Throughput optimality of delay-driven MaxWeight scheduler for a wireless system with flow dynamics Bilal Sadiq and Gustavo de Veciana Dept. of Electrical and Computer Engineering The University of Texas at Austin Abstract—We consider a wireless downlink shared by a dy- namic population of flows. The flows of random size (bits) arrive at the base station at random times, and leave when they have been completely transmitted. The transmission rate supported by the wireless channel of each flow while the flow awaits transmission varies randomly over time and is independent of that of the other flows. The scheduling problem in this context is to select a flow for transmission based on the current system state (e.g., backlogs, wait times, and channel states of the contending flows). It has recently been shown that for such a system, the well- known (backlog-driven) MaxWeight scheduler is not throughput optimal. That is to say, the MaxWeight scheduler will not stabilize a given system even though it is possible to construct a stabilizing scheduler using the various flow- and channel-related statistics. However, in this paper, we show that the delay-driven MaxWeight scheduler is, nevertheless, throughput optimal for such a system. The delay-driven MaxWeight, like its backlog-driven version, does not require any knowledge of the flow- or channel-related statistics. I. I NTRODUCTION The time-varying nature of wireless channels provides an opportunity to schedule the flows/users when they see a favorable channel states – this is referred to as opportunistic scheduling [1]–[3]. In a dynamic system, i.e., one where users’ data or even new users arrive into the system as a random process, the opportunistic or channel-aware schedulers may not be stable, i.e., keep the data/user queues bounded, unless they are carefully designed, e.g., possibly using prior knowledge of the arrival and channel processes [4], [5]. For systems with time-varying channels but a fixed number of users whose data packets arrive as a stationary random process, there are well-known queue- and channel- aware schedulers that are provably throughput optimal in a variety of network settings. A scheduler in this context is said to be throughput optimal, if without the knowledge of arrivals and channel statistics it is able to stabilize the system, if at all possible under some other scheduler. Examples of such schedulers are the Longest Connected Queue [6], MaxWeight [7], Exponential rule [8], and Log rule [9]; see [5] for more details. A typical application of these schedulers in, e.g., deciding downlink packet transmissions from a wireless base station, would be to (try to) achieve low packet delays of the order of few tens to few hundreds of milliseconds [10] [11], This research was supported in part by grants AFOSR FA9550-07-1-0428 and NSF CNS-0721532. when there is a given fixed number of users/flows which might correspond to real-time voice/video sessions etc. However for best effort flows, the relevant performance metrics are defined over longer time scales, i.e., the time scales of flow-level dynamics, e.g., file transfer delays or web browsing interactivity. Unlike a system where there is a fixed number of users/flows and each flow generates a stationary packet arrival process, in this setting the arrivals correspond to new flows and users, i.e., files to be transferred associated with different users, and thus the number of ongoing flows in the system is dynamic. Each flow can be viewed as having its own queue associated with the residual data that needs to be transmitted in order to successfully transfer a file or web page. In this context, [5] recently showed that the queue-driven MaxWeight scheduler is not throughput optimal 1 . In this paper, we show that the delay-driven MaxWeight still is throughput optimal in the dynamic flow setting. The critical observation which explains why the queue-driven version is not throughput optimal but the delay-driven version is, is as follows. In the setting where there is a fixed number of flows, a linear relation can be established between the head- of-line packet delay and the queue length of a flow (Little’s law) as either one gets large. By contrast, in the setting with dynamic number of flows, while the head-of-line delay of an un-served flow will continue to increase, its queue length will not due to the finite size of flows. So, in the former setting, the queue-driven and the delay-driven versions of MaxWeight are equivalent in some sense, whereas, in the latter, the queue-driven version may perpetually fail to exploit good channel states of small queues (i.e. files with few residual bits) irrespective of how long these small files wait while the bigger newer files may get scheduled (because of their longer queue lengths) even when their channels are poor; see excellent illustrative examples in [5]. II. SYSTEM MODEL Let random A(t) ∈ Z + denote the number of files arriv- ing in time slot [t, t + 1), these files will not be available for service until the next time slot. We assume A(·) are i.i.d. with finite mean λ ≡ EA(0). For 0 <i ≤ A(t), let B i (t) denote the file size in bits of the i th arriving file. 1 Queue-driven Exponential rule and Log rule can similarly be shown to be not throughput optimal in the dynamic flow setting.