published_or_final_version / abstract / Mathematics / Doctoral / Doctor of Philosophy

A platoon commander has a helicopter to support two squads, which encounter two types of missions -- critical or routine --on a daily basis. During a mission, a squad always benefits from having the helicopter, but the benefit is greater during a critical mission than during a routine mission. Because the commander cannot verify the mission type beforehand, a selfish squad would always claim a critical mission to compete for the helicopterâ which leaves the commander no choice but to assign the helicopter at random. In order to encourage truthful reports from the squads, we design a token system that works as follows. Each squad keeps a token bank, with tokens deposited at a certain frequency. A squad must spend either 1 or 2 tokens to request the helicopter, while the commander assigns the helicopter to the squad who spends more tokens, or breaks a tie at random. The two selfish squads become players in a two-person non-zero-sum game. We find the Nash Equilibrium of this game, and use numerical examples to illustrate the benefit of the token system. / US Navy (USN) author.

A thesis submitted to the Faculty of Science, University of the Witwatersrand, Johannesburg, in fulfilment of the requirements for the degree of Doctor of Philosophy February 1976 / 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.

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It was first pointed out by Doob and Kakutani the connection between classical potential theory and Brownian motion. In [10] one finds that if P(t,x,A) is the probability transition function, i.e. P(t,x,A) = probability that a particle moves from the point x to the Borel subset A of a set I in time t, then the potential kernal, K(x,A), is defined as follows [...]

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