Page 31 - A First Course In Stochastic Models

22 THE POISSON PROCESS AND RELATED PROCESSES

α and scale parameter β, the sum D 1 + · · · + D n has a gamma distribution with
shape parameter nα and scale parameter β. The computation of the gamma distribu-
tion offers no numerical difficulties; see Appendix B. The assumption of a gamma
distribution is appropriate in many inventory applications with X(t) representing
the cumulative demand up to time t.

1.3 NON-STATIONARY POISSON PROCESSES

The non-stationary Poisson process is another useful stochastic process for counting
events that occur over time. It generalizes the Poisson process by allowing for an
arrival rate that need not be constant in time. Non-stationary Poisson processes
are used to model arrival processes where the arrival rate fluctuates significantly
over time. In the discussion below, the arrival rate function λ(t) is assumed to be
piecewise continuous.

Definition 1.3.1 A counting process {N(t), t ≥ 0} is said to be a non-stationary
Poisson process with intensity function λ(t), t ≥ 0, if it satisfies the following
properties:

(a) N(0) = 0
(b) the process {N(t)} has independent increments

(c) P {N(t + t) − N(t) = 1} = λ(t) t + o( t) as t → 0
(d) P {N(t + t) − N(t) ≥ 2} = o( t) as t → 0.
The next theorem proves that the total number of arrivals in a given time interval
is Poisson distributed.
Theorem 1.3.1 For any t, s ≥ 0,
[M(t + s) − M(t)] k
−[M(t+s)−M(t)]
P {N(t + s) − N(t) = k} = e , (1.3.1)
k!
x
for k = 0, 1, . . . , where M(x) = λ(y) dy, x ≥ 0.
0
Proof The proof is instructive. Fix t ≥ 0. Put for abbreviation
p k (s) = P {N(t + s) − N(t) = k}, k = 0, 1, . . . .
Consider now p k (s + s) for s small. Since the probability of two or more
arrivals in a small time interval of length s is negligibly small compared with
s as s → 0, it follows that the only possibility for the process to be in state k
at time t + s + s is that the process is either in state k − 1 or in state k at time
t + s. Hence, by conditioning on the state of the process at time t + s and given
that the process has independent increments,

p k (s + s) = p k−1 (s)[λ(t + s) s + o( s)] + p k (s)[1 − λ(t + s) s + o( s)]