%0 Journal Article
%T A Batch Arrival Single Server Queue with Server Providing General Service in Two Fluctuating Modes and Reneging during Vacation and Breakdowns
%A Monita Baruah
%A Kailash C. Madan
%A Tillal Eldabi
%J Journal of Probability and Statistics
%D 2014
%I Hindawi Publishing Corporation
%R 10.1155/2014/319318
%X We study the behavior of a batch arrival queuing system equipped with a single server providing general arbitrary service to customers with different service rates in two fluctuating modes of service. In addition, the server is subject to random breakdown. As soon as the server faces breakdown, the customer whose service is interrupted comes back to the head of the queue. As soon as repair process of the server is complete, the server immediately starts providing service in mode 1. Also customers waiting for service may renege (leave the queue) when there is breakdown or when server takes vacation. The system provides service with complete or reduced efficiency due to the fluctuating rates of service. We derive the steady state queue size distribution. Some special cases are discussed and numerical illustration is provided to see the effect and validity of the results. 1. Introduction In this paper, we study a single server queue where the server is providing service in two different modes with probabilities and . In real life, service offered to each arriving unit may not be at the same rate. Sometimes the service offered by the server may be fast, mostly normal, and at times slow. Thus the efficiency of a queuing system is completely affected by different modes of service. In the current age of electronics, most of the servers in a queuing system are electronic devices. It is very natural that such machines may experience sudden mechanical breakdown which cause the stoppage of service until the machine is repaired. In such cases, the customer whose service is interrupted returns back to the head of the queue and waits until repair process is completed. In the real world, we see such breakdown occurring in machines used in production and manufacturing units, communication systems, traffic intersections, automatic teller machines, and so forth. The server while providing service may take a pause or break which is referred to as vacation in the queuing literature. We have also considered the phenomenon of customers¡¯ impatience called reneging. Persons may renege or leave the queue after joining as they become impatient when the server breaks down or during server vacation. Here we assumed reneging to happen during breakdowns and vacation. This is a very realistic situation where we come across situations where customers prefer to leave without receiving service when there is a failure in the service system or the server is not available for a certain period of time. There have been extensive studies in queues with vacations by prominent researchers since
%U http://www.hindawi.com/journals/jps/2014/319318/