Steady state analysis of an M/D/1 queue with two stages of heterogeneous server vacations (M/D/G1,G2/1 queue)

A single server queue with Poisson arrivals, deterministic service of constant duration b(> 0) and two stages of heterogeneous server vacations having different general (arbitrary) distributions is studied. This model is designated as (M/D/G1,G2/1). After completion of each service, the server may take a vacation with probability p or may continue working in the system with probability 1−p. Closed explicit forms for the steady state system size probability generation functions of various states of the server as well as the average number and the average waiting time in the system and the queue are obtained. Some new useful special cases including the known results of the M/D/1 queue are derived. Finally a numerical illustration is discussed.