Stochastic Modeling and Analysis of Telecoms Networks

Chapter 8Systems with Delay

 

 

 

All actual telecommunication systems are loss systems, since all the buffers have finite, and hence limited capacity. By system with delay, we mean a system in which the dimensioning is such that the loss caused by the overflow is negligible, and for which the relevant criterion for assessing the performances, is the waiting time.

Before we start a detailed study of these systems, we first introduce a well-known, general and very useful relation called Little's Formula.

8.1. Little's Formula

We consider a system with delay, in which the customers arrive at times (T_n, n ≥ 1), spend in the system sojourn times given by (W_n, n ≥ 1) and leave the system at times (D_n = T_n + W_n, n ≥ 1). We denote N as the point process of arrivals, D as the departure process and X, the process counting the number of customers in the system. At time 0, the system is assumed to be empty, i.e. X (0) = 0. The key point is that the system is conservative : all the incoming work is processed by the server(s).

THEOREM 8.1 (Little's Formula).– We assume that N is asymptotically linear, i.e. there exists λ > 0 such that

and that the sequence W is ergodic, i.e.

Under these assumptions, we have

The existence of the latter limit is shown in the following proof.

Proof. Let us fix ...