A Single Server Queue with Two Phases of Heterogeneous Service under Bernoulli Schedule and a General Vacation Time

AbstractWe consider a single server queuewith Poisson input, two phases of heterogeneous servicewith Bernoulli schedule and a general vacation time, where the server provides two phasesof heterogeneous service one after the other to the arriving customers. After completion ofboth phases of service the server either goes for a vacation with probability (0    1) ormay continue to serve the next unit, if any, with probability (1 ). Otherwise, it remainsin the system until a customer arrives. For this model, we rst obtain the steady stateprobability generating functions for the queue size distributions at a random epoch as wellas at a departure epoch. Next, we derive the Laplace Stieltjes transform of the waitingtime distribution. Finally, we obtain some system performance measures and discuss someimportant particular cases of this model. Keywords: M=G=1 Queue, Two Phases of Heterogeneous Service, Bernoulli Schedule,Generalized Vacation Time, Queue Size, Waiting Time and Busy Period.1. IntroductionThe single server queueing system with Bernoulli vacation is not new. Keilson andServi [8] were rst to study such a model, where after each service completion the servertakes a vacation with probability  and starts a new service with probability (1 ).Subsequently, Keilson and Servi [9], Ramaswamy and Servi [14], Doshi [4, 5] and Takagi[15] among others have studied this and the models of similar nature due to its numerousapplications in many real life situations.Recently Madan [11, 12] has studied two similar types of vacation models for theM=G=1 queueing system. In both the models, he introduced the concept of two stage