Note on a series for M/G/1 queues

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.