Resource Allocation in Multiple Access Channels

arXiv:0810.1248v1 [cs.IT] 7 Oct 2008 1 Resource Allocation in Multiple Access Channels Ali ParandehGheibi, Atilla Eryilmaz, Asuman Ozdaglar, and Muriel M´edard Abstract— We consider the problem of rate allocation in a Gaussian multiple-access channel, with the goal of maximizing a utility function over transmission rates. In contrast to the literature which focuses on linear utility functions, we study general concave utility functions. We present a gradient projection algorithm for this problem. Since the constraint set of the problem is described by exponentially many constraints, methods that use exact projections are computationally intractable. Therefore, we develop a new method that uses approximate projections. We use the polymatroid structure of the capacity region to show that the approximate projection can be implemented by a recursive algorithm in time polynomial in the number of users. We further propose another algorithm for implementing the approximate projections using rate-splitting and show improved bounds on its convergence time. I. INTRODUCTION Dynamic allocation of communication resources such as bandwidth or transmission power is a central issue in multiple access channels in view of the time varying nature of the channel and interference effects. Most of the existing literature on resource allocation in multiple access channels focuses on specific communication schemes such as TDMA (time-division multiple access) [1] and CDMA (code-division multiple access) [2], [3] systems. An exception is the work by Tse et al. [4], who introduced the notion of throughput capacity for the fading channel with Channel State Information (CSI) and studied dynamic rate allocation policies with the goal of maximizing a linear utility function of rates over the throughput capacity region. In this paper, we consider the problem of rate allocation in a multiple access channel with perfect CSI. Contrary to the linear case in [4], we consider maximizing a general utility function of transmission rates over the capacity region. General concave utility functions allow us to model different performance metrics and fairness criteria (cf. Shenker [5], Srikant [6]). In view of space restrictions, we focus on the non-fading channel in this paper. In our companion paper [7], we extend our analysis to the fading channel. Our contributions can be summarized as follows. We introduce a gradient projection method for the problem of maximizing a concave utility function of rates over the capacity This research was partially supported by the National Science Foundation under grant DMI-0545910, and by DARPA ITMANET program. A. ParandehGheibi is with the Laboratory for Information and Decision Sys- tems, Electrical Engineering and Computer Science Department, Massachusetts Institute of Technology, Cambridge MA, 02139 (e-mail: parandeh@mit.edu) A. Eryilmaz is with the Electrical and Computer Engineering, Ohio State University, OH, 43210 (e-mail: eryilmaz@ece.osu.edu) A. Ozdaglar and M. M´edard are with the Laboratory for Information and Decision Systems, Electrical Engineering and Computer Science Depart- ment, Massachusetts Institute of Technology, Cambridge MA, 02139 (e-mails: asuman@mit.edu, medard@mit.edu) region of a non-fading channel. We establish the convergence of the method to the optimal solution of the problem. Since the capacity region of the multiple-access channel is described by a number of constraints exponential in the number of users, the projection operation used in the method can be compu- tationally expensive. To reduce the computational complexity, we introduce a new method that uses approximate projections. By exploiting the polymatroid structure of the capacity region, we show that the approximate projection operation can be implemented in polynomial time using submodular function minimization algorithms. Moreover, we present a more efficient algorithm for the approximate projection problem which relies on rate-splitting [8]. This algorithm also provides the extra information that allows the receiver to decode the message by successive cancelation. Other than the papers cited above, our work is also related to the work of Vishwanath et al. [9] which builds on [4] and takes a similar approach to the resource allocation problem for linear utility functions. Other works address different criteria for resource allocation including minimizing the weighted sum of transmission powers [10], and considering Quality of Service (QoS) constraints [11]. In contrast to this literature, we consider the utility maximization framework for general concave utility functions. The remainder of this paper is organized as follows: In Section II, we introduce the model and describe the capacity region of a multiple-access channel. In Section III, we consider the utility maximization problem in non-fading channel and present the gradient projection method. In Section IV, we address the complexity of the projection problem. Final

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