IEEE TRANSACTIONS ON COMPUTERS, VOL. 56, NO. 10, OCTOBER 2007 1329 Combinatorial Reverse -Based Scheduling 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 , 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 ð1 þ log mÞ with 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

1INTRODUCTION

HE concept of opportunistic scheduling in wireless net- which demand that packets be delivered within certain Tworks was first introduced in [23]. The basic idea is to delay bounds so as to comply with the application-level continuously monitor the uncertainty of the underlying quality of service (QoS). We justify that time constraint wireless channel and take decisions opportunistically so as scheduling is a necessity for delay-sensitive applications by to optimize the objective functions under consideration. explaining the timing requirements of VoIP applications. Extensive research has been conducted with varying According to the International Telecommunication Union objectives such as maximizing the system throughput [24], (ITU-T) G.114 specifications [10], for good and pleasing maintaining both long and short-term fairness among users voice quality, the end-to-end delay for both the forward and [26], [27], and maximizing the user utility [23]. In general, reverse paths should not be more than 150 ms. This delay is the goal has been to maximize a concave utility function contributed by various sources: representing the specified objective function. Unfortunately, such concave functions fail to capture the importance of the 1. the voice coder, with a processing delay of 10 ms, timelineness of decision making in user scheduling. 2. the bit compression module, with a delay of up to On the other hand, the next-generation multirate wire- 7.5 ms, less data networks, such as Evolution-Data Optimized 3. the packetization scheme, which introduces a delay (1xEV-DO) [1], High Data Rate (HDR) [2], and Enhanced between 20 and 60 ms, Data Rates for Global Evolution (EDGE) [3], promise to 4. serialization, with varying delay between 0.20 and provide data services and applications with strict timing 15 ms, constraints. Examples of such applications include stream- 5. a queuing/buffering and network switching delay of ing multimedia, voice over Internet Protocol (VoIP), instant around 65 ms, and 6. a dejitter buffer, with a worst-case delay figure of messaging (IM), and real-time videoconferencing, all of 40 ms. Summing up these figures, it is easy to observe that the . S. Pal, S.R. Kundu, and S.K. Das are with the Center for Research in delay budget is already exceeds the acceptable ITU-G.114 Wireless Mobility and Networking (CReWMaN), Department of Compu- ter Science and Engineering, The University of Texas at Arlington, requirements. That, too, is without taking into account the 416 Yates Street Nederman Hall, Room 300, Arlington, TX 76019. last-hop wireless link, where additional delay may occur E-mail: {spal, kundu, das}@cse.uta.edu. due to the uncertainty associated with the underlying . M. Chatterjee is with the School of Electrical and Computer Science, University of Central Florida, Orlando, FL 32816. wireless channel. Thus, to keep the end-to-end delay within E-mail: [email protected]. acceptable limits, the wireless delivery system must Manuscript received 1 May 2006; revised 23 Jan. 2007; accepted 30 Jan. 2007; schedule user data delivery within a strict timing constraint. published online 22 May 2007. Therefore, the objective of scheduling is not only to Recommended for acceptance by A. Zomaya. improve the throughput of the system and enforce fairness For information on obtaining reprints of this article, please send e-mail to: [email protected], and reference IEEECS Log Number TC-0168-0506. among participating users but also to meet the minimum Digital Object Identifier no. 10.1109/TC.2007.1082. data requirements of users at each scheduling time slot. It is

0018-9340/07/$25.00 ß 2007 IEEE Published by the IEEE Computer Society 1330 IEEE TRANSACTIONS ON COMPUTERS, VOL. 56, NO. 10, OCTOBER 2007 not possible to provide such delay-sensitive scheduling with system. In [12], an auction-based algorithm was proposed, the help of existing scheduling techniques. It is worth which allowed users to compete for time slots in a fading pointing out that the challenges associated with delay- wireless channel. Using the second-price auction mechan- sensitive scheduling have been extensively studied in the ism, the users in the system were allocated channel slots context of wired networks (see [20] and references within). and the existence of a Nash equilibrium for such a strategy However, the solutions applicable to wired networks cannot was proven. Later, in [13], the Nash equilibrium strategy be directly ported to wireless networks because of the was found when the channels for two users are uniformly fundamental differences in transmission behavior, which distributed. stem from the physical-layer transmission characteristics. To summarize, existing opportunistic scheduling algo- Moreover, the wireless data systems support incremental rithms aim at maximizing the overall system throughput error-correction mechanisms, medium access control and do not focus on the delay-sensitive requirements of the (MAC) layer retransmission of lost packets, and multirate applications. transmission capabilities, all of which significantly impact the dynamics of the underlying wireless channel. Before 1.2 Contributions of This Paper proceeding further, let us review the related work on In this paper, we take a fresh approach to the delay- multirate wireless systems for multiple users. sensitive scheduling problem by borrowing techniques 1.1 Related Work from the [22]. We consider a cellular network with one base station and multiple users. The resources Most of the existing opportunistic scheduling schemes available to the base station (for example, time slots, suffer from a syndrome, popularly referred to as the frequency bands, and codes) form the goods, which are exposure problem [5] in auction theory. This refers to the sold to the users in a marketlike environment. The users phenomenon where a bidder who bids straightforwardly value these goods distinctively and express the values in according to his demand schedule is exposed to the terms of a common transaction unit called money.1 We also possibility that he may end up winning a collection of slots consider a time-slotted wireless packet data system where that he does not want at the prices that he bid because the complementary slots have become too expensive. Such a the duration of an individual time slot is smaller than the situation arises when the minimum data requirement of the average fading duration of the received signal. Thus, during users is not met. Since opportunistic scheduling algorithms symbol transmission, we can assume that the underlying make their decisions on a slot-by-slot basis, they fail to wireless channel exhibits time-invariant properties. provide the users with the minimum amount of requested Each user demands a certain number of slots (called a data until the very end oftheschedulecycle.Such bundle and denoted as S) in order to satisfy the minimum limitations in scheduling decisions negatively impact the data requirement within a specific schedule cycle. The performance of delay-sensitive applications. The scheduling number of such slots depends on the condition of the algorithm is an important component that determines the underlying wireless channel. Since the market has multiple performance of multirate wireless systems supporting real- indivisible goods and each user’s individual valuation of time data streams. The scheduler needs to be aware not the goods depends on the bundle of goods received, we only of the wireless channel conditions but also of the QoS formulate the scheduling problem as a specific case of requirements of the users. . This is due to the fact that a single item In the literature, significant research has focused on transaction of the goods does not suffice since the user is varied issues such as user fairness [21], [26], throughput more interested in the sum total of the data received. This maximization [4], [14], and efficiency [27]. Existing oppor- underlying condition is exactly the reason that the single-slot tunistic scheduling algorithms exploiting time-varying allocation approach is not appropriate for delay-sensitive channel conditions concentrate mainly on throughput applications in multirate wireless systems. Consequently, maximization while satisfying other QoS requirements. schedulers based on the principles of opportunistic For example, it has been shown in [4], [14] how the system scheduling are unable to satisfy the minimum data rate throughput in code division multiple access (CDMA)-based constraints demanded by the users. HDR systems can be maximized while maintaining “pro- In contrast, scheduling based on combinatorial auction portional fairness” among users. Similarly, in [27], it has deals with multislot allocation. Our proposed scheme can been shown how we can formulate the opportunistic be used to satisfy the minimum data rate constraint of problem for a multichannel scenario with resource con- individual users. To model our system, we use both forms straints, along with a scheduling scheme that aims to of combinatorial auction: forward and reverse. In the forward provide fairness among users. The work reported in [23] auction, there exists a single seller who wants to sell considered techniques that exploit the wireless channel multiple distinct goods to multiple buyers, whereas, in the conditions while guaranteeing each user a predetermined reverse auction, there exists a single buyer who wants to time share in a schedule cycle. In [16], a bandwidth pricing procure goods from multiple sellers. In the former case, the mechanism was proposed which solves congestion-related intention of the seller is to maximize the total money problems in wireless networks. Based on the second-price received, whereas, in the latter, the buyer tries to choose auction, this scheme shows how the allocation of resources from sellers who quote the minimum price. maximizes social welfare. This work was subsequently extended in [15] for designing a pricing mechanism for the 1. We use the concept of “money” as a tool for defining the resource downlink transmission power in a CDMA-based wireless allocation problem and, as such, this has no significance in real life. PAL ET AL.: COMBINATORIAL REVERSE AUCTION-BASED SCHEDULING IN MULTIRATE WIRELESS SYSTEMS 1331

In our study, we establish that existing opportunistic problem. We also define the objective functions for optimal schedulingalgorithmsareatbestequivalenttoour scheduling. For the sake of completeness, we start by briefly proposed scheme. We first formulate a combinatorial describing the basics of auction theory, which forms the forward-auction-based multiple-slot scheduling scheme basis of our proposed scheduling scheme. that guarantees the minimum data requirements of the 2.1 Preliminaries on Auction Theory users. However, such an approach is shown to be NP- complete [22] and, hence, computationally intractable. To An auction is the process of buying and selling goods by design a tractable solution, we therefore reformulate the offering them up for bid (that is, an offered price), accepting problem based on the reverse auction and propose an bids, and then selling the item to the highest bidder [22]. In approximation algorithm. economics, an auction is a method to determine the value of The main contributions of this paper are summarized as a commodity that has an undetermined or variable price. In follows: some cases, there is a minimum or reserve price and, if the does not reach the minimum price, then no . We demonstrate that most of the existing scheduling transaction between buyers and sellers is executed. Most algorithms suffer from the exposure problem and, of the are primarily forward auctions which involve hence, fail to guarantee the minimum data require- a single seller and multiple buyers. The buyers compete ments of the admitted users. among themselves in order to procure the goods of their . We use combinatorial reverse auction to formulate the choice by placing an initial bid that they feel is an scheduling problem with two different objectives: appropriate price for the item under consideration. How- 1) toguarantee the minimum data rate of the users ever, in reverse auctions, the role of the buyers and the seller and 2) to maximize the overall system throughput. are reversed. A buyer places a request to purchase a . By mathematical analysis, we show that the pro- particular item and multiple sellers bid to sell the requested posed scheme is capable of supporting more users item. The winner of a reverse auction is the seller who offers with hard real-time requirements than the existing the lowest price. Sometimes, the bidders are interested in schedulers. Our approach also leads to a significant bidding for multiple items at the same time. In such a gain in the system throughput. combinatorial bid, the bidder offers a price for the collection . We prove that the worst-case performance of the of goods according to the choice of the bidder rather than by proposed approximate algorithm is bounded by a placing a bid on each individual items separately. This multiplicative factor ð1 þ log mÞ corresponding to results in combinatorial auction, where the auctioneer selects the optimal solution, where m denotes the number a set of combinatorial bids that provides the maximum of slots in a schedule cycle. We have also derived the return in revenue without assigning any item to more than time complexity of the algorithm. one bidder. . We conduct simulation experiments to evaluate the performance of our proposed algorithm with respect 2.2 Scheduling in Wireless Networks: A Qualitative to two extreme scheduling disciplines: round-robin Formulation and throughput maximization. It is observed that Wireless users derive utility from the services received from our approach can schedule more users whose the wireless service providers. The utility perceived is a minimum QoS requirements are met than existing function of the amount of data received in a specific time schemes. epoch. In our study, we define a nonzero minimum utility, . Finally, we propose a design parameter that Umin, that must be met for user satisfaction. Corresponding determines the trade-off between guaranteeing the to Umin, there exists a certain minimum amount of data Dmin user utility level (a measure for user satisfaction) and that must be made available to each user within a specific system throughput. The variation of system capacity deadline. Failure to transmit the “entire” Dmin to the user with the number of satisfied users for different within the deadline or the schedule epoch results in a scheduling algorithms is also shown. twofold penalty that not only leaves the user dissatisfied, The rest of the paper is organized as follows: In Section 2, but also penalizes the system throughput since partial we formulate the scheduling problem for a multirate transmission of the data ð

otherwise, wi;j ¼ 0. We also introduce a value function V ðÞ that maps the data received in a particular slot to the corresponding satisfaction or utility UðÞ of the user receiving in that time slot. 2.4 Optimal Scheduling of Wireless Users Let N be the set of satisfied users whose minimum data requirements, Dmin, have been met at the end of a schedule cycle. The primary objective is to maximize the size jNj in each cycle. In addition, the secondary objective is to maximize the utility of those jNj users and, hence, the system throughput. After the scheduler has allocated time slots to the users, there might exist residual slots which are insufficient to Fig. 1. Illustration of the schedule cycle for a multirate wireless system. satisfy the minimum data requirement of any additional unallocated user. In order to maximize the system users for the particular schedule cycle has been decided, the throughput, these slots are distributed among the allocated schedule then endeavors to maximize the utility among users. Thus, the Optimal Scheduling Policy can be constructed those users. In general, we argue that the throughput as follows: maximization assuming the “pay-per-byte” philosophy is maximize jNjð1Þ detrimental to maximizing the revenue of the service provider since it does not maximize the number of satisfied 8 P < n users whose minimum data rate ðDminÞ is guaranteed. It is j¼1 wi;j ¼ m thus rational to assume that the generated revenue is such that: Uj Umin for 1 j n ð2Þ proportional to the number of users who are satisfied in the wi;j 0: long run if the service provider wants to keep the churn rate In order to achieve the optimal schedule, we first formulate (measure of the user attrition rate) under control [8], [9]. the problem in terms of linear programming (LP). In the 2.3 Wireless System Model next section, we show that the LP is equivalent to the We consider a single-cell multirate time-division multiple optimal scheduling policy. 2 access (TDMA) wireless data system supporting n users. 2.5 LP Formulation Downlink scheduling of the wireless frames is realized by Considering that wi;j determines the schedule matrix and the base station in a time-division manner, whereby, in each r determines the data rate of user j for slot i, the system time slot, the data is transmitted to only one user, as in i;j throughput ðTpÞ for a schedule cycle is given by HDR-based systems [14]. The schedule cycle, the rate X X supported by each user, and the slot allocation for the Tp ¼ wi;jri;j: ð3Þ multirate wireless system are illustrated in Fig. 1. Table 1 i j lists the various parameters for system description and is used by our proposed algorithm in Section 5. Through However, since the optimal schedule does not maximize the channel-state prediction and feedback mechanisms, the throughput for each slot, the system suffers throughput loss base station is made aware of the channel quality and the governed by a penalty function P . The penalty function corresponding data rate experienced by each user for a measuring the system throughput loss for every schedule specific time window corresponding to the schedule cycle. cycle is given by For each user, the slots in the schedule cycle comprise the X X schedule vector. P ¼ ðmax ri;jÞwi;jri;j : ð4Þ i j Among the admitted users, let rij denote the possible transmission bit rates for slot i and experienced by user j. Consequently, the total utility U derived by the users is Consequently, ri;j 2f0;r1; ...;rRg, where R denotes the given by total number of feasible transmission bit rates and 0 signifies that the user is not allocated any slot in the XN U ; 5 schedule vector. Scheduling decisions are periodic and made U ¼ j ð Þ j¼1 every m slots (the actual value of m is implementation specific). We also denote the length of each slot as ts ms. where Uj denotes the utility of user j as a function of the Thus, if a schedule decision is performed at time instance data received. Since the effective objective function is to Td ¼ a, then subsequent decisions are made at times Td ¼ obtain the joint performance measure of all of the user a þ i m ts for i 1. Associated with every scheduling utilities as well as the system throughput loss, we employ a decision is the schedule matrix Sv ¼½wi;j , where 1 i value function V to map both the penalty and the user utility m and 1 j n. If user j is granted slot i, then wi;j ¼ 1; to a common unit (for example, money metric) so that the joint optimization can be achieved. We define the value 2. In this study, we assume that the n users have already been admitted P by the session admission control algorithm, the specific details of which are penalty ðV Þ as a function of p and the value user U beyond the scope of this paper. satisfaction ðV Þ as a function of U . Thus, PAL ET AL.: COMBINATORIAL REVERSE AUCTION-BASED SCHEDULING IN MULTIRATE WIRELESS SYSTEMS 1333

TABLE 1 Notations Used in Auction-Based Scheduling

8 P V ¼ fðP Þ; ð6Þ < 00

3MODELING USER UTILITY The methodology of quantifying the user satisfaction derived from the services received using the concept of utility functions has been established in [8], [9], and [18]. We assume that the utility is an increasing function of the data received ðDrÞ. However, the utility remains zero unless and until a minimum amount of data ðDminÞ is received; that is, even for nonzero Dr, the utility is zero if Dr Dmax. In other words, the utility does an equilibrium state. The market equilibrium corresponds to not significantly increase if the received data exceeds Dmax,as the optimal schedule, as defined in Section 2.4, where the illustrated in Fig. 2. Also, in all of our analyses, the user utility goods map to the time slots and the objective is to satisfy (8). Under such circumstances, the combinatorial auction has been normalized between 0 and 1, where Umin and Umax are the corresponding threshold utilities. problem can be formulated as follows: Let M ¼f1; 2; ;mg denote the set of goods available 3.1 Utility Function and the Exposure Problem for auction and let ujðSÞ denote the utility that a seller Let us now understand the interdependencies between the derives if the buyer j acquires the bundle S. Consequently, utility function and the exposure problem. According to the utility is formulated as auction theory terminology [19], the exposure problem X arises because the users’ valuations for the number of ujðSÞ¼ xVj;xðSÞ; ð11Þ x2X available slots are not additive. This implies that there exists complementarity or substitutability among the slots. Although where X is the set of factors determining the overall utility a user might be allocated slots according to the wireless Pof a bundle S, x is the weight for a given factor x, and channel condition as in opportunistic scheduling or some x2X x ¼ 1. The term Vj;xðSÞ is the value of the factor x by fixed number of slots based on temporal fairness, the allocating the bundle S to buyer j. We define ðS; jÞ as minimum data requirements might not be satisfied. In 1 if bundle S is allocated to buyer j auction terminology, the user might be enticed to bid a ðS; jÞ¼ ð12Þ 0 otherwise: higher price for a subset of the desired bundle, with the hope of acquiring the total bundle, but ends up gaining Thus, the forward combinatorial auction can be formulated nothing since the minimum requirement is not satisfied. as an optimization problem: We first define and then employ the complementarity X X and substitutability effects to demonstrate that the exposure maximize ujðSÞ ðS; jÞð13Þ problem depends on whether the user utility function is j2N SM linear or nonlinear. As outlined in Section 2.2, the exposure 8 P P problem helps us to identify if the slot allocation (that is, the < Pi3S j2N ðS;jÞ1 8i 2 M scheduling) is being performed effectively. In the comple- such that: SM ðS; jÞ1 8j 2 N ð14Þ mentarity effect, the value maximization for the system is ðS;jÞ¼f0; 1g8S M;8j 2 N: only achieved by allocating a particular bundle of slots, but not any subset of it. The substitutability effect encompasses The first condition ensures that the overlapping sets of the scenario when the value maximization of the system is items are never assigned, whereas the second one ensures achieved only when the right bundle of slots is allocated that no bidder receives more than one subset. The reverse and not any superset of it. Note that opportunistic combinatorial auction can be formulated in a similar scheduling does not solve the exposure problem for fashion. Note that, in the reverse auction, there exists a nonlinear utility functions. This is because a nonlinear single buyer intending to procure items from multiple utility function displays subadditivity or superadditivity sellers who quote the minimum price. In both the forward over various ranges, that is, and reverse scenarios, it is assumed that the slots display complementarity and substitutability in terms of utility and ðxÞþðyÞðx þ yÞ or ðxÞþðyÞðx þ yÞ: ð10Þ costs, respectively. A careful observation of (13) reveals that the formulation is identical to the system objective problem Also, most utility functions for users are nonlinear since defined in (8). Finding the solution to (13) is known as the user demand for real-time applications is inherently non- winner determination problem [19] for both combinatorial linear. Opportunistic scheduling mechanisms concentrate forward and reverse auctions. on the current slot to be scheduled and base their decision 4.1 for Multirate Slot Allocation 3. In economics, “marginal utility” is the additional utility (satisfaction or benefit) that a user derives from an additional unit of service, such as time Consider the scenario where the wireless system represents slot in our case. the seller and the users represent the buyers. Recall that the PAL ET AL.: COMBINATORIAL REVERSE AUCTION-BASED SCHEDULING IN MULTIRATE WIRELESS SYSTEMS 1335 primary system objective of scheduling is to maximize the the deprivation function is essential for deciding the best set number of users whose minimum utility, Dmin, is satisfied. of slots for any user. Thus, the quoted price for any bundle The secondary objective is to maximize the system is a function of the deprivation function. throughput once it is no longer possible to add a user 5.1 Deprivation Function whose Dmin can be satisfied. Such conditions require that the auction be done in two stages. In the first stage, bids The objective of the user is to dispose of the set of slots and obtain the desired utility. The buyer (base station) buys the satisfying the minimum utility are determined. Let BI be D the initial feasible bid set for the forward auction. It is set of slots only if a specific minimum value of min is achieved during the bidding process. Failure to sell the slots defined as the set of all sets (bids) of slots which are deprivestheuseroftheminimumutility.Hence,a maximal sets satisfying the minimum utility for all the deprivation value is associated with the set of slots not users. The initial phase is thus described by X X acquired by the base station from that user. The deprivation function depends on two factors: maximize ujðSÞ ðS;jÞð15Þ j2N S2BI 1. the utility derived by the user by giving the bundle 8 P P to the base station and < Pi2S j2N ðS;jÞ1 8i 2BI 2. the throughput loss that the base station may such that wðS;jÞ1 8j 2 N ð16Þ : S2BI experience while procuring the bundle. ðS;jÞ¼f0; 1g8S 2BI ; 8j 2 N: The utility that the user gets from a bundle of slots S The termination criterion for the first phase of slot allocation depends on the type of applications. Following (9), the V U V U V occurs when no additional user can be granted Dmin amount utility function can be defined as j ¼ ð jÞ, where ðÞ is of data. The second phase consists of allocating the residual a value function mapping the utility to an equivalent money slots which are available at the end of the first phase of the metric. Similarly, the monetary equivalent of the through- scheduling operation. Depending on the objective and utility put loss that the system experiences by acquiring bundle S L functions, either a second round of auctions or any standard from user j is calculated using (6) and is denoted by Vj ðSÞ. opportunistic scheduling algorithm can be employed to Therefore, we can define the deprivation function for a slot disseminate the residual slots among the users. Though there bundle S as exists a solution for the above formulation, finding the set of U L winners is shown by [11] to be an NP-complete problem and PjðSÞ¼Vj ðSÞþð1 ÞVj ðSÞ: ð19Þ cannot even be approximated to a ratio of n1 in polynomial Here, is a control or tunable design parameter that time, where n is the total number of users. As a result, it is controls the relative weight of the two attributes. For ¼ 1, infeasible to implement forward-combinatorial-auction- the deprivation function basically boils down to guarantee- based scheduling for real-time multirate wireless systems. ing only user satisfaction, whereas, for ¼ 0, the system This motivates us to explore reverse-auction-based schedul- considers only throughput maximization. ing in the next section. 5.2 Mechanism for Reverse Auction 5MULTIPLE-SLOT SCHEDULING THROUGH We use the simple single-round sealed-bid-first price combi- REVERSE AUCTIONS natorial reverse auction mechanism. All of the “asks” (or quotes) are submitted prior to a deadline and the slot In this section, the delay-sensitive multirate scheduling allocation is achieved based on the set of “asks” received. problem is reformulated based on the reverse combinatorial The throughput would have been drastically penalized had auction. In such a scenario, the wireless base station is the the auction been nonincentive-based. However, the equili- buyer who wants to procure m slots and the set of N users brium is not guaranteed for nonincentive combinatorial are the sellers, each having m slots of different values (that auctions [5]. In general, whether the mechanism is incentive is, data rates). The prices that the users quote for the bundle p S j of slots depend on the utility derived by the user when the compatible or not, the price jð Þ quoted by user for the base station procures those slots. The problem can be bundle S is a function of the deprivation function Pj. Thus, formally stated as p ðSÞ¼f ðSÞP ðSÞ; ð20Þ X X j j j minimize pjðSÞ ðS;jÞð17Þ where fjðSÞ is the price mapping function that defines the j2N SM relationship between the price and the deprivation function. 8 P P Since we have assumed an incentive-compatible auction < Pi2S j2N ðS; jÞ1 8i 2 M mechanism, fjðSÞ¼1 for all j and for all S. such that: SM ðS;jÞ1 8j 2 N ð18Þ The solution to the winner determination provides the ðS; jÞ¼f0; 1g8S M;8j 2 N: desired schedule vector. It has been shown in [11] that, in reverse auction, approximate solutions can be developed in Here, pjðSÞ denotes the price that user j quotes for the slot spite of the fact that the problem is NP-complete. Hence, we bundle S. The solution to the above problem is nothing but develop our slot procurement algorithm along the lines of the winner determination problem. In this framework, the reverse auction. However, since the primary and secondary users compete against each other to sell the set of slots to the objectives have conflicting goals, we decouple the algorithm base station. They are deprived of some value if they cannot into two phases. In the first phase, the restricted phase, we get the base station to buy the slots from them. Identifying compute the set of users whose minimum requirement is 1336 IEEE TRANSACTIONS ON COMPUTERS, VOL. 56, NO. 10, OCTOBER 2007

Fig. 3. Restricted phase: the algorithm for winner set determination. satisfied. That is, slots are acquired from as many users as Once the minimum deprivation value of a user has been possible while requiring that each user is able to get rid of satisfied in a certain round, the user is barred from taking the minimum deprivation value. During the second phase, part in subsequent rounds of the auction process in the the unrestricted phase, the residual slots are allocated, which restricted phase. Let ACQUIREDSET denote the set of cannot satisfy Dmin for any additional user. These two accepted “asks,” and each “ask” Aj is represented by a set j j j j phases are described below. of vector <1;2; ;m > , where i is 1 if the ith slot is in the “ask” for user j; otherwise, it is 0. Let PERMITTEDSET 5.3 Restricted Phase be the set of permitted “asks.” Let us define i such that In this phase, multiple single-round reverse auctions are held i ¼ 1 if the ith slot has not been acquired; otherwise, i ¼ 0. until no additional user is able to get rid of the minimum Let SATISFIEDSET be the set of users whose minimum deprivation value. In each round, the system considers “asks” deprivation value has been satisfied. The algorithm for the from the users on the remaining minimal unallocated restricted phase is described in Fig. 3. bundles. This means that the bundle should just be able to 5.4 Unrestricted Phase get rid of the minimum deprivation value. Each user is The residual slots aid in achieving the secondary objective of allowed to provide an “ask” for only one bundle of slots Aj. maximizing the utility of allocated users, as well as From these initial “asks,” the initial feasible bundle or set, maximizing the system throughput during the unrestricted ðAkÞ, is constructed for round k of the restricted phase. phase. The allocated users strive to further minimize the For each round, the reverse auction takes the following deprivation value by selling their slots. However, unlike the formulation: X X restricted phase, there is no restriction on the size of slot bundle. Note that none of the users whose minimum utility minimize pjðSÞ ðS; jÞð21Þ j2N S2Ak (that is, the minimum deprivation value) has not been satisfied is allowed to compete in this phase. Additionally, 8 P P < k the slots may not exhibit a complementarity/substitutability Pi S j N ðS; jÞ1 8i 2A 3 2 relationship. Consequently, the exposure problem ex- such that k ðS;jÞ1 8j 2 N : S2A plained earlier will not occur. Under such conditions, when ðS; jÞ¼f0; 1g8S 2Ak; 8j 2 N: the utility is assumed to be linear, scheduling the residual PAL ET AL.: COMBINATORIAL REVERSE AUCTION-BASED SCHEDULING IN MULTIRATE WIRELESS SYSTEMS 1337

6PERFORMANCE ANALYSIS In this section, we analyze the proposed algorithms. Theorem 1. The worst-case running time for the restricted phase slot procurement algorithm is Oðn2mÞ, where m is the number of slots in each schedule cycle and n is the number of users. Proof. Assume that m>>n. The complexity of the algorithm in the restricted phase depends on the “ask” construction (line 2) in Fig. 3 and the selection of appropriate j (line 6). In the worst case, line 2 of the algorithm takes ðn kÞðm kÞ operations, where k is the current round. Line 6 takes n k operations in the worst case. The maximum number of possible rounds is n. Thus, the worst-case complexity can be given by ! Xn1 O ½ðn kÞðm kÞþðn kÞ ¼) k¼0 Oðnmðn 1Þnð1 þ 2 þ ...þðn 1ÞÞ mð1 þ 2 þþðn 1ÞÞ þ ð12 þ 22 þ ...þðn 1Þ2Þ¼) ðn 1Þðn 2Þ ðn 1Þðn 2Þ Oðnmðn 1Þn m 2 2 nðn þ 1Þð2n þ 1Þ þ ’Oðn2mÞ 6 since n<

Fig. 4. Unrestricted phase: the residual slot allocation algorithm. Corollary 1. The worst-case complexity of the unrestricted phase slot procurement algorithm is Oðn2mÞ. slots can be performed by employing one of the existing opportunistic scheduling algorithms. Lemma 1. Let the effective price for each slot be defined pj as PðoÞ¼ , where pj is the price paid by user j for On the contrary, if the complementarity/substitutability jAjj effect exists between the residual slots, then the allocation acquiring the set of slot Aj.IfOPT is the total cost that the should be performed using combinatorial reverse auction so base station pays for the optimal solution, then as to overcome the exposure problem. For the unrestricted OPT phase, the “asks” are based on the further reduction of the PðokÞ ; l0 k þ 1 deprivation value. The objective for the buyer (that is, the system) is now set to choose the “asks” from the users, where foig, i ¼ 1; ;l, is an ordering of the slots based on the which minimizes its total price. This guarantees throughput sequence in which they are acquired by the base station, l is the maximization for both the system and the chosen users. The total number of slots procured by the base station in the 0 auction proceeds similar to the restricted phase, but restricted phase, and l l. continues until all of the slots have been exhausted. Let Proof. Let ok be covered (that is, these slots are taken up) k A denote the set of all remaining slots after round k of the when the “ask” Aj was picked by the algorithm. After ok, unrestricted phase. The mathematical formulation for there are at least l0 k þ 1 slots to be covered. Since the round k is given by optimal cost OPT covers all of the l slots, it can also X X cover the remaining l0 k þ 1 slots. Thus, there must be minimize pjðSÞ ðS; jÞð22Þ at least one “ask” whose average cost of covering is at j2N S2Ak OPT most l0kþ1 . As our algorithm chooses the slots from the 8 P P lowest average cost to the highest average cost per slot, < k OPT Pi2S j2N ðS; jÞ1 8i 2A PðokÞl0kþ1 . tu such that k S; j 1 j N 23 : S2A ð Þ 8 2 ð Þ k Theorem 2. The restricted-phase slot procurement algorithm ðS; jÞ¼f0; 1g8S 2A ; 8j 2 N: finds a solution that is within a factor ð1 þ log mÞ of the Note that the difference between (22) and (23) is the type of optimal solution, where m is the total number of slots to be “asks” possible and the set of slots that are part of the procured. reverse auction. The ACQUIREDSET obtained in the Proof. The proof for the bound is similar to the one previous algorithm is used in the unrestricted phase. The presented in [19]. Let l be the total number of slots that algorithm is described in Fig. 4. could be covered by the optimal solution and l0 be the After the execution of the unrestricted phase algorithm, total number of slots that are covered by our algorithm. the ACQUIREDSET is updated, which provides the distribu- From Lemma 1, the proof of Theorem 2 can be outlined tion of the slots for the schedule cycle under consideration. as follows: Let the “asks” which were picked in the 1338 IEEE TRANSACTIONS ON COMPUTERS, VOL. 56, NO. 10, OCTOBER 2007

restricted phase that are able to get rid of the minimum deprivation value be denoted by Aj1;Aj2; ;Ajs, where js is the last user whose bundleP of slotsP was chosen. The s l0 total cost is given by x¼1 pjx ¼ k¼1 PðokÞ.Using Lemma 1, the total cost can be written as

0 Xl 1 l Pðo ÞOPTð1 þ þþ ÞOPTH 0 ; k 2 l0 l k¼1 0 where Hl0 is the ðl Þth harmonic number. Since

0 Hl0 1 þ ln l 1 þ ln l 1 þ ln m; the cost is bounded by ð1 þ log mÞ of the optimal. tu

Lemma 2. Let n~i be the number of slots obtained by user i in order to satisfy the minimum utility using our scheme and n^i be the number of slots obtained using an opportunistic scheme. Fig. 5. Throughput versus number of users in the system. Notice how Then, n^i n~i, 8i 2 N. the throughput decreases with the increase in Dmin. Proof. In our scheme, the slot allocation always tries to give the best available slots to any user so as to satisfy the This is clearly not possible since it would mean that our minimum utility requirement at the minimum cost. auction-based scheme would be able to accommodate at ~ Here, n~i is the minimum number of slots required to least one more user by using the kis. Hence, jNcjjNoj.tu satisfy the minimum utility. Now, consider an opportu- nistic scheme where the decision is based on a slot by slot basis. Consider a user whose minimum utility has been 7SIMULATION STUDY satisfied. If the user’s best available slots come in descending order of their individual utility values, then This section studies the effectiveness of our proposed the user will reach the minimum utility level with the scheduling scheme through simulation experiments. We smallest number of slots. In this case, n^i ¼ n~i. Otherwise, also compare how the auction-based scheme fares with the user may get another slot, which is not the user’s respect to two extreme scheduling disciplines—round-robin available slot. Therefore, to satisfy the minimum utility, and throughput maximization—that serve as the basis for the user will require at least the minimum number of comparing the fairness and maximum system throughput, slots. In either case, n^i n~i. tu respectively. We study how each scheme performs in terms of the number of satisfied users and global system Theorem 3. Let Nc be the set denoting the maximum number of throughput. users whose minimum utility has been satisfied by the combinatorial-reverse-auction-based scheduling and let No be 7.1 System and Channel Model the set of users who have been satisfied by the opportunistic We consider a single-cell wireless data network for our scheme. Then, jNcjjNoj. simulation study due to the fact that the scheduling Proof. From Lemma 2, n^i n~i for all i whose minimum schemes under evaluation are designed to work best in utility has been satisfied. By contradiction, let us assume the presence of a single base station. We also assume that all that jNcj < jNoj. Then, of the users under consideration are receiving real-time streaming multimedia traffic. In order to support multi- XjNoj XjNcj media traffic (MPEG-4 or H.263) of various qualities (low, ðn^ k~ Þþl ¼ ðn^ k~ Þ; ð24Þ i i i i medium, and high), as given in [7], we consider three i¼1 i¼1 values for D : 16, 64, and 128 kilobits per second (Kbps). ^ ~ min where ki and ki are the extra slots given to user i after We model our simulation based on the HDR system that is satisfying the minimum utility and l is the total number capable of supporting 11 different data rates, with each of slots given in the case of opportunistic scheduling to schedule cycle consisting of 1,000 slots. We assume that user users whose minimum utility could not be satisfied. The mobility is random (both speed and direction) and employ above equation can be rewritten as the path-loss model and the slow log-normal model [25] for wireless channels. XjNcj XjNoj XjNoj XjNcj ^ ~ ^ ðn^i n~iÞþ n^i þ l þ ki ¼ ðki kiÞ: ð25Þ 7.2 Simulation Results i¼1 i¼jNcjþ1 i¼jNcj i¼1 For our proposed auction-based scheduling scheme, the However, this implies that variation of the system throughput with the number of users for different values of Dmin is shown in Fig. 5. As XjNcj XjNoj expected, the system throughput initially increases, but k~ n^ : ð26Þ i i ultimately gets saturated with the increase in the number of i¼1 i¼jNcj users. Next, we identify the maximum system capacity in PAL ET AL.: COMBINATORIAL REVERSE AUCTION-BASED SCHEDULING IN MULTIRATE WIRELESS SYSTEMS 1339

TABLE 2 Dmin versus Maximum Number of Users

terms of satisfied users by setting Dmin to different values. For each value of Dmin, we obtained a range of users who are satisfied by the system. This is shown in Table 2. It is Fig. 7. Throughput versus the number of users in the system for different logical that, for smaller Dmin, a greater number of users can scheduling algorithms. be satisfied (see Fig. 6). The comparison of the system throughput achieved by the various schemes is shown in the residual slots after the restricted phase and does not Fig. 7. As expected, the system throughput is the best for the allocate any more slots if Dmax is attained, as in the case throughput maximization scheme and the worst for the with user 10 in this example. Careful observation reveals round-robin scheduling algorithm. In the case of opportu- that, although all of the allocated users were receiving D nistic scheduling with temporal fairness, the throughput is min or more data, user 7 was receiving lesser slots in each penalized. The throughput performance of the proposed succeeding schedule cycle such that, in the fourth cycle, auction-based scheme is better than both the round-robin scheduling and opportunistic scheduling with temporal user 7 was eliminated by user 9 in the restricted phase. fairness and is very close to the throughput maximization Thus, the scheduler is intelligent enough to identify and scheme. In order to visualize the working of the proposed allocate the user to achieve the system objective. In each scheduling scheme, we consider a hypothetical scenario schedule cycle, all of the allocated users are guaranteed D with 15 users in the system. The users are represented as ui min amount of data. in Fig. 8. For the purpose of explaining the workings of our Next, we investigate the variation of the system algorithm, a temporal snapshot of four successive schedul- throughput and the number of satisfied users with the ing decision cycles is also presented. tunable parameter , as defined in (19). The value of With Dmin ¼ 128Kbps, the first three schedule cycles depends on the objective of the wireless service providers yield the schedule vector as ½1; 2; 4; 5; 6; 7; 10. However, for that maximizes the throughput, guarantees user utility, or a the fourth cycle, user 7 is replaced by user 9. Although the combination of both. Hence, we evaluate the system allocated users are receiving Dmin, the variation of the slot throughput and the number of satisfied users by varying distribution between users is due to the varying channel from 0 to 1. For ¼ 0, the deprivation function totally conditions. The scheduling scheme judiciously distributes becomes a function of the throughput maximization, whereas, for ¼ 1, the deprivation function only cares about the user satisfaction. As expected, the throughput

Fig. 6. Performance of each scheduling scheme measured using satisfied users as a percentage of the total users. Fig. 8. Slot distribution of users in schedule cycle. 1340 IEEE TRANSACTIONS ON COMPUTERS, VOL. 56, NO. 10, OCTOBER 2007

[2] P. Bender, P. Black, M. Grob, R. Padovani, N. Sindhushayana, and A. Viterbi, “CDMA/HDR: A Bandwidth-Efficient High-Speed Wireless Data Service for Nomadic Users,” IEEE Comm. Magazine, pp. 70-77, July 2000. [3] General Packet Radio Services (GPRS) service description, 3GPP TS 03.60, http://www.3gpp.org, 2007. [4] A. Jalali, R. Padovani, and R. Pankaj, “Data Throughput of CDMA-HDR: A High-Efficiency High Data Rate Personal Com- munication Wireless System,” Proc. 51st IEEE Vehicular Technology Conf. (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 Energy Transmission Scheduling Subject to Deadline Constraints,” Proc. Third IEEE Modeling and Optimization in Mobile, Ad Hoc, and Wireless Networks (WiOPT ’05), pp. 67-76, 2005. [7] F. Fitzek and M. 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Hastad, “Clique Is Hard to Approximate within n1,” Acta In this paper, we have proposed an auction-based scheduling Mathemitica, vol. 182, pp. 105-142, 1999. algorithm for allocating the slots in a time-division multirate [12] J. Sun, L. Zheng, and E. Modiano, “Wireless Channel Allocation wireless system. We have justified that opportunistic Using an Auction Algorithm,” Proc. 41st Allerton Conf. Comm., Control, and Computing, vol. 24, pp. 1085-1096, Oct. 2003. scheduling algorithms that aim to maximize the system [13] J. Sun, E. Modiano, and L. Zheng, “A Novel Auction Algorithm throughput are unable to address the exposure problem. We for Fair Allocation of a Wireless Fading Channel,” Proc. 38th Ann. have formalized the slot allocation problem in the form of a Conf. Information Sciences and Systems (CISS ’04), Mar. 2004. market, where multiple users bid for the number of slots to [14] P. Bender, P. Black, M. Grob, R. Padovani, N. Sindhushayana, and A. Viterbi, “CDMA/HDR: A Bandwidth-Efficient High-Speed satisfy their minimum QoS requirements. With the help of Wireless Data Service for Nomadic Users,” IEEE Comm. Magazine, combinatorial auctions, we have shown how the exposure pp. 70-77, July 2000. problem can be successfully eliminated. In the process, we [15] P. Maille, “Auctioning for Downlink Transmission Power in have been able to achieve the primary objective of maximiz- CDMA Cellular Systems,” Proc. Seventh ACM Int’l Symp. Modeling, Analysis and Simulation of Wireless and Mobile Systems (MSWiM ing the number of users whose minimum slot requirements ’04), pp. 293-296, Oct. 2004. are satisfied. The remaining slots, if any, are allocated with a [16] P. Maille and B. Tuffin, “Multi-Bid Auctions for Bandwidth view to maximizing the system throughput. In our study, we Allocation in Communication Networks,” Proc. IEEE INFOCOM have applied the reverse auction theory in order to deal with ’04, Mar. 2004. [17] S.S. Kulkarni and C. 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ACKNOWLEDGMENTS [20] T. Ling and N. Shroff, “Scheduling Real-Time Traffic in ATM Networks,” Proc. IEEE INFOCOM ’96, vol. 1, pp. 198-205, 1996. The authors would like to thank the anonymous referees for [21] V. Bharghavan, S. Lu, and T. Nandagopal, “Fair Queuing in valuable suggestions to improve the quality of the paper. Wireless Networks: Issues and Approaches,” IEEE Personal Comm., vol. 6, pp. 44-53, Feb. 1999. They would also like to thank Kalyan Basu for helpful [22] V. Krishna, Auction Theory. Academic Press, 2002. discussions and Shuvendu Dang for his contributions to [23] X. Liu, E.K.P. Chong, and N.B. Shroff, “Transmission Scheduling this work. This work is partially supported by the US for Efficient Wireless Utilization,” Proc. IEEE INFOCOM ’00, National Science Foundation Information Technology Re- vol. 3, pp. 776-785, 2000. search (NSF ITR) Grant IIS-0326505. [24] X. Liu, E.K.P. Chong, and N.B. Shroff, “Optimal Opportunistic Scheduling in Wireless Networks,” Proc. 58th IEEE Vehicular Technology Conf. (VTC ’03 Fall), vol. 3, pp. 1417-1421, Oct. 2003. [25] EFERENCES X. Liu, K.P. Chong, and N.B. Shroff, “Opportunistic Transmission R Scheduling with Resource-Sharing Constraints in Wireless Net- [1] IS-856 cdma2000 high rate packet data air interface specification, works,” IEEE J. Selected Areas in Comm., vol. 19, no. 10, pp. 2053- 3GPP2 C.SO024 version 4.0, http://www.3gpp2.org, Oct. 2002. 2064, Oct. 2001. PAL ET AL.: COMBINATORIAL REVERSE AUCTION-BASED SCHEDULING IN MULTIRATE WIRELESS SYSTEMS 1341

[26] Y. Liu, S. Gruhl, and E. Knightly, “WCFQ: An Opportunistic Sajal K. Das is a distinguished scholar professor Wireless Scheduler with Statistical Fairness Bounds,” IEEE Trans. of computer science and engineering and the Wireless Comm., vol. 2, pp. 1017-1028, Sept. 2003. founding director of the Center for Research in [27] Y. Liu and E. Knightly, “Opportunistic Fair Scheduling over Wireless Mobility and Networking (CReWMaN) Multiple Wireless Channels,” Proc. IEEE INFOCOM ’03, vol. 2, Laboratory at the University of Texas at Arling- pp. 1106-1115, 2003. ton (UTA). He is also a visiting professor at the Indian Institute of Technology (IIT), Kanpur, and Sourav Pal received the BE degree from the IIT Guwahati, an honorary professor of Fudan Bengal Engineering College, Shibpur, India, and University, Shanghai, and a visiting scientist at the MS degree from the University of Texas at the Institute of Infocomm Research (I2R), Arlington (UTA). He is currently working toward Singapore. He is frequently invited as keynote speaker at international the PhD degree at the Center for Research in conferences and symposia. He serves as the founding editor in chief of Wireless Mobility and Networking Laboratory at the Pervasive and Mobile Computing journal (Elsevier) and an associate UTA. His interests and expertise are wireless editor of the IEEE Transactions on Mobile Computing, ACM/Springer and multimedia systems. He is a student Wireless Networks, IEEE Transactions on Parallel and Distributed member of the IEEE. Systems, and Journal of Peer-to-Peer Networking. He is the founder of the IEEE International Symposium on a World of Wireless, Mobile, and Multimedia Networks (WoWMoM) and a cofounder of the annual IEEE International Conference on Pervasive Computing and Communications (PerCom). He has served as the general chair, the Technical Program Sumantra R. Kundu received the BTech Committee (TPC) chair, or a TPC member for numerous IEEE and ACM degree from the Indian Institute of Technology, conferences. He is a member of the executive committees of the IEEE Kharagpur, India, and the MS degree from the CS Technical Committee on Computer Communications (TCCC) and University of Iowa, Iowa City. He is currently a Technical Committee on Parallel Processing (TCPP). His current PhD candidate at the Center for Research in research interests include the design and modeling of smart environ- Wireless Mobility and Networking Laboratory at ments, sensor networks, security, mobile and pervasive computing, the University of Texas at Arlington. His resource and mobility management in wireless networks, wireless research interests are operating system (OS) multimedia, mobile Internet, mobile grid computing, biological network- internals, hardware architecture, efficient data ing, applied graph theory, and game theory. He has published more than structures, and performance evaluation using 400 papers in international conference proceedings and journals and statistical principals and queuing theory. He is a student member of more than 30 invited book chapters. He is the holder of five US patents the IEEE. in wireless mobile networks and coauthored the book Smart Environ- ments: Technology, Protocols, and Applications (John Wiley, 2005). He Mainak Chatterjee received the BSc degree received the Best Paper Awards from PerCom 2006, ACM MobiCom ’99, (Hons) in physics from the University of Calcutta Information Networking, Wireless Communications Technologies and in 1994, the ME degree in electrical commu- Network Applications, International Conference (ICOIN) 2002, Third nication engineering from the Indian Institute of ACM International Workshop on Modeling, Analysis, and Simulation of Science, Bangalore, in 1998, and the PhD Wireless and Mobile Systems (MSwiM 2000), and ACM/IEEE 11th degree from the Department of Computer Workshop on Parallel and Distributed Simulation (PADS 1997). He is Science and Engineering at the University of also a recipient of the 2006 UTA Academy of Distinguished Scholars Texas at Arlington in 2002. He is currently an Award, 2005 University Award for Distinguished Record of Research, assistant professor in the School of Electrical 2003 College of Engineering Research Excellence Award, and 2001 and Engineering and Computer Science at the 2003 Outstanding Faculty Research Award in Computer Science. He is University of Central Florida. He serves on the executive and technical a member of the IEEE. program committee of several international conferences. His research interests include economic issues in wireless networks, applied game theory, resource management and quality-of-service provisioning, ad hoc and sensor networks, code division multiple access (CDMA) data . For more information on this or any other computing topic, networking, and link-layer protocols. He is a member of the IEEE. please visit our Digital Library at www.computer.org/publications/dlib.