9.2.1.1. Multi-bit rate Erlang model

The aim here is to evaluate the handling capacity of a system that concentrates traffic from sources with different bit rates. This is typically the case of ISDN user concentrators, and synchronous network links, which have to handle calls with different bit rates (at n 64 kbit/s for example). But above all, as we shall see later, analysis of this case forms the basis of the modelling of statistical multiplexing for ATM and IP traffic.

The aim is to evaluate the probability of blocking B (call rejection), for offered traffic A.

Let c ₁ be the bit rate (or capacity) required by the source with the lowest bit rate. It is assumed that c ₁ = 1, ( c ₁ is taken as the bit rate unit). The bit rates c _i of the other sources will be approximated to multiples of c ₁. C is the total capacity of the concentrator in bit rate units c ₁.

The blocking probability is then simply given by the generalised Erlang formula applied to a system with N = C servers,

The calls are grouped into bit rate classes: x is the number of bit rate classes, and a _i is the activity in erlangs of a bit rate class c _i.

The probability P( n ₁, ..., n _x) of having ( n ₁