Analysis of the M/G/1 queue with exponentially working vacations: a matrix analytic approach

In this paper, an M/G/1 queue with exponentially working vacations is analyzed. This queueing system is modeled as a two–dimensional embedded Markov chain which has an M/G/1–type transition probability matrix. Using the matrix analytic method, we obtain the distribution for the stationary queue length at departure epochs. Then, based on the classical vacation decomposition in the M/G/1 queue, we derive a conditional stochastic decomposition result. The joint distribution for the stationary queue length and service status at the arbitrary epoch is also obtained by analyzing the semi–Markov process. Furthermore, we provide the stationary waiting time and busy period analysis. Finally, several special cases and numerical examples are presented.