ned,asinEquation 3.54. (3.46) (3.54) Var = R = 1 - n (3.47) In fact by a manipulation the waiting time transform can be inverted. as in Equation 3.55. obtain the remaining probabilities, in the case when m > 1, ..., R compute the probability . of there being j w"(s) = departures between two customer arrivals when there are n cus- tomers initially. This has been done already in the case n - j N m. Suppose n V m, then Equation 3.48 may be obtained, and if n>m >n-j,thenEquation 3.49 is obtained. (3.55) (3.48) However the Laplace transform of the residual service time den- sity r(x)=11(1-H(x)) is given by expression 3.56. SO that Equation 3.57 gives the equilibrium waiting time density (Benes, 1956) in an M/G/1 queue. (3.49) (1-H'(s)) (3.56) 3/12 Queuing theory (3.62) otherwise (3.57) (3.63) For the G/M/1 queue the equilibrium distribution of queue length on arrival (and therefore on departure as well) is geometric, as in The customers' routes through the network are determined by an Equation 3.58. ordered set of nodes, which may be regarded as the list of nodes which the customer will visit in the order in which he will visit Pr(Q=k|=(1-n)n* (3.58) them. A list is denoted r = (r(1) .,r(S(r))) where S(r) is the number of nodes visited. Repeat visits are permitted. It is supposed that customers following route r arrive as a Poisson From this it follows immediately that the equilibrium G/M/1 stream with rate independently. Furthermore it will be supposed waiting time distribution is exponential, as in Equation 3.59. that the number of routes is finite although this is not a necessary (3.59) assumption. Apart from specifying which order the customers of a particular routing class visit a queue it is also necessary to specify how they The probability of not waiting at all is given by setting X to be 0, enter each queue. This is determined by the random entrance vector, 1-n. This will not be the probability that the server is idle in 8, which is independent of routing class. Given that there are n general with an exceptio