On the Correlation Structure of the Departure Process of the M/Eλ/1 Queue

Abstract

The joint distribution in the stationary case of four random variables is studied in this paper. Two of these are discrete and arise as the numbers of customers left behind in the queue by two successive departing customers in the imbedded Markov chain analysis. The other two are continuous, being the time intervals between three successive departures. The marginal joint distribution of the latter two random variables is found; in particular, the autocorrelation of lag 1 for intervals in the departure process in a stationary system is evaluated. By extending the analysis to the study of three successive departure intervals the autocorrelation of lag 2 for the process is evaluated. Burke (1956) andFinch (1959) proved that with Poisson arrivals, and service times independent for a single-server queue in equilibrium, the departure intervals are independent if and only if service time is exponentially distributed. This paper gives a measure of the dependence in the important case when the service times are special Erlangian.

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

“…Numerical results for corr (Do, D a ) as well as for corr (Do,D 1 ) and corr (Do, DJ are presented in Table 1. Observe that as in Jenkins (1966) these correlations are all quite small. The author wishes to thank the referee for suggesting the method of proof of Theorem I.…”

“…Equation (1) remains true for T = 1 by interpreting plO) as the identity matrix. In the case N = co Jenkins (1966) has derived the joint density functions of Doand D 1 , and of Do, D 1 and D 2 assuming the existence of a service time density H'(x), and Daley (1968) has given an expression similar to equation (I).…”

Random Flow in Queueing Networks: A Review and Critique

“…In 1966, Jenkins [82] followed the same study as Cox for the case of an M/Ek/l queue. He derived the autocorrelations of lag one and lag two for the process by considering the joint densities of successive departure intervals for both cases.…”

Covariance Properties for the Departure Process of M/Ek/1/N Queues

“…Makino [9] has given a generating function for F(x,) for M/G/l/=queues and has obtained coefficients of variation for several service distributions (including E,). Section 3 extends the results of Jenkins [7] to N < =. Indeed Fig.…”