The miracle of insensitivity for loss networks and the critical regime for load balancing in servers farms
Matthieu Jonckherre
FCEyN-UBA

The study of loss neworks (modelling for instance telephone networks) was very successfull in great part because of a mathematical miracle called insensitivity (robustness to statistical assumptions), which we explain in details. Nowadays, the proliferation of network architectures with millions of servers (amazon, google, CERN,...) makes load balancing (the possibility of routing incoming tasks to less loaded servers) one of the hottest topics in applied mathematics. If strict insensitivity (unfortunately) does not hold for many models of networks with load balancing, we show that some insensitive models can still give crucial insights. In particular, they allow to identify critical loads depending both on the buffer depths and the number of servers. We can show that there is a phase transition for the blocking probability: before the critical load, the blocking is exponentially small and becomes of polynomial order at the critical load. This generalizes the well-known Jagerman-Halfin-Whitt regime for a one-dimensional queue. It also gives a generalized staffing rule for a given target blocking probability. The question of universality of this phase transition is still open.