The Covariance Structure of the Departure Process from M/G/1 Queues with Finite Waiting Lines

Abstract

Summary This paper contains a study of the departure process from M/G/1 queues with a waiting space of size N>0. The property of the departure process which receives the most attention is the covariance of pairs of departure intervals, although a more thorough investigation of the probabilistic structure is undertaken for the case N = 1. A main result of the paper is that in the M/G/1 queue with N = 1, departure intervals separated by one or more intervals are independent. When G is the deterministic server, the departure stream is a renewal process. The paper concludes with an expression for the covariance of any pair of intervals in the departure process for any value of N, and numerical values for correlations in the case of a deterministic server with N = 2.

Random Flow in Queueing Networks: A Review and Critique

“…For the E2/M/ 1 I queue, he found that the idle period and the time measured from a departure instant, at which the queue size was exactly one, to the next arrival instant were identically distributed. Some of these results appear in King [87].…”

Independent Inter-Departure Times in M/G/1/N Queues

“…Finch (1959) proved the necessity part of (2°) under the additional assumption that B"(·) exists and is continuous in (0, (0). The initial steps of our argument below are similar to those of Finch, who assumed D n and D n +1 to be independent, and deduced that B(·) must be of a form similar to (3°), but then omitted further discussion of such B(') because its first derivative is discontinuous at x = g. King (1971) proved both parts of (1°), and Disney et al (1973) essentially proved part (b). The argument below for (3°), which in passing proves the necessity parts of (10) and (2°), answers the question left open in Disney et al whose conclusion concerning D n , D n +1 and D n +2 one of us reached independently in unpublished work begun in Shanbhag and Sharma (1972).…”

“…The bulk of this theorem is proved in various places in the literature: the new part is case (3°). Specifically, Burke (1956) (see also Reich, 1957) proved the sufficiency part of (20), and King (1971) proved the sufficiency part of (10), and since our method has little bearing on the proofs of these results, we say no more of them. Finch (1959) proved the necessity part of (2°) under the additional assumption that B"(·) exists and is continuous in (0, (0).…”

Queueing output processes

“…Yarovitskii gives the M/G/1 system as an example, observing that the output is then a renewal process. King (1971) showed that the M / D /1 system with a waiting room of size 1 has a renewal output process. Since 1To = (1To + 1Tl){3 (A) for all M / G /1 systems with finite waiting room, 1To = e -P for this M / D /1 system and hence from (2.6),…”

“…To my knowledge these are the only systems for which anything like 'explicit expressions have been found for the whole sequence {COy(Do, D; )}. King (1971) shows how in principle the sequence may be found for M / G /1 systems with finite waiting room, and gives values of Var (Do), COy (Do, D 1 ) and COy (Do, D z ) for some M / D /1 systems. These terms are also given by Jenkins (1966) for some stable M / E; /1 systems with infinite waiting room.…”