1 Delay Analysis for Wireless Networks with Single Hop Traffic and General Interference Constraints Gagan Raj Gupta, Ness B. Shroff, Fellow, IEEE Abstract—We consider a class of wireless networks with general interference constraints on the set of links that can be served simultaneously at any given time. We restrict the traffic to be single-hop, but allow for simultaneous transmissions as long as they satisfy the underlying interference constraints. We begin by proving a lower bound on the delay performance of any scheduling scheme for this system. We then analyze a large class of throughput optimal policies which have been studied extensively in the literature. The delay analysis of these systems has been limited to asymptotic behavior in the heavy traffic regime and order results. We obtain a tighter upper bound on the delay performance for these systems. We use the insights gained by the upper and lower bound analysis to develop an estimate for the expected delay of wireless networks with mutually independent arrival streams operating under the well-known Maximum Weighted Matching (MWM) scheduling policy. We show via simulations that the delay performance of the MWM policy is often close to the lower bound, which means that it is not only throughput optimal, but also provides excellent delay performance. Index Terms—Wireless Networks, Scheduling, Delay Analysis, Interference, Lyapunov function. I. I NTRODUCTION In a wireless system, users compete for accessing a shared transmission medium. Since link transmissions cause mutual interference, the medium access layer (MAC) is needed to schedule the links carefully so that packets can be transmitted with minimal collisions. Many scheduling policies have been studied at the MAC layer with the objective of maximiz- ing throughput. These schemes are often called throughput- optimal scheduling schemes. However, the delay analysis of these systems has largely been limited. Our focus in this paper is to analyze the expected delay for this system. To that end, we will derive upper and lower bounds on the expected delay, and also provide an accurate estimate of the expected delay for a well-known and extensively-studied (e.g., [1]–[4]) throughput-optimal scheme called the Maximum Weighted Matching (MWM). To simplify the analysis we, in common with related work [3], [5], [6], restrict the traffic model to single-hop traffic. Under the single-hop traffic model, all packets transmitted on a link (s,d) are generated by an exogenous arrival process A d s at the source node s. As shown in Figure 1, the exogenous arrivals waiting to be transmitted at each link are queued in their respective queues. This approach has also been adopted Gagan Raj Gupta is with the School of Electrical and Computer En- gineering, Purdue University, West Lafayette, IN 47907 USA e-mail: gr- gupta@purdue.edu Ness B. Shroff is with the Departments of ECE and CSE, The Ohio State University, Columbus, Ohio, USA e-mail: shroff@ecn.osu.edu. This work was supported by ARO MURI Award W911NF-08-1-0238, and NSF Award 0721236-CNS. a d i f h e c g i h A c a A i f A Set of links that interfere with link (g,h) Q (a, c) Q (f, i) Q (h, i) b a A (a, b) Q d f A Q (f, d) b Fig. 1. Figure showing a wireless network with single-hop traffic. All packets A d s , transmitted on link (s,d) are exogenous and are queued (Q s,d denotes the queue length). All the links that interfere with link (g,h) are shown. in the literature while studying the throughput performance of scheduling policies for wireless networks. This allows us to study the effect of scheduling policy on the delay of the system, independent of routing. We note that this model allows for simultaneous transmissions as long as they satisfy the underlying interference constraints. Such systems are more general than the cellular type systems where the system is divided into non-interfering cells. The results presented here work for any underlying model for interference constraints. The design of scheduling policies which stabilize the system even under single-hop traffic is a challenging task. Intuitively, the scheduler must schedule as many links as possible in every time slot. Such schedulers are called maximal schedulers (as opposed to maximum weighted schedulers that also take the queue length into account). However, even with max- imal scheduling, some of the queue lengths may become unbounded. The reason is that if the scheduler does not use the queue length information, some of the queues may grow large, while others remain very small or become empty. This, in turn, does not allow the scheduler to schedule a large number of queues and leads to instability. Thus a throughput optimal policy like MWM, carefully uses the information of the queue lengths while scheduling the links. The above behavior caused by throughput-efficient sched- ulers significantly complicates the delay analysis of these systems, because the service process of each link is governed not only by the interference constraints, but also by its queue length. For example, in a wireless network operating under a throughput optimal policy, such as the MWM policy, the expected delay at a link may be large even if the arrival rate