A Cross-Layer Study of the Scheduling Problem
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This thesis is inspired by the need to study and understand the interdependence between the transmission powers and rates in an interference network, and how these two relate to the outcome of scheduled transmissions. A commonly used criterion that relates these two parameters is the Signal to Interference plus Noise Ratio (SINR). Under this criterion a transmission is successful if the SINR exceeds a threshold. The fact that this threshold is an increasing function of the transmission rate gives rise to a fundamental trade-off regarding the amount of time-sharing that must be permitted for optimal performance in accessing the wireless channel. In particular, it is not immediate whether more concurrent activations at lower rates would yield a better performance than less concurrent activations at higher rates. Naturally, the balance depends on the performance objective under consideration. Analyzing this fundamental trade-off under a variety of performance objectives has been the main steering impetus of this thesis. We start by considering single-hop, static networks comprising of a set of always-backlogged sources, each multicasting traffic to its corresponding destinations. We study the problem of joint scheduling and rate control under two performance objectives, namely sum throughput maximization and proportional fairness. Under total throughput maximization, we observe that the optimal policy always activates the multicast source that sustains the highest rate. Under proportional fairness, we explicitly characterize the optimal policy under the assumption that the rate control and scheduling decisions are restricted to activating a single source at any given time or all of them simultaneously. In the sequel, we extend our results in four ways, namely we (i) turn our focus on time-varying wireless networks, (ii) assume policies that have access to only a, perhaps inaccurate, estimate of the current channel state, (iii) consider a broader class of utility functions, and finally (iv) permit all possible rate control and scheduling actions. We introduce an online, gradient-based algorithm under a fading environment that selects the transmission rates at every decision instant by having access to only an estimate of the current channel state so that the total user utility is maximized. In the event that more than one rate allocation is optimal, the introduced algorithm selects the one that minimizes the transmission power sum. We show that this algorithm is optimal among all algorithms that do not have access to a better estimate of the current channel state. Next, we turn our attention to the minimum-length scheduling problem, i.e., instead of a system with saturated sources, we assume that each network source has a finite amount of data traffic to deliver to its corresponding destination in minimum time. We consider both networks with time-invariant as well as time-varying channels under unicast traffic. In the time-invariant (or static) network case we map the problem of finding a schedule of minimum length to finding a shortest path on a Directed Acyclic Graph (DAG). In the time-varying network case, we map the corresponding problem to a stochastic shortest path and we provide an optimal solution through stochastic control methods. Finally, instead of considering a system where sources are always backlogged or have a finite amount of data traffic, we focus on bursty traffic. Our objective is to characterize the stable throughput region of a multi-hop network with a set of commodities of anycast traffic. We introduce a joint scheduling and routing policy, having access to only an estimate of the channel state and further characterize the stable throughput region of the network. We also show that the introduced policy is optimal with respect to maximizing the stable throughput region of the network within a broad class of stationary, non-stationary, and anticipative policies.