Time dependent solution of Non-Markovian queue with two phases of service and general vacation time distribution

Abstract

We consider an \(M^{[x]} / G / 1\) queue with two phases of service, with different general (arbitrary) service time distributions. The first phase of service is essential for all customers, as soon as the first service of a customer is completed, then with probability $\theta$, he may opt for the second service or else with probability \((1-\theta)\), he leaves the system. At each service completion, the server will take compulsory vacation. The vacation period of the server has two heterogeneous phases. Phase one is compulsory and phase two follows the phase one vacation in such a way that the server may take phase two vacation with probability \(p\) or return back to the system with probability \((1-p)\). The service and vacation periods are assumed to be general. The time dependent probability generating functions have been obtained in terms of their Laplace transforms and the corresponding steady state results have been obtained explicitly. Also the average number of customers in the queue and the waiting time are also derived.