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

Summary 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) and Finch (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.