IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 53, NO. 11, NOVEMBER 2005
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Wireless Packet Scheduling Based on the Cumulative Distribution Function of User Transmission Rates Daeyoung Park, Member, IEEE, Hanbyul Seo, Student Member, IEEE, Hojoong Kwon, Student Member, IEEE, and Byeong Gi Lee, Fellow, IEEE
Abstract—In this paper, we present a new wireless scheduling algorithm based on the cumulative distribution function (cdf) and its simple modification that limits the maximum starving time. This cdf-based scheduling (CS) algorithm selects the user for transmission based on the cdf of user rates, in such a way that the user whose rate is high enough, but least probable to become higher, is selected first. We prove that the CS algorithm is equivalent to a scheduling algorithm that regards the user rates as independent and identically distributed, and the average throughput of a user is independent of the probability distribution of other users. So, we can evaluate the exact user throughput only if we know the user’s own distribution, which is a distinctive feature of this proposed algorithm. In addition, we try a modification on the CS algorithm to limit the maximum starving time, and prove that the modification does not affect the average interservice time. This CS with starving-time limitation (CS-STL) algorithm turns out to limit the maximum starving time at the cost of a negligible throughput loss. Index Terms—Interservice time, multiuser diversity, starving time, wireless scheduling.
I. INTRODUCTION
T
HE rapid growth of the needs for high-rate data applications demands efficient wireless data-access schemes. Data traffic, in contrast to voice traffic, usually has a bursty arrival pattern and can sustain a certain amount of packet delay, given that the long-term throughput is sufficiently high. These characteristics call for the potential use of scheduling for data transmissions. On the other hand, the channel condition of mobile communications is time-varying in wireless networks due to path loss, shadowing, and multipath fading. For wireless data services, users with better channel conditions may be served at higher data rates by adopting adaptive modulation and coding (AMC). Therefore, if we can properly combine the scheduling and AMC schemes, more efficient wireless data services would be possible. Based on the notion above, the high-data-rate (HDR) system was proposed [1]. In HDR, the system allocates a time slot to
Paper approved by M. Hamdi, the Editor for Network Architecture of the IEEE Communications Society. Manuscript received April 17, 2004; revised March 22, 2005. This work was supported in part by the Samsung Advanced Institute of Technology, and in part by the University IT Research Center Project. This paper was presented in part at IEEE Globecom, San Francisco, CA, December 2003. D. Park was with the School of Electrical Engineering and Computer Science, Seoul National University, Seoul 151-742, Korea. He is now with Samsung Advanced Institute of Technology, Suwon 440-600, Korea (e-mail:
[email protected]). H. Seo, H. Kwon, and B. G. Lee are with the School of Electrical Engineering and Computer Science #029, Seoul National University, Seoul 151-742, Korea (e-mail:
[email protected];
[email protected];
[email protected]). Digital Object Identifier 10.1109/TCOMM.2005.858675
the currently best user having the maximum service rate that can be supported in the current channel condition. Since each user is served when its channel condition falls in the relatively best time slot, the overall throughput of the system becomes a maximum. In order to fully exploit such multiuser diversity [2], however, it is important to set a good scheduling algorithm that can maximize the system throughput and, in addition, can accomplish a fair resource share among users as well. The simplest scheduling algorithm may be the one that selects the user for which the supportable rate is the highest, but in this case, it can lead to selecting some particular users more frequently than other users due to uncontrollable channel conditions, thus causing a critical fairness problem. In order to improve such a maximum-rate algorithm [3], [4], there have been several other scheduling algorithms proposed for use in HDR systems [5]–[7]. Each of those algorithms intended to optimize the system performance while satisfying its own-defined fairness criterion. The scheduling algorithm by Jalali et al. [5] was proposed to satisfy the so-called proportional fairness, the algorithm by Liu et al. [6] tried to maximize the long-term average total throughput for a given time fraction, and the scheduling algorithm by Borst et al. was made to be Pareto-optimal [7], [8]. All those existing scheduling algorithms guarantee fairness by adjusting some control parameters, such as adding offsets and weighting factors. However, it is hard to set the control parameters appropriately to guarantee prespecified throughputs. In this sense, those existing scheduling algorithms provide relative fairness among users rather than guarantee a prespecified performance goal. In this paper, we are going to introduce a new wireless scheduling algorithm, called the cumulative distribution function (cdf)-based scheduling (CS) algorithm, that provides a fair allocation of the transmission time fraction. The CS algorithm schedules the user whose channel condition is at its best state, and is most unlikely to be better. We will investigate the properties of the CS algorithm, including that the average throughput of a user is independent of the other users’ probability distribution. In addition, we will show that the exact average throughput can be derived analytically, which is useful to guarantee prespecified user throughputs. Further, we will consider extending the CS algorithm for use in delay-sensitive applications by limiting the number of consecutively unserved slots without sacrificing much average throughput. This paper is organized as follows. We first describe the system model and the existing memoryless scheduling algorithms in Section II. Then we present the CS algorithm and investigate its properties in Section III. In Section IV, we
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IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 53, NO. 11, NOVEMBER 2005
examine the CS algorithm for the case when user rates are Gaussian distributed. Finally, in Section V, we extend the CS algorithm to limit the maximum starving time. II. SYSTEM MODEL AND SCHEDULING ALGORITHMS To begin with, we describe the system model and the existing scheduling algorithms for reference in later comparisons. We also introduce the concept of equivalence of scheduling algorithms to support the analytical discussions that follow in the subsequent sections. A. System Model and Existing Scheduling Algorithms We consider the HDR downlink channel [1], in which user , measures the downlink channel, computes at time slot that the downthe maximum possible rate link can support, and feeds this rate information back to the base station (BS). Based on the rate information, the BS determines which user to serve at the next time slot, and transmits a packet to that user. We assume that in the BS, the queue for each user’s traffic is filled with an infinite number of packets, such that no queue depletion occurs. be the user selected for data transmission at slot Let the average throughput of user until slot ; and the long-term average throughput of user . We assume that each is an independent and stationary random sequence, and each user can feed back this channel information to the BS in an error-free manner. Then the existing scheduling algorithms can be characterized as follows. • Maximum rate (MR) algorithm [3], [4]: It selects the user , to be the user that for the next data transmission, has the largest value of the maximum available transmission rate, i.e., (1)
•
It maximizes the throughput at each slot, as well as the , but does not meet any fairness total throughput criteria. Proportional fairness (PF) algorithm [5]: It selects to be the user for which the ratio of over is the largest, i.e., (2)
•
It turns out to maximize the product of the long-term av, which means that it achieves erage throughputs, PF. Algorithm by Liu et al. [6]: It selects the user for which and an offset is the largest, the sum of the rate i.e., (3) is determined to satisfy a given where the offset time-fraction assignment requirement. With this requirement met, this algorithm maximizes the average total throughput.
•
Algorithm by Borst et al. [7]: It selects the user for which the weighted rate is the largest, i.e., (4)
for the weight determined to make each normalized long-term average throughput identical, i.e., . With this requirement met, it maximizes the average total throughput, and hence, achieves the Pareto-optimal throughput. Multiuser diversity takes advantage of the fact that it is highly probable that at each time slot, there exists a user for which the transmission rate is significantly higher than its average value [2]. The MR algorithm allocates the transmission time slot to that particular user, and hence, the overall total throughput gets maximized. The other algorithms introduce an offset or a weight to transform user rates into some different form to incorporate fairness while maximizing the throughput, so the user selected for transmission at a time slot may be different from that selected by the MR algorithm. B. Memoryless Scheduling Algorithms and Equivalence Relation The PF algorithm selects a user for which the ratio of the current rate to the current throughput is the largest, so each selection depends on the past selection results. In contrast, the other scheduling algorithms depend not on the history of the user selection, but on the current user rate only. We may express such memoryless scheduling algorithms in the general form1 (5) is a nondecreasing function that does not depend where on the history of the user selection.2 So, a memoryless scheduler may be characterized by the set of the functions . In some cases, there may exist two or more memoryless algorithms that yield the same scheduling results, but are not necessarily to be distinguished. In order to handle such algorithms collectively, we define the term “equivalent” as follows. Definition 1: Two memoryless scheduling algorithms and characterized by , respectively, are said to be equivalent to each for all possible other if ’s other than those that lead to a tie.3 The equivalence above corresponds to the equivalence relation in [9] applied on the set of memoryless scheduling algorithms. By applying the equivalence concept, we can put together all the scheduling algorithms that are equivalent to a particular scheduling algorithm S to form an equivalence class determined by S. This enables us to partition the whole memoryless scheduling algorithms into disjoint subsets, with each subset consisting of equivalent scheduling algorithms. 1When a tie occurs (that is, when two or more users yield the same maximum value), one of the users is selected in random fashion. 2A function ( ) is said to be monotonically increasing if ( ) ( ) for and nondecreasing if ( ) ( ) for . 3If we apply a random-selection rule whenever a tie occurs, a scheduler can ’s. So, we exclude possibly yield many different scheduling results for some the ’s that lead to a tie.
x
