MOSCOW MATHEMATICAL JOURNAL Volume 8, Number 1, January–March 2008, Pages 159–180 PHASE TRANSITIONS IN THE QUEUING NETWORKS AND THE VIOLATION OF THE POISSON HYPOTHESIS ALEXANDER RYBKO AND SENYA SHLOSMAN Abstract. We present examples of queuing networks that never come to equilibrium. That is achieved by constructing certain non-linear Markov evolutions, which are non-ergodic and possess eternal transience property. 2000 Math. Subj. Class. Primary 82C20; Secondary 60J25. Key words and phrases. Self-averaging, mean field, eternal transience. 1. Introduction The Poisson Hypothesis is a device to predict the behavior of large queuing net- works. It was formulated first by L. Kleinrock in [K], and concerns the following situation. Suppose we have a large network of servers, through which many cus- tomers are traveling, being served at different nodes of the network. If the node is busy, the customers wait in the queue. Customers are entering into the network from the outside via some nodes, and these external flows of customers are Poisso- nian. The service time at each node is random, depending on the node. The PH prediction about the (long-time, large-size) behavior of the network is the following: • consider the total flow F of customers to the node N . Then F is approx- imately equals to some Poisson flow, P . Moreover, its rate function λ(t) tends to a limit as t →∞. Also, the flows to different nodes are almost independent. Note that the distribution of the service time at the nodes of the network can be arbitrary, so PH deals with quite a general situation. The range of validity of PH is supposed to be the class of networks where the internal flow to every node N is a sum of flows from many other nodes, and each of these flows constitute only a small fraction of the total flow to N . If true, PH provides one with means to make easy computations of quantities of importance in network design. In our paper [RS] we have proven the Poisson Hypothesis for some simple net- works in the infinite volume limit, under some natural conditions. The present paper is devoted to the cases of violations of the Poisson Hypoth- esis. We will show that once the distribution of the service time does not decay Received April 26, 2006. c 2008 Independent University of Moscow 159