An M|G|1 Retrial Queue with Nonpersistent Customers and Orbital Search

Abstract

The M|G|1 retrial queue with nonpersistent customers and orbital search is considered. If the server is busy at the time of arrival of a primary customer, then with probability 1 − H 1 it leaves the system without service, and with probability H 1 > 0, it enters into an orbit. Similarly, if the server is occupied at the time of arrival of an orbital customer, with probability 1 − H 2, it leaves the system without service, and with probability H 2 > 0, it goes back to the orbit. Immediately after the completion of each service, the server searches for customers in the orbit with probability p > 0, and remains idle with probability 1 − p. Search time is assumed to be negligible. In the case H 2 = 1, the model is analyzed in full detail using the supplementary variable method. The joint distribution of the server state and the orbit length in steady state is studied. The structure of the busy period and its analysis in terms of Laplace transform is discussed. We also provide a direct method of calculation for the first and second moment of the busy period. In the case H 2 < 1, closed form solution is obtained for exponentially distributed service time, in terms of hypergeometric series.