Averaging Principles for a Diffusion-Scaled, Heavy-Traffic Polling Station with K Job Classes

Abstract

This paper provides heavy traffic limit theorems for a polling station serving jobs from K exogenous renewal arrival streams. It is a standard result that the limiting diffusion-scaled total workload process is semimartingale reflected Brownian motion. For polling stations, however, no such limit exists in general for the diffusion-scaled, K-dimensional queue length or workload vector processes. Instead, we prove that these processes admit averaging principles, the natures of which depend on the service discipline employed at the polling station. Parameterized families of exhaustive and gated service disciplines are investigated. Each policy under consideration has K stochastic matrices associated with it—one for each job class—that describe the transition of the workload vector while a given job class is being polled. These matrices give rise to K probability vectors that are vertices of a K - 1-simplex. Loosely speaking, each point of this simplex acts as an instantaneous lifting operator, converting the one-dimensional limiting total workload process to the K-dimensional workload vector. The averaging principles—needed because the workload vector traverses a Hamiltonian cycle over K edges of the simplex infinitely fast under diffusion scaling—are limits of cumulative cost functions of the diffusion-scaled workload vector, evaluated over a fixed time interval. The “averaging” takes place over the edges of the simplex.