### Poisson hypothesis for mean-field models of generalised Jackson networks with countable set of nodes

Rybko, A *(Russian Academy of Sciences)*

Wednesday 05 May 2010, 16:00-17:00

Seminar Room 1, Newton Institute

#### Abstract

In some of our previous papers we considered closed queueing "mean-field" systems. The mean-field condition for such a system means the evolution invariance under the action of corresponding permutation group. For example, for closed networks on complete graph with M servers this condition means that after being served, the customer is allowed to go for his next service to any of the M servers (queues) with uniform probability 1/M. In addition, the service time distribution is the same for all customers in all nodes. Now we consider countable symmetric generalised Jackson networks in thermodynamic limit. We will show that under general conditions of not being overloaded such systems always satsify the Poisson hypothesis (PH). We will relax the mean-field symmetry of our system. Namely, we will allow different probabilities to go to different classes of servers, as well as different service time distributions, depending on these classes. The rough idea why PH always holds for our open systems is the following: as we know, the main reason for the possible violation of PH is that the memory about the initial state of the system is preserved to some degree. Since however every customer of our open system spends in it only a finite time, the memory of the initial state falls away as the number of customers initially present in the system goes to zero with time.
This talk is based on the joint work with S Shlosman.

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