Refining Diffusion Models for State-Dependent Queues

Abstract

A diffusion model is developed for a class of queues with Poisson arrivals and general services, both of which are state-dependent. This class covers many practical queueing systems with multiple servers, finite waiting spaces, finite sources, discouraged arrivals and more. For each of these systems, a large number of diffusion models have been developed individually. Our diffusion model unifies and refines those previous models. The main focus is on the steady-state distribution of the number of customers in system. The discrete process of the number of customers is approximated by a continuous diffusion process with piece wise-constant infinitesimal parameters defined on a closed interval in the non-negative real line. A conservation law together with a consistent discretization generate an approximate formula for the steady-state distribution, which is more accurate than the previous models.