rates U and a equilibrium distribution cannot depend on the dummy arrival rates. respectively. The service rate at each queue is Q. The normalisation constant cancels out the dependence of the Let n. denote the number of customers going from A to D at each equilibrium distribution on the dummy arrival rates. stage i and m. the number at each stage going from B to C then, again As an example, consider a simple model of a series of trans- using Equation 3.64, the equilibrium distribution is as in Equation 3.82. mission links access to which is regulated by window flow control, see Figure 3.10. As depicted, a packet gains access by obtaining one of the window flow control tokens. This token is carried with the packet over the three transmission links before the token is released. It is supposed to return immediately to be available for other packets to use. As Figure 3.10 shows this can be represented by a closed (3.82) cyclic queuing network with the tokens as customers. These are 'served' by the arrivals which take place at rate a and then by the transmission links. Arrivals which find no tokens are lost. Suppose that there are N tokens and that the service rate at each Note the combinatorial term for the centre queue. The state re- of the transmission links is Q. The network state is represented by flects only the number of customers of each type at each queue and Equation 3.78. the combinatorial term accounts for all possible arrangements, which are all equally likely. (Equation 3.83.) (n123)+a (3.78) (3.83) Using Equation 3.64, the equilibrium distribution is then given by Equation 3.79, where G is given by Equation 3.80. (3.79) 3.4.2 Fixed point models for closed queuing networks (3.80) In many communication applications, the distributions of system random variables may be known only rather imperfectly. Consider the distribution of sojourn (queuing plus transmission) time of a packet at a node within a communication network. This varies with The probability a packet is lost because there is no token available 