##### Abstract

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This dissertation studies systems of "competing"
discrete random walks as discrete and continuous time
processes. A system is thought of as containing n
imaginary particles performing random walks on lines
parallel to the x-axis in Cartesian space. The particles
act completely independently of each other and have,
in general, different starting coordinates.
In the discrete time situation, the motion of the
n particles is governed by n independent streams of
Bernoulli trials with success probabilities PI' P2' · .. ,
and p n respectively. A success for partie at a
trial causes that particle to move one unit toward the
origin, and a failure causes it to take a "zero-'step U
(i.e. remain stationary). A probabilistic description
is first given of the positions of the particles at
arbitrary points in time, and this is extended to provide
time dependent and independent probabilities of which
particle is the winner, that is to say, of which particle first reaches the origin.
In this case "draws" are
possible and the relevant probabilities are derived.
The results are expressed, in particular, in tenns
of Generalized Hypergeometric Functions. In addition,
formulae are given for the duration of what may now be
regarded as a race with winning post at the origin.
In the continuous time situation, the motion of
the n particles is governed by n independent Poisson
streams, in general, having different parameters. A
treatment similar to that for the discrete time situation
is given with the exception of draw probabilities which
in this case are not possible.
Approximations are obtained in many cases. Apart
from their practical utility, these give insight into
the operation of the systems in that they reveal how
changes in one or more of the parameters may affect the
win and draw probabilities and also the duration of the
race.
A chapter is devoted to practical applications.
Here it is shown how the theory of random walks racing
toward the origin can be utilized as a basic framework
for explaining the operation of, and answering pertinent
questions concerning several apparently diverse situations.
Examples are Lanchester Combat theory, inventory control,
reliability and queueing theory.