Note on a series for M/G/1 queues

Percy H. Brill

International Journal of Operational Research, 2009, vol. 5, issue 3, 363-373

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. (search for similar items in EconPapers)
Date: 2009
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