Communication Networks An Optimization, Control, and Stochastic Networks Perspective

Stationary distributions of queueing systems with general arrivals and service-time distributions are difficult to compute. In this chapter, we will develop techniques to understand the behavior of such systems in certain asymptotic regimes. We consider two such regimes: the heavy-traffic regime and the large-deviations regime. The analysis in the heavy-traffic regime can be thought of as the analog of the central limit theorem for random variables, and the analysis of the large-deviations regime can be thought of as the analog of the Chernoff bound. Traditionally, heavy-traffic analysis is performed by scaling arrival and service processes so that they can be approximated by Brownian motions, which are the stochastic process analogs of Gaussian random variables. Here, we take a different approach: we will show that heavy-traffic results can be obtained by extending the ideas behind the discrete-time Kingman bound presented in Chapter 3. The analysis of the large-deviations regime in this chapter refines the probability of overflow estimates obtained using the Chernoff bound, also in Chapter 3.

Roughly speaking, heavy traffic refers to a regime in which the mean arrival rate to a queueing system is close to the boundary of the capacity region. We will first use a Lyapunov argument to derive bounds on the moments of the scaled queue length ∈q(t) for the discrete-time G/G/1 queue; where ∈ = μ — λ; μ is the service rate and λ is the arrival rate.