On a class of packet-switching networks on an infinite graph

A class of open queueing networks with packet switching is discussed. The configuration graph of the network may be finite or infinite. The external messages are divided into standard pieces (packets) each of which is transmitted as a single message. The messages are addressed, as a rule, to nearest neighbours and thereby the network may be treated as a small perturbation of the collection of isolated servers. The switching rule adopted admits overtaking: packets which appeared later may reach the delivery node earlier. The transmission of a message is completed when its last packet reaches the destination node. The main result of this paper is that the network possesses a unique stationary working regime. Weak dependence properties of this regime are established as well as the continuity with respect to the parameters of the external message flows.