(An) analysis of general input queues with working vacations

Abstract

In this dissertation, general input queues with single working vacation are analysed. General queues encompass the continuous time GI/M/1 queue and the discrete time GI/Geo/1 queue. The working vacation is a newly devised vacation scheme. This scheme is different from the classical vacation schemes in that a server serves customers even when the server is on a vacation, but service rate is assumed to be different (in most cases, lower) from the nominal one. Using embedded Markov chain, the queue length distribution at arrival epochs of the discrete time GI/Geo/1/SWV queue is obtained. Then using this queue length distribution, the sojourn time in the system is presented. Lastly, using semi-Markov process, the queue-length distribution at arbitrary epochs is derived. Then a similar analysis is applied to the continuous time GI/M/1/SWV queue, same performance indices are presented. After all these steps, the GI/Geo/1/SWV queue and the GI/M/1/SWV queue is contrasted by taking the limit that makes a slot size of the GI/Geo/1/SWV queue goes to zero. We can obtain the result of GI/M/1/SWV queue from the GI/Geo/1/SWV queue. We also present analyses of a busy period of three general input queues with vacations: the n-policy GI/M/c queue, the GI/M/1/MWV queue, and the GI/Geo/1/MWV queue. A busy period is defined as time duration from when the system size becomes one to the instant when the system size becomes zero. For the n-policy GI/M/c queue, the length of a busy period, the number of customers served during the busy period, and the consecutive idle period are given in terms of the joint transform. For the queue with MWV, the length of a period when a service rate is normal, the length of a period with a lower service rate, the number of customers served in each period, and the length of consecutive idle period are given in terms of the joint transform. By utilizing this joint transform, we can derive various results related to moments, i.e., higher moments a...