A two-stage batch arrival queueing system with a modified bernoulli schedule vacation under N-policy

We consider a batch arrival queueing system, where the server provides two stages of heterogeneous service with a modified Bernoulli schedule under N-policy. The server remains idle till the queue size becomes N (≥ 1). As soon as the queue size becomes at least N, the server instantly starts working and provides two stages of service in succession to each customer, i.e., the first stage service followed by the second stage service. However, after the second stage service, the server may take a vacation or decide to stay in the system to provide service to the next customer, if any. We derive the queue size distribution at a random epoch as well as a departure epoch under the steady state conditions. Further, we demonstrate the existence of the stochastic decomposition property to show that the departure point queue size distribution of this model can be decomposed into the distributions of three independent random variables. We also derive some important performance measures of this model. Finally, we develop a simple procedure to obtain optimal stationary operating policy under a suitable linear cost structure.