Diffusion Approximations for Queues with Markovian Bases

Author: Kimura T.

Source: Annals of Operations Research, Volume 113, Numbers 1-4, July 2002 , pp. 27-40(14)

Publisher: Springer

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Abstract:

Consider a base family of state-dependent queues whose queue-length process can be formulated by a continuous-time Markov process. In this paper, we develop a piecewise-constant diffusion model for an enlarged family of queues, each of whose members has arrival and service distributions generalized from those of the associated queue in the base. The enlarged family covers many standard queueing systems with finite waiting spaces, finite sources and so on. We provide a unifying explicit expression for the steady-state distribution, which is consistent with the exact result when the arrival and service distributions are those of the base. The model is an extension as well as a refinement of the M/M/s-consistent diffusion model for the GI/G/s queue developed by Kimura [13] where the base was a birth-and-death process. As a typical base, we still focus on birth-and-death processes, but we also consider a class of continuous-time Markov processes with lower-triangular infinitesimal generators.

Keywords: piecewise-constant diffusions; Markovian queues; finite capacity; finite sources; steady-state distribution; birth-and-death processes; lower-triangular infinitesimal generators

Language: English

Document Type: Research article

Affiliations: 1: Graduate School of Economics, Hokkaido University, Sapporo 060-0809, Japan kimura@econ.hokudai.ac.jp

Publication date: 2002-07-01