M/G/1 queue with multiple working vacations

We study an M/G/1 queue with multiple vacations and exhaustive service discipline such that the server works with different service times rather than completely stopping service during a vacation. Both service times in a vacation and in a service period are generally distributed random variables. It is assumed that the Laplace–Stieltjes transform (LST) for the distribution of the vacation length is a rational function. We derive the distributions for the queue size and the system time for an arbitrary customer in the steady state. Several special cases, namely, exponentially distributed vacation lengths and/or exponentially distributed service times in a vacation, are considered. Finally some numerical examples are presented.