Maximizing Utility via Random Access without Message Passing

It has been an intensively sought-after goal to achieve high throughput and fairness in wireless scheduling through simple and distributed algorithms. Many recent papers on the topic have relied on various types of message passing among the nodes. The following question remains open: can scheduling without any message passing guarantee throughput-optimality and fairness? Over the last year, it has been suggested in three papers [1]–[3] that random access without message passing may be designed and proved to be optimal in terms of throughput and utility. In this paper, we first extend the algorithm in [2] and provide a rigorous proof of utility-optimality for random access without message passing for Poisson clock model. Then we turn to the more difficult discrete contention and backoff model with collisions, study its optimality properties, and control a tradeoff between long-term efficiency and short-term fairness that emerges in this model. I. I There have been growing interests in the design of scheduling algorithms which efficiently and fairly exploit the radio resources in wireless networks in recent years. In their seminal work [4], Tassiulas and Ephremides developed a centralized scheduling algorithm, the Max-Weight scheduler, achieving throughput optimality. The traffic scenario considered in [4] is that of infinite buffers fed by exogenous random packet arrivals with fixed rates, and being throughput optimal means that the proposed algorithm achieves the maximum stability region. In this paper, we are interested in a different traffic scenario, more appropriate to represent the elasticity of traffic in data networks. We considered saturated users (i.e., with fully back-logged buffers), who perceive performance as a function of the average service rate, and the problem is to design a scheduling algorithm that achieves the desired trade-off between efficiency and fairness. Specifically, the proposed scheduling algorithm aims at maximize the sum of user utilities, where the utility is a non-decreasing and concave function of the user average service rate. This optimization problem has received a lot of attention recently, for it appears not only as a MAC layer problem but also in joint rate control and scheduling through dual decomposition, e.g., in [5]–[8]. There has been a long series of work on distributed scheduling, indeed too long to list here, involving randomization, maximal weight matching, and random access with message passing. They usually require some information of the queues to be passed around among the nodes, e.g., in [9]–[14]. These signaling overhead reduces the effective throughput and makes the algorithms not fully distributed. This naturally leads to the following question that turns out to be very challenging: what about the performance of random access algorithms without any message passing? In this paper, we focus on such algorithms. In recent papers, it has been demonstrated that such algorithms could also achieve strong throughput performance. For example, in [1], [15], [16], it has been shown that non-adaptive CSMA (Carrier Sense Multiple Access) algorithms, where each link accesses the channel with a fixed probability, are able to provide average throughputs close to throughput-optimality. Turning to random access with adaptive channel access rate, [17] first proposed a simulated-annealing based approach to distributed scheduling. Similar idea has been developed this year in two papers at similar time: in [3], Rajagopalan and Shah suggested that users can adapt their access channel rate depending on their buffer size, so that the system dynamics under the random CSMA algorithm solves the Max-Weight problem. As discussed in [18], one issue with this approach is that when the buffer of a given user becomes large, its channel access rate should also become large. Consequently, to ensure buffer stability and to control the system behavior for arbitrarily large buffers, one should be able to design a CSMA protocol with arbitrarily large access rates. This is made possible in [3] by implementing idealized CSMA algorithms, where Poisson clocks are used to control the channel accesses, and to ensure zero collisions. By simply limiting the virtual buffer sizes, the problem of large buffers and stability in the implementation of the simulated annealing technique may be avoided. In [2], Jiang and Walrand also use the idea of simulated annealing technique as in [3], [17], and they propose an adaptive CSMA algorithm (without message passing) to maximize utility. In this paper, we provide a further detailed analysis of such promising algorithms. The contributions of this paper are three-fold: • We first extend the algorithms presented in [2], and provide a generalized framework for random access without message passing with two styles of algorithms that combine the simulated-annealing approach in [17] and the loss network model in [19]. • We develop a rigorous proof of the convergence of these algorithms (such a proof is not presented in [2]). The proof of convergence is conducted by analyzing the behavior of stochastic sub-gradient algorithms modulated by a Markov chain. • We then turn from the Poisson clock model used by the references above to the more challenging discrete-time contention and backoff model. There, the effect of collisions cannot be ignored and a tradeoff between long-