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2.1 Introduction: A Brief History

,
t
jump from the state n n 1 in the time interval ( t ) t is , t
n
whereas, the probability for the transition n n 1 in the interval
( t ) t is . t This is not all. There is also the probability that no
t
,
n
transitions from the state n will occur, given by 1 It can then
n
be shown, under these assumptions, which ignores the probability for
all other neighbouring types of transitions, since they are highly
improbable as well as of negligible order ( t that the entire birth-
),
o
and-death process can be expressed as a Taylor series, as elucidated in
Bailey (1990) [1],
p (t t p t t
n n 1 n 1 n n
(2)
( )
n 1 n 1
where the probability coefficients, , and , are arbitrary functions
n n n
of both the index n and the time .t
Generally, for linear birth-and-death processes, the linear
dependence of the probability coefficients , and with respect
n n n
to n is specified but their dependence on t is still arbitrary, so that the
character of these coefficients persevere as continuous-time functions.
More precisely,

n (t ), n (t ), and n (t ). (3)
n n n
Of course, what appears from Eq. (2), when the limit t 0
is invoked and the definitions above for the coefficients , , and ,
n n n
applied, are the exalted birth-and-death equations:

dp ( )
t
n
t
t
(n 1) ( )p n 1 ( ) n ( ) ( ) ( 1) ( ) n 1 ( ), ( 1)
n
dt
(4)
dp ( )
t
0
( ) ( ).
1
dt
If the terminology of probability theory, as consumed in
Chapter 9 of Bailey (1990) [1], is remembered, then, it is not uncommon
to refer to Eq. (4) as a time inhomogeneous or non-homogenous birth-and-death
process, since (t ), (t )and (t are time-dependent coefficients, in
),
general. These quantities, in particular, “meter” the birth-and-death
rates in Eq. (4); that is, the probability for a random process to
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