Stochastic Modeling and Analysis of Telecoms Networks

Chapter 6Poisson Process

 

 

 

The modeling of a physical system must comply with two constraints. On the one hand, it must reflect the reality as accurately as possible, and on the other hand, it must have a predictive role, in other words it must provide computational tools for the analysis. Beyond the difficulty to qualitatively and quantitatively determine the pertinent parameters of a physical system, the experience shows that the more one wants an accurate model, the less it will be tractable in practice.

Within the framework of queuing systems, we must, in the first place, model the process of arrivals of the requests. The Poisson process which we study in this Chapter, is the most frequently used model, primarily because it is one of the rare models with which we can make computations. This modeling is found to be highly pertinent for the telephone calls to a commutator. Unfortunately, this is not the same for other types of network, where the traffic is much more versatile. However, as we will see at the end of this chapter, the Poisson process can be modified, so as to reflect this versatility to a certain extent.

The definition of a point process and the associated notations are given in A.5.2. Let us recall that an integrable point process is a strictly increasing sequence of positive random variables (T₁, T₂, …) such that T_n → ∞ a.s.. By convention, we adjoin the random variable T₀ = 0 a.s. to this sequence. These random variables will represent the arrival times of ...