IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 24, NO. 8, PP. 1489-1501, AUG. 2006 1 Super-Fast Delay Tradeoffs for Utility Optimal Fair Scheduling in Wireless Networks Michael J. Neely Abstract— We consider the fundamental delay tradeoffs for utility optimal scheduling in a general network with time varying channels. A network controller acts on randomly arriving data and makes flow control, routing, and resource allocation deci- sions to maximize a fairness metric based on a concave utility function of network throughput. A simple set of algorithms are constructed that yield total utility within O(1/V ) of the utility- optimal operating point, for any control parameter V > 0, with a corresponding end-to-end network delay that grows only logarithmically in V . This is the first algorithm to achieve such “super-fast” performance. Furthermore, we show that this is the best utility-delay tradeoff possible. This work demonstrates that the problem of maximizing throughput utility in a data network is fundamentally different than related problems of minimizing average power expenditure, as these latter problems cannot achieve such performance tradeoffs. Index Terms— Fairness, flow control, wireless networks, queue- ing analysis, optimization, delay, network capacity I. I NTRODUCTION We consider the fundamental tradeoff between network utility and network delay for a wireless network with time varying channels. Traffic arrives to the network randomly, and we assume input rates exceed network capacity. Such a situation is typical for modern networks where growing user demands can quickly overload physical system resources. It is essential to establish simple solutions that maintain low network congestion and delay while providing fair access to all users. In this paper, we evaluate fairness according to a general concave utility function of the long term throughput of each user. The goal is to design a controller that drives total utility towards its maximum value, considering all possible methods of flow control, routing, and resource allocation, while ensuring an optimal tradeoff in network delay. In our previous work on the network fairness problem, we constructed a set of algorithms indexed by a control parameter V> 0 that yield total network utility within O(1/V ) of the utility-optimal operating point, with a corresponding end-to- end delay tradeoff that is linear in V [2] [3]. This result suggests that delay necessarily increases when utility is pushed toward optimality, although the existence of such a tradeoff and the form of the optimal utility-delay curve were left as open questions. In this paper we explore these questions and characterize the fundamental tradeoff curve. Specifically, we consider a particular class of overloaded systems, where user Michael J. Neely is with the Department of Electrical Engineering, Uni- versity of Southern California, Los Angeles, CA 90089 USA (email: mjneely AT usc.edu, web: http://www-rcf.usc.edu/∼mjneely). This work was presented in part at the IEEE INFOCOM conference, Barcelona, Spain, April 2006. This material is based upon work supported in part by the National Science Foundation under grant OCE 0520324. ε ε ε ε λ Fig. 1. An example network of 5 nodes, and an illustration of a capacity region (shown in two dimensions) with an input rate vector that strictly dominates the optimal operating point. data rates strictly dominate the optimally fair operating point. We then develop a novel algorithm that deviates from the optimal utility by no more than O(1/V ) while ensuring that average network delay is less than or equal to O(log(V )). Further, for the special case of one-hop networks, we prove that no algorithm can achieve a better asymptotic tradeoff. This establishes a fundamental relationship between utility and delay, and demonstrates the unexpected result that logarithmic delay is possible for systems with any concave utility metric. Related work in [5] considers the tradeoff between energy and delay for a single queue that stores data for transmission over a single fading channel. There, it is shown that any scheduling policy yielding average energy expenditure within O(1/V ) of the minimum energy required for stability must also have average queueing delay greater than or equal to Ω( √ V ). Strategies for achieving this tradeoff are proposed in [5] using the concept of buffer partitioning, and a recent result in [33] shows that the same square-root tradeoff applies to the minimum energy problem for multi-user networks. In this paper, we combine the technique of buffer parti- tioning with the recently developed technique of performance optimal Lyapunov scheduling [2] [3] [4] [34]. Specifically, in [2] [3] [4] [34] Lyapunov drift theorems are developed that treat stability and performance optimization simultaneously, leading to simple and robust control strategies. Here, we extend the theory to treat optimal utility-delay tradeoffs. The result is a novel set of Lyapunov scheduling algorithms that can be used for general networks, without requiring a-priori knowledge of traffic rates or channel statistics. The algorithms use weights that aggressively switch ON and OFF in order to achieve optimal delay tradeoffs. We find that the special structure of the long-term fairness objective allows for a “super-fast” logarithmic delay tradeoff that cannot be achieved in related problems of minimizing energy expenditure. It is important to distinguish the tradeoffs we explore here to the capacity-delay tradeoffs recently explored for ad-hoc mobile networks in [2], [6]-[10]. These tradeoffs are quite