Title: Note on a series for M/G/1 queues

Authors: Percy H. Brill

Addresses: Department of Management Science and Department of Mathematics and Statistics, University of Windsor, 401 Sunset Avenue, Windsor, Ontario N9B 3P4, Canada

Abstract: This paper provides a geometrical (physical) interpretation for a series representing the steady-state probability density function (pdf) of wait in a standard M/G/1 queue. This series was called |intriguing| by a prominent queueing theorist in 1975. The series converges geometrically fast, making it potentially useful for approximating the pdf. We provide an intuitive explanation in terms of sample-path upcrossings of a level of the virtual wait. We also consider a similar series for an M/G/1 variant with zero-wait customers receiving special service. This leads to a generalised explanation of both series in terms of sample-path upcrossings.

Keywords: M/G/1 queues; M/G/1 variants; probability density function; waiting; series representation; renewal theory; excess service time; level crossings; PASTA.

DOI: 10.1504/IJOR.2009.025202

International Journal of Operational Research, 2009 Vol.5 No.3, pp.363 - 373

Published online: 16 May 2009 *

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