lobal balance equations. This is SO because each global balance equation is the sum of two detailed balance equations, excluding the E[D] = 1 first and the last, if there is a last. These two are just a single local balance equation. (Apart from the simplification provided, the sig- 3/8 Queuing theory Table 3.3 Equilibrium queue distribution for some birth-death models Queue type Parameters Equilibrium queue distribution M/M/I Nk=h,kz0 Qk=(1-p)pk,kzl Mk=u,kz1 E|Q]=(1-p) M/M/K//L L>K,N=(L-k) uk=ku,ksk Ku,k>K M/M/1/S Ak=h,k<s Nk=0,kzs Mk=u,k>0 M/M/x Nk=h, Mk=ku en=cupe E[Q] = p (3.20) Fortunately it is the case that, in equilibrium, the first two of these distributions coincide in very general circumstances. This will be demonstrated in a moment. It is even possible that all three of these distributions will coincide in equilibrium, this is the case when the 3.3 Transform methods arrival process is Poisson. For the equality of the first two distributions in equilibrium it is 3.3.1 Customer arrival and departure sufficient that arrivals and departures should take place singly and distributions that at least one of the two equilibrium distributions should exist. This latter condition holds for most queues, even Non-Markovian It is possible to think of the distribution of the number of customers ones, provided that the queue is stable. To see this suppose, for in a queue in three ways: simplicity, that the queue is initially empty. (The argument also works when there is an initial queue of customers but this minor complication is ignored.) 1. The distribution of the number of customers found in queue by an arriving customer. Figure 3.8 shows the kth departure leaving j m customers still 2. The distribution of the number of customers left in queue by a in the system. This means that customer k+m+ 1 is yet to arrive and departing customer. that when he does he will find m or less customers in the system. 3. The distribution of the number of customers in queue at a (Equation 3 particular time. (3.21) Departure Arrival m+