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Markovian Birth-and-Death Equations

T he Markovian birth-and-death equations, or simply referred to here
as birth-death equations, consents to a continuous-time Markov
process that were thought, for some time, to have been solved
exactly. However, there was much skepticism by the present
author with reference to the solution cited in the literature, since it did
not appear to adhere to any kind of periodicity that an evolutionary
cycle, such as that of a birth-death process, is expected to maintain. This
apprehension urges for a reexamination of the birth-death equations,
with the possibility of finding periodic solutions of the birth-death
equations that may have, in fact, been overlooked – which is the main
purpose of the present chapter.
This chapter begins first with a historical discussion of the
Markov equations for birth-and-death, and then builds up to the recent
uncovering of a “hidden symmetry” inherent in these birth-death
equations. Moreover, as will be demonstrated, a unique solution for the
birth-death equations must, inevitably, also satisfy the initial condition
of a Poisson process, .
n ,0

2.1 Introduction: A Brief History
There is no need to be so in-depth here than to indicate, out of all the
various types of Markov processes in probability theory, the one type of
Markov chain likely to be studied most is a linear “birth-and-death”
process. For this process, the transition probabilities, for a random
variable X ( ), are denoted formally by
t

p ( ) X ( ) n , n 0,1, 2, . (1)
t
t
n
as based on the assumptions that at time t the transition probability to