A Single Server Markovian Queue with Single Working Vacation, State Dependent Reneging and Retention of Reneged Customers

Abstract

In this paper an infinite capacity single server Markovian queuing system with single working vacation, state dependent reneging and retention of reneged customers is examined. Whenever a customer arrives at the system, it activates an impatience timer for their service to begun. If it has not begun before the customer’s impatience timer expires, then the customers get impatient and may leave the system without getting service with some probability and may remain in the system with probability by employing certain convincing mechanisms for their service. It is assumed that the impatience timer depends on the server states. The closed form expressions of steady state probabilities when the server is in a regular busy period and in a working vacation period are obtained by using probability generating function approach. Various performance measures such as expected system size, expected sojourn time of a customer served, the proportion of customers served and the average reneging rate due to impatience are derived. Finally, some numerical illustrations generated by MATLAB(R2019a) software were presented in order to show how the various parameters of the model influence the performance measures of the system.