ON SOME RESULTS IN BRANCHING PROCESSES AND GA/G/INFINITY QUEUE WITH AN APPLICATION TO BIOLOGY
Abstract
This thesis deals with a stochastic model suggested by a life situation, the life cycle of bacteria, as well as a queueing model suggested by the life situation, and some new results in the theory of branching processes which arose out of research into the properties of the model. In Chapter I, we prove certain preliminary results for random measures as well as two strong laws of large numbers. These results are especially useful in Chapter II in proving results for the GA/G/(INFIN) queue, while one strong law is used in Chapter III in proving a theorem for age-dependent branching processes. In Chapter II, the GA/G/(INFIN) queue is introduced. The distinguishing feature of this infinite server queue is that the arrival rate converges to infinity though in a "stable" sense suggested by results in Chapter I. Various limit theorems are given for this queueing model. In Chapter III, the almost sure convergence of the age distribution to a non-trivial limit is proved for the case when the mean of the offspring distribution is finite for age dependent branching processes. Two interesting corollaries of this fact are also proved concerning growth properties of the process. Chapter IV contains a summary of previous mathematical models of the bacterial life cycle, as well as the postulates of our own model. Chapter V contains some mathematical development of the model as well as some qualitative comparisons with experimental data. In addition, the strengths and weaknesses of the model are presented along with suggestions for further research.
Degree
Ph.D.
Subject Area
Statistics
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