An M/M/s Queue With Servers’ Vacations

Abstract

In an M/M/s queueing system a server that completes service and finds no waiting units in line leaves for a vacation of an exponentially distributed duration. At the end of the vacation the server returns to the main system. Two models are analysed. In the first, a server returning to an empty queue takes immediately another vacation. In the second, only a single vacation is taken each time. For model 1, formulas for the distribution of the number of busy servers and the mean number of units in system, L, are derived. Numerical calculations indicate that L is very closely a linear function of the mean vacation time. Finally it is shown that model 2 may be analysed similarly to model 1.