Page 10 - A First Course In Stochastic Models

CHAPTER 1

The Poisson Process and

Related Processes

1.0 INTRODUCTION
The Poisson process is a counting process that counts the number of occurrences
of some specific event through time. Examples include the arrivals of customers
at a counter, the occurrences of earthquakes in a certain region, the occurrences
of breakdowns in an electricity generator, etc. The Poisson process is a natural
modelling tool in numerous applied probability problems. It not only models many
real-world phenomena, but the process allows for tractable mathematical analysis
as well.
The Poisson process is discussed in detail in Section 1.1. Basic properties are
derived including the characteristic memoryless property. Illustrative examples are
given to show the usefulness of the model. The compound Poisson process is
dealt with in Section 1.2. In a Poisson arrival process customers arrive singly,
while in a compound Poisson arrival process customers arrive in batches. Another
generalization of the Poisson process is the non-stationary Poisson process that is
discussed in Section 1.3. The Poisson process assumes that the intensity at which
events occur is time-independent. This assumption is dropped in the non-stationary
Poisson process. The final Section 1.4 discusses the Markov modulated arrival
process in which the intensity at which Poisson arrivals occur is subject to a
random environment.

1.1 THE POISSON PROCESS
There are several equivalent definitions of the Poisson process. Our starting point is
a sequence X 1 , X 2 , . . . of positive, independent random variables with a common
probability distribution. Think of X n as the time elapsed between the (n−1)th and
nth occurrence of some specific event in a probabilistic situation. Let
n

S 0 = 0 and S n = X k , n = 1, 2, . . . .
k=1
A First Course in Stochastic Models H.C. Tijms
c 2003 John Wiley & Sons, Ltd. ISBNs: 0-471-49880-7 (HB); 0-471-49881-5 (PB)