IEEE TRANSACTIONS ON COMPUTERS, VOL. 56, NO. 10, OCTOBER 2007 1329 Combinatorial Reverse Auction-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 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 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 auction theory [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. combinatorial auction. 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 bidding 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 auctions 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;jÞ wi;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 fð P Þ; ð6Þ < 00