Perturbation theory approach for a class of hybrid switching networks with small transit flows

A method originating from statistical mechanics (low-density and high-temperature expansions) is used to prove the existence and uniqueness of a stationary regime for switching networks on finite or infinite graphs. The main assumption is that the message (customer) flows circulating through the network are .localized' in the sense that, for any message, the probability of having a long path is rapidly decreasing (and, moreover, a path of a .typical' message consists of one line). The switching rule combines message-switching and circuit-switching principles. The stationary regime for the network under consideration may be treated as a .small perturbation' of the .idealized' regime in the totally decoupled network where all the messages have single line paths.