form of the waiting distribution is then given by Equation 3.39 and Equation 3.40. (3.33) 1=Q(1)= -- l++H`(()) (3.39) (3.34) (3.40) 2-H'(1-UZ) Transform methods 3/11 The form of I is particularly simple for the G/M/m queue. Work The equilibrium probabilities that there are k customers found in the opposite way around from the M/G/1 queue and look at the queue on arrival for k=m-1, m - 2, - are given by a series of number of departures between two arrivals. Suppose the interarrival equations such as Equation 3.50, Equation 3.51, and SO on. density is given by f(x). Then assume that there are n > j customers in the queue. The probability that there are j departures before the next arrival is given by Equation 3.41 if n j N m SO that all servers are busy during this period. (3.50) (3.41) For the R_, n m-1, the equilibrium probability that an arrival finds n customers ahead of him in queue is given by Equation 3.42, (3.51) where the equation includes all the possibilities: the preceding customer may find n -1, n, n + 1, customers in queue and then that there are (), 1, 2, corresponding departures. These equations can be solved recursively by setting R.1 and (3.42) then determining Rm-2'RM.3' in turn. The solution is com- pleted by renormalising the probabilities to sum to 1, as in Equation 3.52. These equations have the solution given by Equation 3.43 pro- (3.52) vided Equation 3.44 holds. (3.43) 3.3.5 Results from the transforms n=F(mu-mun)< (3.44) By differentiating Equation 3.38 and setting S to (), the mean waiting Thus n is determined through the Laplace transform of F. Aunique time in a M/G/1 queue is obtained. This formula is known as the root exists if mua < I where a is the arrival rate. By setting n = Pollazcek-Kinchine formula (Equation 3.53). m - 1 we obtain Equation 3.45. In the special case where m = 1 the normalisation of Equation 3.46 gives Equation 3.47 and the queue (3.53) length distribution is geometric. C = Rm-1 n1-m (3.45) Differentiating again the variance of the equilibrium wait may be determi