Modelling and control of birth and death processes

Abstract

This thesis treats systems of ordinary differential equations that ar*? extracted from ch-_ Kolmogorov forward equations of a class of Markov processes, known generally as birth and death processes. In particular we extract and analyze systems of equations which describe the dynamic behaviour of the second-order moments of the probability distribution of population governed by birth and death processes. We show that these systems form an important class of stochastic population models and conclude that they are superior to those stochastic models derived by adding a noise term to a deterministic population model. We also show that these systems are readily used in population control studies, in which the cost of uncertainty in the population mean size is taken into account. The first chapter formulates the univariate linear birth and death process in its most general form. T i«- prvbo'. i: ity distribution for the constant parameter case is obtained exactly, which allows one to state, as special cases, results on the simple birth and death, Poisson, Pascal, Polya, Palm and Arley processes. Control of a popu= lation, modelled by the linear birth and death process, is considered next. Particular attention is paid to system performance indecee which take into account the cost associated with non-zero variance and the cost of improving initial estimates of the size of the popula” tion under control.