Structural Properties of LDP for Queue-Length Based Wireless Scheduling Algorithms

In this paper, we are interested in wireless scheduling algorithms for the downlink of a single cell that can minimize the queue-overflow probability. Assuming that a samplepath large-deviation principle holds for the backlog process, we first study structural properties of the minimum-cost-pathto-overflow for a class of scheduling algorithms collectively referred to as the “α-algorithms.” For a given α ≥ 1, the α-algorithm picks the user for service at each time that has the largest product of the transmission rate multiplied by the backlog raised to the powerα. We show that when the overflow metric is appropriately modified, the minimum-cost-to-overflow under the α-algorithm can be achieved by a simple linear path, and it can be written as the solution of a vector-optimization problem. Using this structural property, we then show that when α approaches infinity, the α-algorithm asymptotically achieves the largest value of the minimum-cost-to-overflow under all scheduling algorithms. I. I NTRODUCTION Link scheduling is an important functionality in wireless networks due to both the shared nature of the wireless medium and the variations of the wireless channel over time. In the past, it has been demonstrated that, by carefull y choosing the scheduling decision based on the channel state and/or the demand of the users, the system performance can be substantially improved (see the references in [1]). Most studies of scheduling algorithms have focused on optimizin g the long-term average throughput of the users. Similarly, i n the class of stability problems, the goal is to find schedulin g algorithms that can stabilize the network at given offered loads, which also ensures that the long-term average servic e rate is no less than the arrival rate of each user. An importan t result along this direction is the development of the socalled “throughput-optimal” algorithms [2]. An algorithmis called throughput-optimalif, at any offered load that any other algorithm can stabilize the system, this algorithm ca n stabilize the system as well. Therefore, a throughput-opti mal scheduling algorithm is optimal if we only impose stability constraints, i.e., it can stabilize the system over the larg est set of offered loads. While stability (and ensuring that the long-term service rate is no smaller than the arrival rate) is an important first order metric of success, for many delay-sensitive applicat ions it is far from sufficient. Note that a stability objective ens ures that the packet delay does not increase to infinity. For realtime applications such as voice and video, we often need to ensure a stronger condition that the packet delay can be uppe r This work has been partially supported by the National Scien ce Foundation through awards CCF-0635202, CNS-0643145, and CNS-072 1484. The authors are with School of ECE, Purdue University, West L afayette, IN 47907. Email:{vvenkat,linx}@ecn.purdue.edu. bounded with high probability. One approach to quantify the requirements of these delay-sensitive applications is to enforce constraints on the probability of queue overflow. In other words, we need to guarantee that the probability of each user’s queue exceeding a given threshold is no greater than a target value. In this paper, we are interested in scheduling algorithms that are optimal subject to the above type of delay constrain ts. We focus on the downlink of a single cell in a cellular network. The base-station serves multiple users. Due to interference, the base-station can only serve one user at a time. We assume that perfect channel information is availab le at the base-station. The ultimate question that we attempt t o answer is the following: Is there a delay-optimalalgorithm in the sense that, at any given offered load, the algorithm can achieve the smallest probability of queue-overflow. Not e that if we impose a quality-of-service (QoS) constraint on each user in the form of an upper bound on the queueoverflow probability, then the above optimality condition w ill also imply that the algorithm can support the largest set of offered loads subject to the QoS constraint. The above question has well-known to be a difficult one. The closest answer in the literature is provided in [3], wher e the author studied the problem in a large-deviation setting , and showed that the so-called “exponential-rule” is delayoptimal in the case with two-users. In a related result, it was shown that for the case when the service rate is fixed, the largest-weighted-delay-first (LWDF) algorithm is delay optimal in a large-deviation setting [4], [5]. To the best of our knowledge, the general case for wireless networks with an arbitrary number of users is still open. Note that to study the queue-overflow probability, it is natural to use the large-deviation theory because the overflow probabilit y of interest is often very small [6], [7]. The queue-overflow probability can then be mapped to the decay-rate of the taildistribution of the queue, and the delay-optimal schedulin g algorithm will correspond to the one that maximizes this delay-rate. Large-deviation theory has been successfully applied to wireline networks (see, e.g., [8]–[13]) and to wire less scheduling algorithms that only use the channel state to make the scheduling decisions [14]–[16]. However, when applied to wireless scheduling algorithms that use also the queue-length to make scheduling decisions, this approach encounters a significant amount of technical difficulty. Not e than many scheduling algorithms of interest are of this latt er flavor, i.e., they choose the user to serve based on both the channel state and the queue backlog. For example, the maxweight algorithm that is known to be throughput-optimal [17 ] serves at each time the user with the largest product of the queue length and the service rate. Intuitively, this class o f queue-length-based scheduling algorithms will have a lowe r queue-overflow probability compared to the queue-unaware algorithms because they make an effort to suppress longer queues. Indeed, the work in [18] has analytically shown the superiority of queue-length-based scheduling algorithms over queue-unaware algorithms for a symmetric case with ONOFF channels. However, the technical difficulty associated with the queue-length-based scheduling algorithms is that the statistics of the service-rate process for each user is unkn own (because now the service-rate process is tightly coupled wi th the backlog process, the channel variations, and the arriva l process). In order to apply the large-deviation theory to queue-length-based scheduling algorithms, one has to use sample-path large-deviation, and formulate the problem as a multi-dimensional calculus-of-variations (CoV) problem for finding the “minimum-cost path to overflow.” The decay-rate of the queue-overflow probability then corresponds to the cost of this path, which is referred to as the “minimum cost to overflow.” Unfortunately, for many scheduling algorithms o f interest, this multi-dimensional calculus-of-variation s problem is very difficult to solve. In the literature, only some restricted cases have been solved: Either restricted probl em structures are assumed (e.g., symmetric users and ON-OFF channels [18]), or the size of the system is very small (only two users) [19]. Due to the above difficulty, the question of finding the optimal wireless scheduling algorithms under delay constraints becomes very challenging. In this paper, we provide a number of results along this direction. Assuming that a sample-path large-deviati on principle holds, we study the structural property of the minimum-cost-path-to-overflow for a class of queue-length based scheduling algorithms. In particular, we show that when the form of the overflow threshold is appropriately modified, at least one of the minimum-cost-path-to-overflow is linear. This result allows us to convert the calculusof-variations problem (of sample-path large-deviation) t o a vector-optimization problem. Using this structure proper ty, we then show the main result of the paper that, as one of the parameters approaches infinity, these class of queue-lengt hbased scheduling algorithms will asymptotically achieve t h largest minimum-cost-to-overflow among all scheduling algorithms. As an immediate corollary of this result, we can show that with the ON-OFF channel model, the max-weight scheduling algorithm is optimal. The rest of paper is organized as follows. We first present the system model and the class of queue-lengthbased scheduling algorithms (referred to as α-algorithms) in Section II. In Section III, we provide an upper bound on the minimum-cost-to-overflow for any scheduling algorithm . We then study the structural properties of the minimum-cost path-to-overflow forα-algorithms in Section IV. Then in Section V, we prove the main result that, as the parameter α approaches infinity, this class of scheduling algorithms asymptotically achieve the largest possible value of the minimum-cost-to-overflow. Then we conclude. II. T HE SYSTEM MODEL AND ASSUMPTIONS We consider the downlink of a single cell in which a basestation servesN users. We assume a slotted system, and we assume that the state of the channel at each time slot is i.i.d from one ofM possible states. Let C(t) denote the state of the channel at timet = 1, 2, . . . , and let pj = P[C(t) = j], j = 1, 2, . . . ,M. Let ~ p = [p1, ..., pM ]. We assume that the base-station can serve one user at a time. Let F i m denote the service rate for user i when it is picked for service at statem. We assume that data for user i arrives as fluid at a constant rateλi. Let ~λ = [λ1, . . . , λN ]. Let Xi(t) denote the backlog of user i at time t, and let ~ X(t) = [X1(t), . . . ,XN (t)]. In general, the decision of picking which user to serve is a function of the global backlog~ X(t) and the channel state C(t). Let U(t) denote the index of the user picked for service at timet. The evolution o