N-Policy for State-Dependent Batch Arrival Queueing System with l-Stage Service and Modified Bernoulli Schedule Vacation

Abstract

This paper deals with a batch arrival queueing system with modified Bernoulli vacation under N-policy, where the server starts the service of the customers only when the queue size becomes at least N (> 1) otherwise remains idle. The customers arrive in batches to the system in Poisson fashion but may also balk in case of long queue, when server is in working state. The customer needs l-stage of service in succession i.e. the first stage service (FSS) is followed by the second stage service (SSS), the second stage service followed by third stage service (TSS) and so on up to l-stages of service. It is assumed that after completion of lh stage service either the server goes on vacation with probability p or decides to stay in the system with probability (1 − p) to give the service to the next customer. We determine the queue size distribution and other performance indices by using the generating function method to solve the governing equations constructed after introducing the supplementary variables. We propose a method to find the optimal value of the threshold parameter to minimize the total expected cost.