Network traffic matrix: how can one learn the prior distributions from the link counts only?

The origin destination (OD) traffic matrix in a network is the volume of traffic for each OD pair. It is very useful for capacity planning, routing, operations and management, SLAs, customer reports, etc... But this matrix cannot be measured directly, it would require very costly upgrades of the existing infrastructures. On the contrary SNMP reports give a periodic account on the volume of traffic for each link. The traffic matrix must then be estimated by mathematical methods from the link counts only. This inverse problem is typically ill-posed since the number of OD pairs is much greater than the number of links. One of the existing techniques is Bayesian: it runs Markov chain Monte Carlo (MCMC) algorithms to simulate the joint distribution of the OD pairs given the link counts. It requires a prior distribution for each OD pair and, most of the time, those priors are arbitrary. A challenging issue would be to train the priors from the only available data, that is to say the link counts themselves. This is exactly what we do in this paper. We prove the validity of our approach on a network on which direct measurements of the OD counts were made available.