1132 IEEE/ACM TRANSACTIONS ON NETWORKING, VOL. 17, NO. 4, AUGUST 2009 Understanding the Capacity Region of the Greedy Maximal Scheduling Algorithm in Multihop Wireless Networks Changhee Joo, Member, IEEE, Xiaojun Lin, Member, IEEE, and Ness B. Shroff, Fellow, IEEE Abstract—In this paper, we characterize the performance of an important class of scheduling schemes, called greedy maximal scheduling (GMS), for multihop wireless networks. While a lower bound on the throughput performance of GMS has been well known, empirical observations suggest that it is quite loose and that the performance of GMS is often close to optimal. In this paper, we provide a number of new analytic results characterizing the performance limits of GMS. We first provide an equivalent characterization of the efficiency ratio of GMS through a topo- logical property called the local-pooling factor of the network graph. We then develop an iterative procedure to estimate the local-pooling factor under a large class of network topologies and interference models. We use these results to study the worst-case efficiency ratio of GMS on two classes of network topologies. We show how these results can be applied to tree networks to prove that GMS achieves the full capacity region in tree networks under the -hop interference model. Then, we show that the worst-case efficiency ratio of GMS in geometric unit-disk graphs is between and . Index Terms—Capacity region, communication systems, greedy maximal scheduling (GMS), longest queue first, multihop wireless networks. I. INTRODUCTION O VER the last few years there has been significant in- terest in studying the scheduling problem for multihop wireless networks [1]–[8]. In general, this problem involves determining which links should transmit (i.e., which node-pairs should communicate) at what times, what modulation and coding schemes should be used, and at what power levels should communication take place. While the optimal solution of this scheduling problem has been known for a long time [1], the resultant solution has high computational complexity and is Manuscript received July 01, 2008; revised January 21, 2009. First pub- lished July 21, 2009; current version published August 19, 2009. This work was supported in part by NSF awards CNS-0626703, CNS-0721236, ANI-0207728, CCF-0635202, and CNS-0721484, by ARO MURI Award W911-NF-08-1-0238, by AFOSR Grant FA 9550-07-1-0456, and by the IT Scholarship Program supervised by IITA and MIC, Korea. A preliminary version of this work was presented at IEEE INFOCOM, Phoenix, AZ, April 13–18, 2008. C. Joo and N. B. Shroff are with The Ohio State University, Columbus, OH 43210 USA (e-mail: cjoo@ece.osu.edu; shroff@ece.osu.edu). X. Lin is with Purdue University, West Lafayette, IN 47907 USA (e-mail: linx@ecn.purdue.edu). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TNET.2009.2026276 difficult to implement in multihop networks. For example, con- sider the simplest one-hop interference model (also known as the node-exclusive or primary interference model), where two links interfere with each other only if they are within a one-hop distance. Under this model, the throughput-optimal policy of [1] corresponds to a maximum weighted matching (MWM) policy, and its complexity is roughly [9], where is the total number of nodes in the network. While the one-hop interference model has been used as a reasonable approxima- tion to Bluetooth or FH-CDMA networks [2], [10], [11], a large class of systems can be modeled using the more general -hop interference models, in which any two links within a -hop distance cannot be activated simultaneously. For example, the ubiquitous IEEE 802.11 distributed coordination function (DCF) wireless networks is often modeled using the two-hop interference model [12], [13], when the carrier-sensing range is equal to the transmission range. On the other hand, when the carrier-sensing range is times the transmission range, we can model these networks with -hop interference models [14]. The complexity of the throughput-optimal policy of [1] for the -hop interference model is NP-Hard [14], and hence, it is difficult to implement in practice. In this paper, we are interested in a well-known suboptimal scheduling policy called the greedy maximal scheduling (GMS) [2], [15] (also known as longest queue first (LQF) in [16], [17]), which determines a schedule by choosing links in a de- creasing order of the backlog while conforming to interference constraints. GMS has low complexity [2], [15], [16] and may be implemented in a distributed manner [18]. However, to date, its performance is not well understood. We characterize the performance of GMS through its efficiency ratio , which is defined as the achievable fraction of the optimal capacity region (see Definition 2 for a precise definition). Under the one-hop interference model, it is relatively straightforward to show that the efficiency ratio of GMS is at least ; i.e., GMS can sustain at least a half of the throughput of the optimal policy. However, simulation results suggest that the performance of GMS is often much better than this lower bound in most network settings [6]. For the -hop interference model, the known performance guarantees of GMS are also quite pessimistic [12], [14], [19]. Recently, Dimakis and Walrand [17] have shown that if the network topology satisfies the so-called local-pooling condi- tion, then GMS can in fact achieve the full capacity region. The idea is extended in [20] and [21] to find network topologies that maximize the throughput under GMS. Unfortunately, realistic network topologies may not satisfy the local-pooling condition. 1063-6692/$26.00 © 2009 IEEE Authorized licensed use limited to: Politecnico di Torino. Downloaded on December 22, 2009 at 08:12 from IEEE Xplore. Restrictions apply.