This paper examines the steady state behaviour of a batch arrival queue with two phases of heterogeneous service along and Bernoulli schedule vacation under multiple vacation policy, where after two successive phases service or first vacation the server may go for further vacations until it finds a new batch of customer in the system. We carry out an extensive stationary analysis of the system, including existence of stationary regime, queue size distribution of idle period process, embedded Markov chain steady state distribution of stationary queue size, busy period distribution along with some system characteristics.
We consider an M X /G/1 queueing system with a second optional service channel under N -policy. The server remains idle until the queue size reaches or exceeds N (≥1). As soon as the queue size becomes at least N , the server immediately begins to serve the first essential service to all the waiting customers. After the completion of which, only some of them receive the second optional service. For this model, our study is basically concentrated in obtaining the queue size distribution at a random epoch as well as at a departure epoch. Further, we derive a simple procedure to obtain optimal stationary policy under a suitable linear cost structure. Moreover, we provide some important performance measures of this model with some numerical examples.
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