IEEE/ACM TRANSACTIONS ON NETWORKING, VOL. 18, NO. 3, JUNE 2010 973 Fast Algorithms for Resource Allocation in Wireless Cellular Networks Ritesh Madan, Stephen P. Boyd, Fellow, IEEE, and Sanjay Lall, Senior Member, IEEE Abstract—We consider a scheduled orthogonal frequency di- vision multiplexed (OFDM) wireless cellular network where the channels from the base-station to the mobile users undergo flat fading. Spectral resources are to be divided among the users in order to maximize total user utility. We show that this problem can be cast as a nonlinear convex optimization problem, and describe an algorithm to solve it. Computational experiments show that the algorithm typically converges in around 25 iterations, where each iteration has a cost that is , with a modest constant. When the algorithm starts from an initial resource allocation that is close to optimal, convergence typically takes even fewer iterations. Thus, the algorithm can efficiently track the optimal resource allocation as the channel conditions change due to fading. We also show how our techniques can be extended to solve resource allocation problems that arise in wideband networks with frequency selective fading and when the utility of a user is also a function of the resource allocations in the past. Index Terms—Fast computation, resource allocation, sched- uling, wireless cellular networks. I. INTRODUCTION R ESOURCE allocation in wireless networks is fundamen- tally different than that in wireline networks due to the time-varying nature of the wireless channel [1]. There has been much prior work on scheduling policies in wireless networks to allocate resources among different flows based on the channels they see and the flow state [1], [2]. The flow state can consist of the average rate seen by the flow in the past [3], [4], the delay of the head-of-line packet [5], or the length of the queue [6]. Much prior work in this area can be divided into two categories: Manuscript received September 16, 2009; revised September 28, 2009; ap- proved by IEEE/ACM TRANSACATIONS ON NETWORKING Editor S.Borst. First published November 24, 2009; current version published June 16, 2010. This work was funded in part by the MARCO Focus Center for Circuit and System Solutions (C2S2, www.c2s2.org) under Contract 2003-CT-888, the AFOSR under Grant AF F49620-01-1-0365, the NSF under Grant ECS-0423905, the NSF under Grant 0529426, the DARPA/MIT under Grant 5710001848, the AFOSR under Grant FA9550-06-1-0514, the DARPA/Lockheed under Contract N66001-06-C-2021, the AFOSR/Vanderbilt under Grant FA9550-06-1-0312, the Stanford URI Architecture for Secure and Robust Distributed Infrastruc- tures (AFOSR DoD award 49620-01-1-0365), and the Sequoia Capital Stanford Graduate Fellowship. R. Madan was with the Department of Electrical Engineering, Stanford Uni- versity, Stanford, CA 94305 USA. He is now with Qualcomm-Flarion Tech- nologies, Bridgewater, NJ 08870 USA (e-mail: rkmadan@stanfordalumni.org). S. P. Boyd is with the Department of Electrical Engineering, Stanford Uni- versity, Stanford, CA 94305 USA (e-mail: boyd@stanford.edu). S. Lall is with the Department of Electrical Engineering and the Department of Aeronautics and Astronautics, Stanford University, Stanford, CA 94305 USA (e-mail: lall@stanford.edu). Digital Object Identifier 10.1109/TNET.2009.2034850 1) Scheduling for elastic (non real-time) flows: The end-user experience for a elastic flow is modeled by a concave increasing utility function of the rate experienced by the flow [7]. The proportional fair algorithm (see, for example, [8]) where all the resources are allocated to the flow with the maximum ratio of instantaneous spectral efficiency (which depends on the channel gain) to the average rate has been analyzed in [3], [9], [10]; roughly speaking this algorithm maximizes the sum of log utilities of average rates over an asymptotically large time horizon. A more general scheduling rule where potentially multiple users can be scheduled simultaneously has been considered in [11], [12]. Most of the above work assumes that the queues have infinite backlogs, i.e., packets are always available in the buffers of all the queues; extensions to finite queues are provided in, for example, [3]. Joint design of scheduling and congestion control with modeling of queue dynamics has been considered in, for example, [4], [13]–[15]; in this case, packets are always assumed to be available at the congestion controller. 2) Scheduling for Real-Time Flows: Real-time flows are typ- ically modeled by a predetermined but unknown arrival process and a delay deadline for each packet. For such flows, we can roughly define the stability region as fol- lows: The stability region for a set of queues is defined as the set of arrival rates at the queues for which there exists a scheduling policy such that the length of any queue does not grow without bound over time (see, for example, [16]). A stabilizing policy is one which ensures that the queue lengths do not grow without bound. Stabilizing policies for a vector of arrival rates within the stability region for different wireless network models have been characterized in, for example, [5], [6], [16]–[19]. The scheduling policy in [5] minimizes the percentage of packets lost because of deadline expiry, while the delay performance of the expo- nential rule (introduced in [6]) was empirically studied in [20]. Work on providing throughput guarantees for such flows includes [21] and [22], and references therein. We note that policies to schedule a mixture of elastic (non real- time) and real-time flows have been considered in [20]. Dis- tributed algorithms for interference management to maximize the sum utilities of user signal-to-noise ratios (SNR) in cellular networks have been studied in [23], [24]. Also, related cross- layer optimization problems for resource allocation in wireless networks with different objectives have been analyzed in, for ex- ample, [25]–[27]. Resource allocation algorithms which focus on maximizing sum rate (without fairness or with minimum rate guarantees) for OFDM systems include [28]–[32]. The above summary is only a representative sample of the work in the gen- eral area of resource allocation in wireless networks. For a more 1063-6692/$26.00 © 2009 IEEE Authorized licensed use limited to: Stanford University. Downloaded on June 29,2010 at 15:03:59 UTC from IEEE Xplore. Restrictions apply.