Page 11 - A First Course In Stochastic Models

2 THE POISSON PROCESS AND RELATED PROCESSES

Then S n is the epoch at which the nth event occurs. For each t ≥ 0, define the
random variable N(t) by

N(t) = the largest integer n ≥ 0 for which S n ≤ t.
The random variable N(t) represents the number of events up to time t.

Definition 1.1.1 The counting process {N(t), t ≥ 0} is called a Poisson process
with rate λ if the interoccurrence times X 1 , X 2 , . . . have a common exponential
distribution function

−λx
P {X n ≤ x} = 1 − e , x ≥ 0.
The assumption of exponentially distributed interoccurrence times seems to be
restrictive, but it appears that the Poisson process is an excellent model for many
real-world phenomena. The explanation lies in the following deep result that is
only roughly stated; see Khintchine (1969) for the precise rationale for the Poisson
assumption in a variety of circumstances (the Palm–Khintchine theorem). Suppose
that at microlevel there are a very large number of independent stochastic pro-
cesses, where each separate microprocess generates only rarely an event. Then
at macrolevel the superposition of all these microprocesses behaves approximately
as a Poisson process. This insightful result is analogous to the well-known result
that the number of successes in a very large number of independent Bernoulli
trials with a very small success probability is approximately Poisson distributed.
The superposition result provides an explanation of the occurrence of Poisson
processes in a wide variety of circumstances. For example, the number of calls
received at a large telephone exchange is the superposition of the individual calls
of many subscribers each calling infrequently. Thus the process describing the over-
all number of calls can be expected to be close to a Poisson process. Similarly, a
Poisson demand process for a given product can be expected if the demands are
the superposition of the individual requests of many customers each asking infre-
quently for that product. Below it will be seen that the reason of the mathematical
tractability of the Poisson process is its memoryless property. Information about
the time elapsed since the last event is not relevant in predicting the time until the
next event.

1.1.1 The Memoryless Property
In the remainder of this section we use for the Poisson process the terminology of
‘arrivals’ instead of ‘events’. We first characterize the distribution of the counting
variable N(t). To do so, we use the well-known fact that the sum of k inde-
pendent random variables with a common exponential distribution has an Erlang
distribution. That is,