A SINGLE SERVER MARKOVIAN QUEUE WITH SINGLE WORKING VACATION AND STATE DEPENDENT RETENTION OF RENEGED CUSTOMERS

Abstract

In this thesis an infinite capacity single server Markovian queuing system with single working vacation, reneging and retention of reneged customers is examined. Whenever a customer arrives at the system, it activates an impatience timer for his 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 con vincing mechanisms for his service. It is assumed that the impatience timer depends on the server’s states. During a working vacation, customers are served at a slower rate than usual service rate. The server serves the customers on a first-come, first served basis. The inter-arrival times, the service times, the impatient times and the vacation times are taken to be independent and exponentially distributed. 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 and the steady state probabilities of the system being in any particular state were obtained by using continued fractions and incomplete gamma function along with some properties of confluent hypergeometric function. Various performance measures such as the expected system size, the expected sojourn time of a customer served, the proportion of customers served and other performance measures are derived. Finally, numerical illustrations generated by MATLAB(R2019a) software were presented in the form of tables and graphs to demonstrate how the various parameters of the model in fluence the performance measures of the system. The model considered in this study can be extended to Markovian queue with state dependent retention of reneged customers, working vacations and Bernoulli scheduled vacation interruption.