Queues Applied to Telecoms

12Chapter 4 Exercise

One of the difficulties of this exercise compared to simple queues is that the state space is not ℕ but ℕ^N, and the indexes are therefore vectors of size N.

To keep notations easy to read, we take n to represent the states (it will be a vector) and n_p for the p-th coordinate. To avoid confusing these notations with those of the distributions, if π is a distribution for the whole system (thus for the N-tuple), π_p represents the marginal distribution of the р-th queue and the probabilities of being in a given state will be denoted π(n) and π_p (n_p).

To determine the infinitesimal stochastic generator, we must decompose the types of transitions that are possible for a given queue р:

• – exogenous arrivals: ;
• – exiting the network: ;
• – routing from queue р to queue .

From this, we can deduce the non-zero terms of the infinitesimal generator:

Let us denote λ_p the expectation of entering queue р, which combines exogenous arrivals and routing from other queues. Since the queues are in equilibrium, everything that enters leaves, so the expectation of an exit from queue ...