ON A STUDY OF POINT PROCESSES ARISING IN CERTAIN LIVE SITUATIONS
Abstract
This thesis deals with three different yet related areas of stochastic modeling inspired by certain live situations. In the first part, a new queueing system with a shorthand name G/G/{p} is introduced and studied. In this queue, unlike the standard queues, the customers after getting served are allowed to become servers themselves. Let X(t) denote the number of customers waiting or being served at time t, with X(0) = X(,0), Y(t) denote the number of servers at time t, with Y(0) = Y(,0), D(t) denote the number of customers having been served and left the system by time t, with D(0) = 0. At the completions of his service each customer is assumed to become a server with probability p or leave the system with probability 1-p, independent of everything else. Among other results, we study the joint distribution of {X(t), Y(t), D(t)} for both their exact as well as asymptotic behavior for large t. In Chapter I such a queueing model is described. The second part deals with certain characterization results concerning point processes arising in Queueing theory. For instance, in Chapter II, we consider the problem that in a queueing system with infinite number of servers, given that the departure process is a Poisson process we show that the arrival process has to be a Poisson process under certain appropriate conditions. We also study certain characterization results for point processes with an exchangeable property in Chapter III. Among others, we prove that a mixed renewal process is a Markov process if and only if it is a mixed Poisson process. The third part deals with a quantal response process, where each time when a release of the drug occurs the amount of the drug to be released is "randomly" proportional to the amount present. Among other results, the probability that the subject never responds and a necessary and sufficient condition for this probability to be zero, are given. This problem will be studied in Chapter IV.
Degree
Ph.D.
Subject Area
Statistics
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