Analysis of a birth and death process with alternating rates and of a telegraph process with underlying random walk
Abstract
My thesis for the Doctoral Programme in Mathematics (November 1,
2008  October 31, 2011) at University of Salerno, Italy, has been oriented
to the analysis of two stochastic models, with particular emphasis on the de
termination of their probability laws and related properties. The discussion
of the doctoral dissertation will be given in 20 March 2012.
The first part of the thesis is devoted to the analysis of a birth and death
process with alternating rates. We recall that an extensive survey on birth
death processes (BDP) has been provided by Parthasarathy and Lenin [3].
In this work the authors adopt standard methods of analysis (such as power
series technique and Laplace transforms) to find explicit expressions for the
transient and stationary distributions of BDPs and provide applications of
such results to specific fields (communication systems, chemical and biolog
ical models). In particular, in Section 9 they use BDPs to describe the time
changes in the concentrations of the components of a chemical reaction and
discuss the role of BDPs in the study of diatomic molecular chains. More
over, the paper by StockMayer et al. [4] gives an example of application of
stochastic processes in the study of chain molecular diffusion. In this work
a molecule is modeled as a freelyjoined chain of two regularly alternating
kinds of atoms. All bonds have the same length but the two kinds of atoms
have alternating jump rates, i.e. the forward and backward jump rates for
even labeled beads are α and β, respectively, and these rates are reversed
for odd labeled beads. The authors obtain the exact timedependent aver
age length of bond vectors. Inspired by this works, Conolly [1] studied an
infinitely long chain of atoms joined by links of equal length. The links
are assumed to be subject to random shocks, that force the atoms to move
and the molecule to diffuse. The shock mechanism is different according to
whether the atom occupies an odd or an even position on the chain. The
originating stochastic model is a randomized random walk on the integers
with an unusual exponential pattern for the interstep time intervals. The
authors analyze some features of this process and investigate also its queue
counterpart, where the walk is confined to the non negative integers.
Stimulated by the above researches, a birth and death process N(t) on
the integers with a transition rate λ from even states and a possibly different
rate μ from odd states has been studied in the first part of the thesis. A de
tailed description of the model is performed, and the ChapmanKolmogorov
equations are introduced. Then, the probability generating functions of even
and odd states are then obtained. These allow to evaluate the transition
probabilities of the process for arbitrary integer initial state. Certain sym
metry properties of the transition probabilities are also pinpointed. Then,
the birth and death process obtained by superimposing a reecting bound
ary in the zerostate is analyzed. In particular, by making use of a Laplace
transform approach, the probability of a transition from state 0 or state 1 to
the zerostate is obtained. Formulas for mean and variance of both processes
are finally provided.
The second part of the thesis is devoted to the analysis of a generalized
telegraph process with an underlying random walk. The classical telegraph
process describes a random motion on the real line characterized by two _nite
velocities with opposite directions, where the velocity changes are governed
by a timehomogeneous Poisson process (see Orsingher [2]). The novelty in
the proposed model consists in the use of new rules for velocity changes,
which are now governed by a sequence of Bernoulli trials. This implies that
the random times separating consecutive changes of direction of the mov
ing particle have a general distribution and form a nonregular alternating
renewal process. Starting from the origin, the running particle performs
an alternating motion with velocities c and v (c; v > 0). The direction
of the motion (forward and backward) is determined by the velocity sign.
The particle changes the direction according to the outcome of a Bernoulli
trial. Hence, this defines a (possibly asymmetric) random walk governing
the choice of the velocity at any epoch. By adopting techniques based on
renewal theory, the general form of probability law is determined as well
as the mean of the process. Furthermore, two instances are investigated in
detail, in which the random intertimes between consecutive velocity changes
are exponentially distributed with (i) constant rates and with (ii) linearly
increasing rates. In the first case, explicit expressions of the transition den
sity and of the conditional mean of the process are expressed as series of
Gauss hypergeometric functions. The second case leads to a damped ran
dom motion, for which we obtain the transition density in closed form. It is
interesting to note that the latter case yields a logistic stationary density.
References
[1] Conolly B.W. (1971) On randomized random walks. SIAM Review, 13,
8199.
[2] Orsingher, E. (1990) Probability law, flow functions, maximum distri
bution of wavegoverned random motions and their connections with
Kirchoff's laws. Stoch. Process. Appl., 34, 4966.
[3] Parthasarathy P.R. and Lenin R.B. (2004) Birth and death process
(BDP) models with applicationsqueueing, communication systems,
chemical models, biological models: the stateof the art with a time
dependent perspective. American Series in Mathematical and Manage
ment Sciences, vol. 51, American Sciences Press, Columbus (2004)
[4] Stockmayer W.H., Gobush W. and Norvich R. (1971) Localjump mod
els for chain dynamics. Pure Appl. Chem., 26, 555561.
NOTE
The thesis consists of four chapters:
Chapter 1. Some definitions and properties of stochastic processes.
Chapter 2. Analysis of birthdeath processes on the set of integers, char
acterized by alternating rates.
Chapter 3. Results on the standard telegraph process.
Chapter 4. Study of the telegraph process with an underlying random
walk governing the velocity changes.
[edited by author]
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