New and Extended Results in Renewal and Queueing Theories

Abstract

This thesis encompasses new and extended results in renewal and queueing theories. In the renewal theory portion of this thesis, the asymptotic result of renewal mass function and new asymptotic moments are found using the method of generating functions. This method is not only simple but also provides the extra constant terms in the asymptotic second moment which are unavailable in the literature. Higher asymptotic moments and their corresponding extra constant terms can also be found using the method of generating functions. Previous results in the existing literature do not have these extra constant terms. Recent work in renewal theory has the extra constant terms in a non-bulk renewal processes. The purpose of this thesis is to extend that recent work to the bulk-renewal processes in discrete-time. In the queueing theory portion of this thesis, the imbedded Markov chain technique is used to determine the distributions of the number of uncompleted service stages, the number of customers in the system, and the waiting-time-in-queue. Single-server queues with a fixed number of service stages have been analyzed by many authors, some of whom state that there is no simple way to analyze the queue 𝐺𝐼/𝐸𝑋/1. The purpose of this thesis is to review and extend the previous work on 𝐺𝐼/𝐸𝑟/1 to the more general model 𝐺𝐼/𝐸𝑋/1 in which the number of stages is randomly distributed.