Time dependent solution of two stages M~X/G/1 queue model server vacation random setup time and balking with Bernoulli schedule

摘要

This paper analyzes a non-Markovian queueing model with setup time and Balking. The arrival pattern of customers is in batches according to compound Poisson process, but the customers are served one by one with first come first served basis. In this model, the server provides two stages of service such as that the service time for each customer follows general(arbitrary)distribution. Upon completion of a service the server may go for a vacation with probability p or stay back in the system to serve a next customer with probability 1 -- p, if any. In extention to this, one of the customers impatience called balking has been incorporated which reflects that a customer may decide to get into the system or not, due to impatience. And also we assume that at the end of a busy period, as soon as a batch of customers arrives, the server does not start giving service, but needs a setup time before actually starting its service of the first customer. We obtain the time dependent probability generating function in terms of their Laplace transforms and the corresponding steady state results explicitly. And also we derive the system performance measures like average number of customers in the queue and the average waiting time in closed form.