TWO HETEROGENEOUS SERVERS MARKOVIAN QUEUE WITH IMPATIENT CUSTOMERS, SYNCHRONOUS VACATIONS AND N-POLICY ON SLOW SERVER

Abstract

In this thesis two heterogeneous servers Markovian queue with server states dependent customers’ impatience (balking and reneging), synchronous vacations and N-Policy on slow server is examined. Whenever a customer arrives at the system, it activates a random impatience timer. If its service has not been begun before the impatience timer expires, the customer may abandon the system. By employing a certain customer reten tion mechanism, impatient customers can be retained in the system. During vacation, customers are not served whereas in regular busy periods, when the number of waiting customers in the queue is below a certain threshold N, only the faster server provides service and the slower one remains idle. At or above this threshold, both servers offer service to customers with different service rates for Server 1 and Server 2, respectively. The inter-arrival times, the service times, the impatient times and the vacation times are taken to be independent and exponentially distributed. The individual steady state probabilities of the system when servers are in regular busy period and in a synchro nous vacation period are obtained recursively through employing continued fractions and incomplete gamma function along with some properties of confluent hyper-geometric function. Various performance measures such as expected system size, expected waiting time of a customer, the proportion of customers served, the average balking rate, the average reneging rate due to impatience and the average retention rate are derived. The finding of this study has shown that the server vacation time and the customer impa tience have a negative impact on the number of customer served as well as the customer waiting time. So, these factors must have to be considered in modeling many real-world service systems so that the system becomes realistic. Finally, numerical illustrations in the form of tables and graphs are presented to demonstrate how the various parameters of the model influence the performance measures of the system. For further work, it will be interesting to consider a multiserver queueing system with heterogeneous ser vice times, server states dependent impatient customers, Synchronous Vacations and multiple threshold levels on the Slower Servers.