network where U is the total traffic offered by the customer class using these random routes. The approach to closed queuing networks is very similar. To Transition begin with customers may be regarded as following a route given by a list of nodes in the order in which they are to be visited. Arrivals Ury(k,ng+1) However when a customer departs the final node in its list it returns Departures Qj(nj)8(k,nj) to the first and cycles through the network again. For each route there is a population of customers which follow it, N. Let N denote Moving between queues the population vector giving the number of customers following each route . The above definitions resemble those for an open network but with no departures or arrivals from the network. A reasonable candidate for the equilibrium is to use the same form as the open network. However there are no arrivals and SO a set of dummy (c)q(c,D) (3.71) arrival rates are assigned to each route u The precise values are not important as long as they are positive. The reason for this freedom in choice of arrival rates is mentioned in a moment. 3/14 Queuing theory Only certain states of the corresponding open network can be attained in the closed network. Thus, to obtain the candidate closed equilibrium distribution, the open distribution is renormalised to the total probability of all the possible states T in the closed queuing network, as in Equation 3.76 where (C) is defined in Equation 3. 64 SO that the equilibrium probability of being in state C is given by expression 3.77. (n2, m2 ) (3.76) C E T Token Figure 3.11 Communication network with two customer routes Arriving Packets Figure 3.10 Window flow control over three consecutive (3.81) transmission links G(N)r(C) (3.77) As a second example consider the communication network shown That this is indeed the equilibrium may be verified using the same in Figure 3.11. Customers move from A to C via the central queue argument based on theorem 1. Since the system is closed the or from B to D via the central queue with arrival 