A single server queueing system with two phases of service subject to server breakdown and Bernoulli vacation

This paper deals with a single server M/G/1 queue with two phases of heterogeneous service and unreliable server. We assume that customers arrive to the system according to a Poisson process with rate λ. After completion of two successive phases of service the server either goes for a vacation with probability p(0 ⩽ p ⩽ 1) or may continue to serve the next unit, if any, with probability q(=1 − p). Otherwise it remains in the system until a customer arrives. While the server is working with any phase of service, it may breakdown at any instant and the service channel will fail for a short interval of time. For this model, we first derive the joint distribution of state of the server and queue size, which is one of the chief objectives of the paper. Secondly, we derive the probability generating function of the stationary queue size distribution at a departure epoch. Next, we derive Laplace Stieltjes transform of busy period distribution and waiting time distribution. Finally we obtain some important performance measures and reliability indices of this model.