Queues Applied to Telecoms

4Product-Form Queueing Networks

Never will you fulfill an expectation.

Johann Wolfgang von Goethe (1749–1832)

We have examined simple queues, but, in practice, we often encounter queueing networks and not simple queues. We here study an important class of queueing networks commonly used in the application of network optimization problems. They are called Jackson¹ networks, from the name of the mathematician who studied them at the end of the 1950s. Later, during the mid-1960s, four specialists of queues, F. Baskett², K.M. Chandy³, R.R. Muntz⁴ and F.G. Palacios⁵, in an article that appeared in the Journal of ACM, generalized Jackson’s results by relaxing certain constraints on the distributions of service. These generalized networks are called BCMP networks, following the initials of the four mathematicians.

This chapter is devoted in particular to the notion of queueing networks. It is only an introduction for those who wish to study further. We will not offer many exercises.

4.1. Jackson networks

4.1.1. Definition of a Jackson network

We define a Jackson network of M/M/1 queues.

Let us consider an open network of I queues:

• – arrivals: in each queue, i customers arrive following a Poisson process with intensity v_i;
• – demand for service: customers in queue i request i.i.d exponential services with an average of 1/μ_i, i = 1, …, I;
• – Jackson routing: after being served from queue i, a customer is directed to queue j with a routing probability of р_ij and leaves the network ...