Further analysis of the expected number of departures in M/G/1 queues with server vacations

In this paper, the departure processes of M/G/1 queueing systems with single server vacation and multiple server vacations are analysed. Expressions for the Laplace–Stieltje (LS) transform of the expected number of departures during the time interval (0, t] are obtained. It is shown that the LS transform of the expected number of departures during (0, t] is invariant under the interchange of arrival and service rates in the initially empty M/M/1 queue in which the server begins vacations at t=T and the length of each vacation has an exponential distribution. Simultaneously the asymptotic expansions of the expected number of departures during (0, t] are also discussed. It is shown that the asymptotic formulae are convenient in application.