A Characterization of M/G/1 Queues with Renewal Departure Processes

Abstract

Burke [Burke, P. J. 1956. The output of a queueing system. Oper. Res. 4 699–704.] showed that the departure process from an M/M/1 queue with infinite capacity was in fact a Poisson process. Using methods from semi-Markov process theory, this paper extends this result by determining that the departure process from an M/G/1 queue is a renewal process if and only if the queue is in steady state and one of the following four conditions holds: (1) the queue is the null queue—the service times are all 0; (2) the queue has capacity (excluding the server) 0; (3) the queue has capacity 1 and the service times are constant (deterministic); or (4) the queue has infinite capacity and the service times are negatively exponentially distributed (M/M/1/∞ queue).

Thinning of point processes—covariance analyses

“…Since then, the overflow queue has been the target of repeated investigation. Disney, et al [1973] characterized renewal departures from M/GI/l/N overflow queues. Disney and Simon [1982] obtained a Markov renewal characterization of the input process in overflow queues.…”

“…However, Markov renewal processes find a natural application in many situations and have been particularly useful in modeling queueing situations and providing useful insights (cf. Disney et al (1973), Disney et al (1981)). Further, the special structure of Markov renewal processes allows for computations without undue difficulty.…”

The M/G/1 queue with instantaneous bernoulli feedback

“…The departure process {t n } can be studied as in Disney , F a r r e l l , deMorais [4] upon using the mapping in section 2.1. T h u s we know that this departure process is a renewal process and is a Poisson process whenever {s} is a renewal process with exponential distribution.…”

“…The result that is needed for the reaults of [4] to follow is that S ' be n exponentiall y d istributed (since it is known that {s '} is a sequence m u t u a l l y independent , identically distributed random variables). t ) is e x p o n e n t ia l .…”

A queue with a pair of instantaneous independent bernoulli feedback processes

“…The result then follows directly using arguments as in [4]. where Then using standard embedded Markov chain results [I], we get the probability generating function where = G*(A-hz).…”