tomer visits node k imme- (3.90) diately after departing node j. A cycle is completed when a given node is returned to. As can be seen the routing rules include those mentioned in the previous section. (3.91) Given routing rules such as these, let the mean number of visits to each node per cycle be given by aji Finally denote the mean rate at which class r complete cycles by 0, and the flow of customers On many occasions the above recursion will not converge. How- through node j by Pj Conservation of flow gives Pi in terms of the of damping gets over this problem. With damping factor as in Equation 3.84. Equation 3.90 can be written as in Equation 3.92. (3.92) (3.84) The model is determined once the mean sojourn time at each node is specified. In the simplest cases this will be a function of mean On some occasions heavy damping may be needed i.e. values of flow and service capacity only. (Equation 3.85.) Y close to (). As an example, consider again the model of a series of trans- (3.85) mission links, access to which is regulated by window flow control, see Figure 3.10. Again there are N tokens which circulate at rate 0. From Figure 3.10 this is determined by the fixed point Equation This definition allows all the queuing models described in Sec- 3.93. tions 3.2.3 and 3.3.3 to 3.3.5. The conservation of flow equation (Equation 3.85), together with (3.93) Little's law applied to each customer class, gives Equation 3.86. which yield J +R equations in J+R unknowns. The rate of transmitting packets is 0 of course and therefore the (3.86) proportion of packets which are lost is given by expression 3.94. (3.94) If Dj is non-negative and an increasing function of Pi then the solution is unique. We now demonstrate this. Define H as in Equation 3.87. 3.5 Queuing models for multi-access channels (3.87) Queuing analysis has been successfully applied to determine the throughput of a wide range of access protocols used in radio com- However H is strictly concave because the D.'s are continuous and munications and elsewhere. T