Queues Applied to Telecoms

1The Poisson Process

Living without memory is an unforgettable experience.

Yolande Villemaire (1949–)

It is with this citation by the Quebecois novelist and poet Yollande Villemaire that we begin this chapter on the memoryless characteristics of the exponential function. We review this distribution, which is a probability tool frequently used in the study of queues. Using properties analogous to the exponential function, we can then introduce the Poisson¹ process in this chapter, describing the moments when random events occur, such as the arrival of customers in a queue.

1.1. Review of the exponential distribution

1.1.1. Definitions

DEFINITION 1.1.– A real positive random variable X follows an exponential distribution with parameter λ > 0 if:

[1.1]

The notation ℙ indicates the probability function. In equation [1.1], the expression ℙ(X > t) therefore designates the probability that the random variable X has a value that is superior to a positive real number t.

Just like any real random variable, it can be characterized by a probability density function.

Its density function is given by:

[1.2]

From the density function, we can also express the expected value [1.3] and the variance [1.4] of the random variable X following an exponential distribution with parameter λ:

[1.3]