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6.5 Counting Processes 187

Example 6.13 (Two-stage chain). Consider the two-stage Markov chain considered in
Examples 6.9 and 6.11. The initial distribution p = (θ, 1 − θ)is stationary with respect to
P whenever

θ = θα + (1 − θ)(1 − β);

that is, whenever,
θ 1 − β
=
1 − θ 1 − α
in which case
1 − β
θ = .
2 − (α + β)

6.5 Counting Processes
A counting process is an integer-valued, continuous time process {X t : t ≥ 0}. Counting
processes arise when certain events, often called “arrivals,” occur randomly in time, with
X t denoting the number of arrivals occurring in the interval [0, t]. Counting processes are
often denoted by N(t) and we will use that notation here.
It is useful to describe a counting process in terms of the interarrival times. Let T 1 , T 2 ,...
be a sequence of nonnegative random variables and define

S k = T 1 + ··· + T k .

Suppose that N(t) = n if and only if
S n ≤ t and S n+1 > t;

Then {N(t): t ≥ 0} is a counting process. In the interpretation of the counting process in
terms of random arrivals, T 1 is the time until the first arrival, T 2 is the time between the first
and second arrivals, and so on. Then S n is the time of the nth arrival.
If T 1 , T 2 ,... are independent, identically distributed random variables, then the process
is said to be a renewal process.If T 1 , T 2 ,... has a stationary distribution then {N(t): t ≥ 0}
is said to be a stationary point process.

Example 6.14 (Failures with replacement). Consider a certain component that is subject
to failure. Let T 1 denote the failure time of the original component. Upon failure, the original
component is replaced by a component with failure time T 2 . Assume that the process of
failure and replacement continues indefinitely, leading to failure times T 1 , T 2 ,... ; these
failure times are modeled as nonnegative random variables. Let {N(t): t ≥ 0} denote the
counting process corresponding to T 1 , T 2 ,.... Then N(t) denotes the number of failures in
the interval [0, t]. If T 1 , T 2 ,... are independent, identically distributed random variables,
then the counting process is a renewal process.
Figure 6.3 gives plots of four randomly generated counting processes of this type in which
T 1 , T 2 ,... are taken to be independent exponential random variables with λ = 1/2, 1, 2, 5,
respectively.