandwidth of more than 4 Mbps, there is noperceptible improvement in the video quality. Thus, inFig. 2, Dmin ¼ 1 Mbps and Dmax ¼ 4 Mbps. The correspond-ing utilities derived by the user are Umin and Umax,respectively. To satisfy Umin, the corresponding resource(that is, the number of slots) must be available. Moreover, asmaller Umin does not necessarily mean that the number ofslots required will be less since the instantaneous channelconditions might be bad, thus requiring more slots to providethe minimum utility. Beyond Dmin, we consider the userutility function along the lines of diminishing returns, that is,the marginal utility3 of the user diminishes as a function of theallocated data. Additionally, the marginal utility is zero ornegligible when Dr > Dmax. In other words, the utility doesnot significantly increase if the received data exceedsDmax, asillustrated in Fig. 2. Also, in all of our analyses, the user utilityhas been normalized between 0 and 1, where Umin and Umaxare the corresponding threshold utilities.

3.1 Utility Function and the Exposure Problem

Let us now understand the interdependencies between theutility function and the exposure problem. According toauction theory terminology [19], the exposure problemarises because the users’ valuations for the number ofavailable slots are not additive. This implies that there existscomplementarity or substitutability among the slots. Althougha user might be allocated slots according to the wirelesschannel condition as in opportunistic scheduling or somefixed number of slots based on temporal fairness, theminimum data requirements might not be satisfied. Inauction terminology, the user might be enticed to bid ahigher price for a subset of the desired bundle, with thehope of acquiring the total bundle, but ends up gainingnothing since the minimum requirement is not satisfied.

We first define and then employ the complementarityand substitutability effects to demonstrate that the exposureproblem depends on whether the user utility function islinear or nonlinear. As outlined in Section 2.2, the exposureproblem helps us to identify if the slot allocation (that is, thescheduling) is being performed effectively. In the comple-mentarity effect, the value maximization for the system isonly achieved by allocating a particular bundle of slots, butnot any subset of it. The substitutability effect encompassesthe scenario when the value maximization of the system isachieved only when the right bundle of slots is allocatedand not any superset of it. Note that opportunisticscheduling does not solve the exposure problem fornonlinear utility functions. This is because a nonlinearutility function � displays subadditivity or superadditivityover various ranges, that is,

�ðxÞ þ �ðyÞ � �ðxþ yÞ or �ðxÞ þ �ðyÞ � �ðxþ yÞ: ð10Þ

Also, most utility functions for users are nonlinear sinceuser demand for real-time applications is inherently non-linear. Opportunistic scheduling mechanisms concentrateon the current slot to be scheduled and base their decision

on an objective function. These schemes do not considerprior or future allocations and, thus, are unable to capturethe complementarity and substitutability effects betweenthe slots.

4 SCHEDULING AND COMBINATORIAL AUCTIONS

In this section, we highlight the equivalence between theoptimal resource allocation problem in multirate wirelesssystems and combinatorial auctions by deriving the map-ping between the two. When the objective in a market isachieved, such as value maximization (respectively, mini-mization), for a seller (respectively, a buyer) in forward(respectively, reverse) auctions, the market is said to be inan equilibrium state. The market equilibrium corresponds tothe optimal schedule, as defined in Section 2.4, where thegoods map to the time slots and the objective is to satisfy(8). Under such circumstances, the combinatorial auctionproblem can be formulated as follows:

Let M ¼ f1; 2; � � � ;mg denote the set of goods availablefor auction and let ujðSÞ denote the utility that a sellerderives if the buyer j acquires the bundle S. Consequently,the utility is formulated as

ujðSÞ ¼Xx2X

�xVj;xðSÞ; ð11Þ

where X is the set of factors determining the overall utilityof a bundle S, �x is the weight for a given factor x, andP

x2X �x ¼ 1. The term Vj;xðSÞ is the value of the factor x byallocating the bundle S to buyer j. We define ðS; jÞ as

ðS; jÞ ¼ 1 if bundle S is allocated to buyer j0 otherwise:

�ð12Þ

Thus, the forward combinatorial auction can be formulatedas an optimization problem:

maximizeXj2N

XS�M

ujðSÞ ðS; jÞ ð13Þ

such that

Pi3SP

j2N ðS; jÞ � 1 8i 2MPS�M ðS; jÞ � 1 8j 2 N

ðS; jÞ ¼ f0; 1g 8S �M; 8j 2 N:

8<: ð14Þ

The first condition ensures that the overlapping sets ofitems are never assigned, whereas the second one ensuresthat no bidder receives more than one subset. The reversecombinatorial auction can be formulated in a similarfashion. Note that, in the reverse auction, there exists asingle buyer intending to procure items from multiplesellers who quote the minimum price. In both the forwardand reverse scenarios, it is assumed that the slots displaycomplementarity and substitutability in terms of utility andcosts, respectively. A careful observation of (13) reveals thatthe formulation is identical to the system objective problemdefined in (8). Finding the solution to (13) is known as thewinner determination problem [19] for both combinatorialforward and reverse auctions.

4.1 Forward Auction for Multirate Slot Allocation

Consider the scenario where the wireless system representsthe seller and the users represent the buyers. Recall that the


3. In economics, “marginal utility” is the additional utility (satisfaction orbenefit) that a user derives from an additional unit of service, such as timeslot in our case.

primary system objective of scheduling is to maximize thenumber of users whose minimum utility, Dmin, is satisfied.The secondary objective is to maximize the systemthroughput once it is no longer possible to add a userwhose Dmin can be satisfied. Such conditions require thatthe auction be done in two stages. In the first stage, bidssatisfying the minimum utility are determined. Let BI bethe initial feasible bid set for the forward auction. It isdefined as the set of all sets (bids) of slots which aremaximal sets satisfying the minimum utility for all theusers. The initial phase is thus described by

maximizeXj2N

XS2BI

ujðSÞ ðS; jÞ ð15Þ

such that

Pi2SP

j2N ðS; jÞ � 1 8i 2 BIPS2BI wðS; jÞ � 1 8j 2 N

ðS; jÞ ¼ f0; 1g 8S 2 BI ; 8j 2 N:

8<: ð16Þ

The termination criterion for the first phase of slot allocationoccurs when no additional user can be granted Dmin amountof data. The second phase consists of allocating the residualslots which are available at the end of the first phase of thescheduling operation. Depending on the objective and utilityfunctions, either a second round of auctions or any standardopportunistic scheduling algorithm can be employed todisseminate the residual slots among the users. Though thereexists a solution for the above formulation, finding the set ofwinners is shown by [11] to be an NP-complete problem andcannot even be approximated to a ratio of n1�� in polynomialtime, where n is the total number of users. As a result, it isinfeasible to implement forward-combinatorial-auction-based scheduling for real-time multirate wireless systems.This motivates us to explore reverse-auction-based schedul-ing in the next section.

5 MULTIPLE-SLOT SCHEDULING THROUGH

REVERSE AUCTIONS

In this section, the delay-sensitive multirate schedulingproblem is reformulated based on the reverse combinatorialauction. In such a scenario, the wireless base station is thebuyer who wants to procure m slots and the set of N usersare the sellers, each having m slots of different values (thatis, data rates). The prices that the users quote for the bundleof slots depend on the utility derived by the user when thebase station procures those slots. The problem can beformally stated as

minimizeXj2N

XS�M

pjðSÞ ðS; jÞ ð17Þ

such that

Pi2SP

j2N ðS; jÞ � 1 8i 2MPS�M ðS; jÞ � 1 8j 2 N

ðS; jÞ ¼ f0; 1g 8S �M; 8j 2 N:

8<: ð18Þ

Here, pjðSÞ denotes the price that user j quotes for the slotbundle S. The solution to the above problem is nothing butthe winner determination problem. In this framework, theusers compete against each other to sell the set of slots to thebase station. They are deprived of some value if they cannotget the base station to buy the slots from them. Identifying

the deprivation function is essential for deciding the best setof slots for any user. Thus, the quoted price for any bundleis a function of the deprivation function.

5.1 Deprivation Function

The objective of the user is to dispose of the set of slots andobtain the desired utility. The buyer (base station) buys theset of slots only if a specific minimum value of Dmin isachieved during the bidding process. Failure to sell the slotsdeprives the user of the minimum utility. Hence, adeprivation value is associated with the set of slots notacquired by the base station from that user. The deprivationfunction depends on two factors:

1. the utility derived by the user by giving the bundleto the base station and

2. the throughput loss that the base station mayexperience while procuring the bundle.

The utility that the user gets from a bundle of slots Sdepends on the type of applications. Following (9), theutility function can be defined as V U

j ¼ V ðUjÞ, where V ð�Þ isa value function mapping the utility to an equivalent moneymetric. Similarly, the monetary equivalent of the through-put loss that the system experiences by acquiring bundle Sfrom user j is calculated using (6) and is denoted by V L

j ðSÞ.Therefore, we can define the deprivation function for a slotbundle S as

PjðSÞ ¼ �V Uj ðSÞ þ ð1� �ÞV L

j ðSÞ: ð19Þ

Here, � is a control or tunable design parameter thatcontrols the relative weight of the two attributes. For � ¼ 1,the deprivation function basically boils down to guarantee-ing only user satisfaction, whereas, for � ¼ 0, the systemconsiders only throughput maximization.

5.2 Mechanism for Reverse Auction

We use the simple single-round sealed-bid-first price combi-natorial reverse auction mechanism. All of the “asks” (orquotes) are submitted prior to a deadline and the slotallocation is achieved based on the set of “asks” received.The throughput would have been drastically penalized hadthe auction been nonincentive-based. However, the equili-brium is not guaranteed for nonincentive combinatorialauctions [5]. In general, whether the mechanism is incentivecompatible or not, the price pjðSÞ quoted by user j for thebundle S is a function of the deprivation function Pj. Thus,

pjðSÞ ¼ fjðSÞPjðSÞ; ð20Þ

where fjðSÞ is the price mapping function that defines therelationship between the price and the deprivation function.Since we have assumed an incentive-compatible auctionmechanism, fjðSÞ ¼ �1 for all j and for all S.

The solution to the winner determination provides thedesired schedule vector. It has been shown in [11] that, inreverse auction, approximate solutions can be developed inspite of the fact that the problem is NP-complete. Hence, wedevelop our slot procurement algorithm along the lines ofreverse auction. However, since the primary and secondaryobjectives have conflicting goals, we decouple the algorithminto two phases. In the first phase, the restricted phase, wecompute the set of users whose minimum requirement is


satisfied. That is, slots are acquired from as many users as

possible while requiring that each user is able to get rid of

the minimum deprivation value. During the second phase,

the unrestricted phase, the residual slots are allocated, which

cannot satisfy Dmin for any additional user. These two

phases are described below.

5.3 Restricted Phase

In this phase, multiple single-round reverse auctions are held

until no additional user is able to get rid of the minimum

deprivation value. In each round, the system considers “asks”

from the users on the remaining minimal unallocated

bundles. This means that the bundle should just be able to

get rid of the minimum deprivation value. Each user is

allowed to provide an “ask” for only one bundle of slots Aj.

From these initial “asks,” the initial feasible bundle or set,

ðAkÞ, is constructed for round k of the restricted phase.For each round, the reverse auction takes the following

formulation:

minimizeXj2N

XS2Ak


such that

Pi3SP

j2N ðS; jÞ � 1 8i 2 AkPS2Ak ðS; jÞ � 1 8j 2 N

ðS; jÞ ¼ f0; 1g 8S 2 Ak; 8j 2 N:

8<:

Once the minimum deprivation value of a user has been

satisfied in a certain round, the user is barred from taking

part in subsequent rounds of the auction process in the

restricted phase. Let ACQUIREDSET denote the set of

accepted “asks,” and each “ask” Aj is represented by a set

of vector < �j1; �j2; � � � ; �jm > , where �ji is 1 if the ith slot is in

the “ask” for user j; otherwise, it is 0. Let PERMITTEDSET

be the set of permitted “asks.” Let us define �i such that

�i ¼ 1 if the ith slot has not been acquired; otherwise, �i ¼ 0.

Let SATISFIEDSET be the set of users whose minimum

deprivation value has been satisfied. The algorithm for the

restricted phase is described in Fig. 3.

5.4 Unrestricted Phase

The residual slots aid in achieving the secondary objective of

maximizing the utility of allocated users, as well as

maximizing the system throughput during the unrestricted

phase. The allocated users strive to further minimize the

deprivation value by selling their slots. However, unlike the

restricted phase, there is no restriction on the size of slot

bundle. Note that none of the users whose minimum utility

(that is, the minimum deprivation value) has not been

satisfied is allowed to compete in this phase. Additionally,

the slots may not exhibit a complementarity/substitutability

relationship. Consequently, the exposure problem ex-

plained earlier will not occur. Under such conditions, when

the utility is assumed to be linear, scheduling the residual


Fig. 3. Restricted phase: the algorithm for winner set determination.

slots can be performed by employing one of the existingopportunistic scheduling algorithms.

On the contrary, if the complementarity/substitutabilityeffect exists between the residual slots, then the allocationshould be performed using combinatorial reverse auction soas to overcome the exposure problem. For the unrestrictedphase, the “asks” are based on the further reduction of thedeprivation value. The objective for the buyer (that is, thesystem) is now set to choose the “asks” from the users,which minimizes its total price. This guarantees throughputmaximization for both the system and the chosen users. Theauction proceeds similar to the restricted phase, butcontinues until all of the slots have been exhausted. LetAk denote the set of all remaining slots after round k of theunrestricted phase. The mathematical formulation forround k is given by

minimizeXj2N

XS2Ak


such that

Pi2SP

j2N ðS; jÞ � 1 8i 2 AkPS2Ak ðS; jÞ � 1 8j 2 N

ðS; jÞ ¼ f0; 1g 8S 2 Ak; 8j 2 N:

8<: ð23Þ

Note that the difference between (22) and (23) is the type of“asks” possible and the set of slots that are part of thereverse auction. The ACQUIREDSET obtained in theprevious algorithm is used in the unrestricted phase. Thealgorithm is described in Fig. 4.

After the execution of the unrestricted phase algorithm,the ACQUIREDSET is updated, which provides the distribu-tion of the slots for the schedule cycle under consideration.

6 PERFORMANCE ANALYSIS

In this section, we analyze the proposed algorithms.

Theorem 1. The worst-case running time for the restricted phaseslot procurement algorithm is Oðn2mÞ, where m is the numberof slots in each schedule cycle and n is the number of users.

Proof. Assume that m >> n. The complexity of thealgorithm in the restricted phase depends on the “ask”construction (line 2) in Fig. 3 and the selection ofappropriate j (line 6). In the worst case, line 2 of thealgorithm takes ðn� kÞðm� kÞ operations, where k is thecurrent round. Line 6 takes n� k operations in the worstcase. The maximum number of possible rounds is n.Thus, the worst-case complexity can be given by

OXn�1

k¼0

½ðn� kÞðm� kÞ þ ðn� kÞ� !

¼)

Oðnmðn� 1Þ � nð1þ 2þ . . .þ ðn� 1ÞÞ�mð1þ 2þ � � � þ ðn� 1ÞÞ þ ð12 þ 22 þ . . .þ ðn� 1Þ2Þ¼)

Oðnmðn� 1Þ � n ðn� 1Þðn� 2Þ2

�m ðn� 1Þðn� 2Þ2

þ nðnþ 1Þð2nþ 1Þ6

’ Oðn2mÞ

since n << m. tu

Corollary 1. The worst-case complexity of the unrestricted phaseslot procurement algorithm is Oðn2mÞ.

Lemma 1. Let the effective price for each slot be definedas PðoÞ ¼ pj

jAjj , where pj is the price paid by user j foracquiring the set of slot Aj. If OPT is the total cost that thebase station pays for the optimal solution, then

PðokÞ �OPT

l0 � kþ 1;

where foig, i ¼ 1; � � � ; l, is an ordering of the slots based on thesequence in which they are acquired by the base station, l is thetotal number of slots procured by the base station in therestricted phase, and l0 � l.

Proof. Let ok be covered (that is, these slots are taken up)when the “ask” Aj was picked by the algorithm. After ok,there are at least l0 � kþ 1 slots to be covered. Since theoptimal cost OPT covers all of the l slots, it can alsocover the remaining l0 � kþ 1 slots. Thus, there must beat least one “ask” whose average cost of covering is atmost OPTl0�kþ1 . As our algorithm chooses the slots from thelowest average cost to the highest average cost per slot,PðokÞ � OPT

l0�kþ1 . tuTheorem 2. The restricted-phase slot procurement algorithm

finds a solution that is within a factor ð1þ log mÞ of theoptimal solution, where m is the total number of slots to beprocured.

Proof. The proof for the bound is similar to the onepresented in [19]. Let l be the total number of slots thatcould be covered by the optimal solution and l0 be thetotal number of slots that are covered by our algorithm.From Lemma 1, the proof of Theorem 2 can be outlinedas follows: Let the “asks” which were picked in the


Fig. 4. Unrestricted phase: the residual slot allocation algorithm.

restricted phase that are able to get rid of the minimumdeprivation value be denoted by Aj1; Aj2; � � � ; Ajs, wherejs is the last user whose bundle of slots was chosen. Thetotal cost is given by

Psx¼1 pjx ¼

Pl0

k¼1 PðokÞ. UsingLemma 1, the total cost can be written as

Xl0k¼1

PðokÞ � OPT ð1þ1

2þ � � � þ l

l0Þ � OPT �Hl0 ;

where Hl0 is the ðl0Þth harmonic number. Since

Hl0 � 1þ ln l0 � 1þ ln l � 1þ ln m;

the cost is bounded by ð1þ log mÞ of the optimal. tu

Lemma 2. Let ~ni be the number of slots obtained by user i inorder to satisfy the minimum utility using our scheme and n̂ibe the number of slots obtained using an opportunistic scheme.Then, n̂i � ~ni, 8i 2 N .

Proof. In our scheme, the slot allocation always tries to givethe best available slots to any user so as to satisfy theminimum utility requirement at the minimum cost.Here, ~ni is the minimum number of slots required tosatisfy the minimum utility. Now, consider an opportu-nistic scheme where the decision is based on a slot by slotbasis. Consider a user whose minimum utility has beensatisfied. If the user’s best available slots come indescending order of their individual utility values, thenthe user will reach the minimum utility level with thesmallest number of slots. In this case, n̂i ¼ ~ni. Otherwise,the user may get another slot, which is not the user’savailable slot. Therefore, to satisfy the minimum utility,the user will require at least the minimum number ofslots. In either case, n̂i � ~ni. tu

Theorem 3. Let Nc be the set denoting the maximum number ofusers whose minimum utility has been satisfied by thecombinatorial-reverse-auction-based scheduling and let No bethe set of users who have been satisfied by the opportunisticscheme. Then, jNcj � jNoj.

Proof. From Lemma 2, n̂i � ~ni for all i whose minimumutility has been satisfied. By contradiction, let us assumethat jNcj < jNoj. Then,

XjNoj

i¼1

ðn̂i � ~kiÞ þ l ¼XjNcj

i¼1

ðn̂i � ~kiÞ; ð24Þ

where k̂i and ~ki are the extra slots given to user i aftersatisfying the minimum utility and l is the total numberof slots given in the case of opportunistic scheduling tousers whose minimum utility could not be satisfied. Theabove equation can be rewritten as

XjNcj

i¼1

ðn̂i � ~niÞ þXjNoj

i¼jNcjþ1

n̂i þ lþXjNoj

i¼jNcjk̂i ¼

XjNcj

i¼1

ð~ki � k̂iÞ: ð25Þ

However, this implies that

XjNcj

i¼1

~ki �XjNoj

i¼jNcjn̂i: ð26Þ

This is clearly not possible since it would mean that ourauction-based scheme would be able to accommodate atleast one more user by using the ~kis. Hence, jNcj � jNoj.tu

7 SIMULATION STUDY

This section studies the effectiveness of our proposedscheduling scheme through simulation experiments. Wealso compare how the auction-based scheme fares withrespect to two extreme scheduling disciplines—round-robinand throughput maximization—that serve as the basis forcomparing the fairness and maximum system throughput,respectively. We study how each scheme performs in termsof the number of satisfied users and global systemthroughput.

7.1 System and Channel Model

We consider a single-cell wireless data network for oursimulation study due to the fact that the schedulingschemes under evaluation are designed to work best inthe presence of a single base station. We also assume that allof the users under consideration are receiving real-timestreaming multimedia traffic. In order to support multi-media traffic (MPEG-4 or H.263) of various qualities (low,medium, and high), as given in [7], we consider threevalues for Dmin: 16, 64, and 128 kilobits per second (Kbps).We model our simulation based on the HDR system that iscapable of supporting 11 different data rates, with eachschedule cycle consisting of 1,000 slots. We assume that usermobility is random (both speed and direction) and employthe path-loss model and the slow log-normal model [25] forwireless channels.

7.2 Simulation Results

For our proposed auction-based scheduling scheme, thevariation of the system throughput with the number ofusers for different values of Dmin is shown in Fig. 5. Asexpected, the system throughput initially increases, butultimately gets saturated with the increase in the number ofusers. Next, we identify the maximum system capacity in


Fig. 5. Throughput versus number of users in the system. Notice how

the throughput decreases with the increase in Dmin.

terms of satisfied users by setting Dmin to different values.For each value of Dmin, we obtained a range of users whoare satisfied by the system. This is shown in Table 2. It islogical that, for smaller Dmin, a greater number of users canbe satisfied (see Fig. 6). The comparison of the systemthroughput achieved by the various schemes is shown inFig. 7. As expected, the system throughput is the best for thethroughput maximization scheme and the worst for theround-robin scheduling algorithm. In the case of opportu-nistic scheduling with temporal fairness, the throughput ispenalized. The throughput performance of the proposedauction-based scheme is better than both the round-robinscheduling and opportunistic scheduling with temporalfairness and is very close to the throughput maximizationscheme. In order to visualize the working of the proposedscheduling scheme, we consider a hypothetical scenariowith 15 users in the system. The users are represented as uiin Fig. 8. For the purpose of explaining the workings of ouralgorithm, a temporal snapshot of four successive schedul-ing decision cycles is also presented.

With Dmin ¼ 128Kbps, the first three schedule cyclesyield the schedule vector as ½1; 2; 4; 5; 6; 7; 10�. However, forthe fourth cycle, user 7 is replaced by user 9. Although theallocated users are receiving Dmin, the variation of the slotdistribution between users is due to the varying channelconditions. The scheduling scheme judiciously distributes

the residual slots after the restricted phase and does not

allocate any more slots if Dmax is attained, as in the case

with user 10 in this example. Careful observation reveals

that, although all of the allocated users were receiving Dmin

or more data, user 7 was receiving lesser slots in each

succeeding schedule cycle such that, in the fourth cycle,

user 7 was eliminated by user 9 in the restricted phase.

Thus, the scheduler is intelligent enough to identify and

allocate the user to achieve the system objective. In each

schedule cycle, all of the allocated users are guaranteed

Dmin amount of data.Next, we investigate the variation of the system

throughput and the number of satisfied users with the

tunable parameter �, as defined in (19). The value of �

depends on the objective of the wireless service providers

that maximizes the throughput, guarantees user utility, or a

combination of both. Hence, we evaluate the system

throughput and the number of satisfied users by varying

� from 0 to 1. For � ¼ 0, the deprivation function totally

becomes a function of the throughput maximization,

whereas, for � ¼ 1, the deprivation function only cares

about the user satisfaction. As expected, the throughput


TABLE 2Dmin versus Maximum Number of Users

Fig. 6. Performance of each scheduling scheme measured using

satisfied users as a percentage of the total users.

Fig. 7. Throughput versus the number of users in the system for different

scheduling algorithms.

Fig. 8. Slot distribution of users in schedule cycle.

maximizes for � ¼ 0, whereas the number of satisfied usersis maximized for � ¼ 1, as illustrated in Fig. 9.

8 CONCLUSIONS

In this paper, we have proposed an auction-based schedulingalgorithm for allocating the slots in a time-division multiratewireless system. We have justified that opportunisticscheduling algorithms that aim to maximize the systemthroughput are unable to address the exposure problem. Wehave formalized the slot allocation problem in the form of amarket, where multiple users bid for the number of slots tosatisfy their minimum QoS requirements. With the help ofcombinatorial auctions, we have shown how the exposureproblem can be successfully eliminated. In the process, wehave been able to achieve the primary objective of maximiz-ing the number of users whose minimum slot requirementsare satisfied. The remaining slots, if any, are allocated with aview to maximizing the system throughput. In our study, wehave applied the reverse auction theory in order to deal withthe real-time scheduling requirements. We have derived theapproximation ratio of the auction-based scheduling algo-rithm, which results in more satisfied users than otherexisting opportunistic scheduling schemes. As part of ourfuture work, we would like to investigate the context of aparallel data scheduler for simultaneous transmission of datato multiple users.

ACKNOWLEDGMENTS

The authors would like to thank the anonymous referees forvaluable suggestions to improve the quality of the paper.They would also like to thank Kalyan Basu for helpfuldiscussions and Shuvendu Dang for his contributions tothis work. This work is partially supported by the USNational Science Foundation Information Technology Re-search (NSF ITR) Grant IIS-0326505.

REFERENCES

[1] IS-856 cdma2000 high rate packet data air interface specification,3GPP2 C.SO024 version 4.0, http://www.3gpp2.org, Oct. 2002.

[2] P. Bender, P. Black, M. Grob, R. Padovani, N. Sindhushayana, andA. Viterbi, “CDMA/HDR: A Bandwidth-Efficient High-SpeedWireless Data Service for Nomadic Users,” IEEE Comm. Magazine,pp. 70-77, July 2000.

[3] General Packet Radio Services (GPRS) service description, 3GPPTS 03.60, http://www.3gpp.org, 2007.

[4] A. Jalali, R. Padovani, and R. Pankaj, “Data Throughput ofCDMA-HDR: A High-Efficiency High Data Rate Personal Com-munication Wireless System,” Proc. 51st IEEE Vehicular TechnologyConf. (VTC ’00 Spring), vol. 3, pp. 1854-1858, 2000.

[5] A. Pekec and M.H. Rothkopf, “Combinatorial Auction Design,”Management Science, vol. 49, pp. 1485-1503, 2003.

[6] A. Tarello, E. Modiano, J. Sun, and M. Zafer, “Minimum EnergyTransmission Scheduling Subject to Deadline Constraints,” Proc.Third IEEE Modeling and Optimization in Mobile, Ad Hoc, andWireless Networks (WiOPT ’05), pp. 67-76, 2005.

[7] F. Fitzek and M. Reisslein, “MPEG-4 and H.263 Video Traces forNetwork Performance Evaluation,” IEEE Network, vol. 15, no. 6,pp. 40-54, 2001.

[8] H. Lin, M. Chatterjee, S.K. Das, and K. Basu, “ARC: An IntegratedAdmission and Rate Control Framework for CDMA Data Net-works Based on Non-Cooperative Games,” Proc. MobiCom 2003,pp. 326-338, 2003.

[9] H. Lin, M. Chatterjee, S.K. Das, and K. Basu, “ARC: An IntegratedAdmission and Rate Control Framework for Competitive WirelessCDMA Data Networks Using Non-Cooperative Games,” IEEETrans. Mobile Computing, vol. 4, no. 3, pp. 243-258, May/June 2005.

[10] Int’l Telecommunication Union, G.114, “One-Way TransmissionTime,” May 2000.

[11] J. Hastad, “Clique Is Hard to Approximate within n1��,” ActaMathemitica, vol. 182, pp. 105-142, 1999.

[12] J. Sun, L. Zheng, and E. Modiano, “Wireless Channel AllocationUsing an Auction Algorithm,” Proc. 41st Allerton Conf. Comm.,Control, and Computing, vol. 24, pp. 1085-1096, Oct. 2003.

[13] J. Sun, E. Modiano, and L. Zheng, “A Novel Auction Algorithmfor Fair Allocation of a Wireless Fading Channel,” Proc. 38th Ann.Conf. Information Sciences and Systems (CISS ’04), Mar. 2004.

[14] P. Bender, P. Black, M. Grob, R. Padovani, N. Sindhushayana, andA. Viterbi, “CDMA/HDR: A Bandwidth-Efficient High-SpeedWireless Data Service for Nomadic Users,” IEEE Comm. Magazine,pp. 70-77, July 2000.

[15] P. Maille, “Auctioning for Downlink Transmission Power inCDMA Cellular Systems,” Proc. Seventh ACM Int’l Symp. Modeling,Analysis and Simulation of Wireless and Mobile Systems (MSWiM’04), pp. 293-296, Oct. 2004.

[16] P. Maille and B. Tuffin, “Multi-Bid Auctions for BandwidthAllocation in Communication Networks,” Proc. IEEE INFOCOM’04, Mar. 2004.

[17] S.S. Kulkarni and C. Rosenberg, “Opportunistic SchedulingPolicies for Wireless Systems with Short-Term Fairness Con-straints,” Proc. IEEE Global Telecomm. Conf. (GLOBECOM ’03),vol. 1, pp. 533-537, Dec. 2003.

[18] S. Pal, M. Chatterjee, and S.K. Das, “User-Satisfaction BasedDifferentiated Services for Wireless Data Networks,” Proc. IEEEInt’l Conf. Comm. (ICC ’05), vol. 2, pp. 1174-1178, May 2005.

[19] T. Sandholm, S. Suri, A. Gilpin, and D. Levine, “WinnerDetermination in Combinatorial Auction Generalizations,” Proc.First Int’l Joint Conf. Autonomous Agents and Multiagent Systems(AAMAS ’02), pp. 69-76, July 2002.

[20] T. Ling and N. Shroff, “Scheduling Real-Time Traffic in ATMNetworks,” Proc. IEEE INFOCOM ’96, vol. 1, pp. 198-205, 1996.

[21] V. Bharghavan, S. Lu, and T. Nandagopal, “Fair Queuing inWireless Networks: Issues and Approaches,” IEEE PersonalComm., vol. 6, pp. 44-53, Feb. 1999.

[22] V. Krishna, Auction Theory. Academic Press, 2002.

[23] X. Liu, E.K.P. Chong, and N.B. Shroff, “Transmission Schedulingfor Efficient Wireless Utilization,” Proc. IEEE INFOCOM ’00,vol. 3, pp. 776-785, 2000.

[24] X. Liu, E.K.P. Chong, and N.B. Shroff, “Optimal OpportunisticScheduling in Wireless Networks,” Proc. 58th IEEE VehicularTechnology Conf. (VTC ’03 Fall), vol. 3, pp. 1417-1421, Oct. 2003.

[25] X. Liu, K.P. Chong, and N.B. Shroff, “Opportunistic TransmissionScheduling with Resource-Sharing Constraints in Wireless Net-works,” IEEE J. Selected Areas in Comm., vol. 19, no. 10, pp. 2053-2064, Oct. 2001.


Fig. 9. System throughput and number of satisfied users versus �.

[26] Y. Liu, S. Gruhl, and E. Knightly, “WCFQ: An OpportunisticWireless Scheduler with Statistical Fairness Bounds,” IEEE Trans.Wireless Comm., vol. 2, pp. 1017-1028, Sept. 2003.

[27] Y. Liu and E. Knightly, “Opportunistic Fair Scheduling overMultiple Wireless Channels,” Proc. IEEE INFOCOM ’03, vol. 2,pp. 1106-1115, 2003.

Sourav Pal received the BE degree from theBengal Engineering College, Shibpur, India, andthe MS degree from the University of Texas atArlington (UTA). He is currently working towardthe PhD degree at the Center for Research inWireless Mobility and Networking Laboratory atUTA. His interests and expertise are wirelessand multimedia systems. He is a studentmember of the IEEE.

Sumantra R. Kundu received the BTechdegree from the Indian Institute of Technology,Kharagpur, India, and the MS degree from theUniversity of Iowa, Iowa City. He is currently aPhD candidate at the Center for Research inWireless Mobility and Networking Laboratory atthe University of Texas at Arlington. Hisresearch interests are operating system (OS)internals, hardware architecture, efficient datastructures, and performance evaluation using

statistical principals and queuing theory. He is a student member ofthe IEEE.

Mainak Chatterjee received the BSc degree(Hons) in physics from the University of Calcuttain 1994, the ME degree in electrical commu-nication engineering from the Indian Institute ofScience, Bangalore, in 1998, and the PhDdegree from the Department of ComputerScience and Engineering at the University ofTexas at Arlington in 2002. He is currently anassistant professor in the School of ElectricalEngineering and Computer Science at the

University of Central Florida. He serves on the executive and technicalprogram committee of several international conferences. His researchinterests include economic issues in wireless networks, applied gametheory, resource management and quality-of-service provisioning, adhoc and sensor networks, code division multiple access (CDMA) datanetworking, and link-layer protocols. He is a member of the IEEE.

Sajal K. Das is a distinguished scholar professorof computer science and engineering and thefounding director of the Center for Research inWireless Mobility and Networking (CReWMaN)Laboratory at the University of Texas at Arling-ton (UTA). He is also a visiting professor at theIndian Institute of Technology (IIT), Kanpur, andIIT Guwahati, an honorary professor of FudanUniversity, Shanghai, and a visiting scientist atthe Institute of Infocomm Research (I2R),

Singapore. He is frequently invited as keynote speaker at internationalconferences and symposia. He serves as the founding editor in chief ofthe Pervasive and Mobile Computing journal (Elsevier) and an associateeditor of the IEEE Transactions on Mobile Computing, ACM/SpringerWireless Networks, IEEE Transactions on Parallel and DistributedSystems, and Journal of Peer-to-Peer Networking. He is the founder ofthe IEEE International Symposium on a World of Wireless, Mobile, andMultimedia Networks (WoWMoM) and a cofounder of the annual IEEEInternational Conference on Pervasive Computing and Communications(PerCom). He has served as the general chair, the Technical ProgramCommittee (TPC) chair, or a TPC member for numerous IEEE and ACMconferences. He is a member of the executive committees of the IEEECS Technical Committee on Computer Communications (TCCC) andTechnical Committee on Parallel Processing (TCPP). His currentresearch interests include the design and modeling of smart environ-ments, sensor networks, security, mobile and pervasive computing,resource and mobility management in wireless networks, wirelessmultimedia, mobile Internet, mobile grid computing, biological network-ing, applied graph theory, and game theory. He has published more than400 papers in international conference proceedings and journals andmore than 30 invited book chapters. He is the holder of five US patentsin wireless mobile networks and coauthored the book Smart Environ-ments: Technology, Protocols, and Applications (John Wiley, 2005). Hereceived the Best Paper Awards from PerCom 2006, ACM MobiCom ’99,Information Networking, Wireless Communications Technologies andNetwork Applications, International Conference (ICOIN) 2002, ThirdACM International Workshop on Modeling, Analysis, and Simulation ofWireless and Mobile Systems (MSwiM 2000), and ACM/IEEE 11thWorkshop on Parallel and Distributed Simulation (PADS 1997). He isalso a recipient of the 2006 UTA Academy of Distinguished ScholarsAward, 2005 University Award for Distinguished Record of Research,2003 College of Engineering Research Excellence Award, and 2001 and2003 Outstanding Faculty Research Award in Computer Science. He isa member of the IEEE.

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Combinatorial Reverse Auction based Scheduling in Multi-rate Wireless Systems

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