n for the case when the arrival process is customers in the queue immediately prior to the entry of the new Poisson. In fact the waiting time distribution of the G/M/m queue, customer the position of the new customer is given by is always exponential with a point probability of not having to wait (k,n,+1). By choice of this vector it is possible to model queues at all as the reader may verify. with last come first served scheduling or service in random order. It remains to specify the state of the network. It is not enough Queuing networks simply to stipulate the number of customers in each queue since this ignores which class of customers are in what positions. The state of 3.4.1 With general customer routes node j is given by a vector c (n) where the kth component of is determined by the routing class of the customer in position k The results of this section can be applied to both open and closed together with the stage which he has reached along his route. The queuing networks. In an open queuing network customers make a state of the network consists of a vector whose components are the sequence of visits to various nodes within the network before individual queue state vectors, C. Let q(C,D) be the transition rate from state C to state D. departing from the network altogether. In closed networks on the There are three kinds of transition within the network, an arrival other hand the customer population within the network remains fixed with no new customers entering the network nor ones within to the network, a customer moving between queues and a departure. in it departing. Examples of these kinds of network with yet more The new state of the network immediately after the arrival of a general servers can be found in (Kelly, 1979). customer, following route r, into position k of its initial queue is The results are similar in both cases but begin by considering the given by A,(C). The state of the network immediately after the open case. The network is supposed to have J nodes. The server, at each node j, appl