 By taking different birth and death rates one can obtain a wide is called the probability flux out of j into state k in equilibrium. range of queuing models, for example as in Table 3.3. Note that in each case the mean sojourn time, E[D], can be found (3.12) by an application of Little's law. However in the case of M/M/K//L there is only a finite number of customers and in the case of M/M/1/S, customers are lost because there is only finite storage. In neither case is the arrival rate of customers entering queue a. 3.2.3 Birth-Death models of queues For M/M/K//L Equation 3.16 may be obtained, where 0 is the mean rate at which the customers complete a cycle of standing idle, The underlying idea of this class of queuing models is to regard queuing and service. arriving customers as births and departures as deaths thus the queue alters by at most one. The most general form of linear birth-death (3.16) process is depicted in Figure 3.7. The global balance equations can be obtained by equating the probability flux out of each state with the probability flux into it as Equation 3.16 follows from Little's law applied to all the cus- already described. (Equation 3.13.) tomers, , 2 is the mean time a customer stands idle before returning to queue for service. The second part comes from Little's law applied to the queuing and service stages, -E[O]. (2k+)k (3.13) Substituting for 0 gives Equations 3.17 and 3.18. However these equations may be solved by using a simpler set of (3.17) equations, known as the local (detailed) balance equations, as in Equation (3.14). (3.14) (3.18) From these equations the equilibrium solution is easily seen to be as in Equation 3.15, and is found from the fact that the prob- For M/M/1/S the mean arrival rate of customers entering queue is abilities sum to 1. given by expression 3,19. (3.15) 2(1-p5) (3.19) (1-pst) It can be seen that if a probability distribution which satisfies the Again from Little's law we have Equation 3.20. detailed balance equations can be found then it must satisfy the g