By Alec M. Lee M.A. (auth.)

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**Example text**

W = _:__p_ Zp. q W I - p =__:__I_ Zp. 27) This model is quite a useful one, as there are numerous real situations in which the coefficient of variation of the service-time distribution is quite small, so that the service-times may be effectively considered to be constant. We shall have occasion to examine at least one case later in this book. M/D/I: (oo/SIRO) In many real-life situations the FIFO rule of queue-discipline is, as we have previously noted, not achievable; in getting aboard buses in Rome during the rush-hour, in trying to buy goods in a store during a sale, and so on.

25) s) By differentiating repeatedly n times with respect to s, dividing by n! and putting s = o, we obtain Po = I - P Pn = {I - p) n P1 = {I - p)(eP - I) 2 (-I)n-kekP k=1 [ (kp)n-k (n - k)! + (kp)n-k-1 ] (n - k - I)! 26) This is the complete specification of the steady-state solution. It follows at once from the probability generating function on differentiation with respect to sand putting s = I, that the mean number of customers in the system is : I p n=I+--- 2 I- p 39 MODELS OF SINGLE-CHANNEL PROCESSES Hence the mean queueing-time and mean waiting-time can be easily deduced.

Only one customer at a time may be receiving service in the 2. service-channel. Every customer must pass through every phase of service, beginning with the first. APPLIED QUEUEING THEORY 3· On completion of one phase, a customer moves immediately into the next. 4· When a customer leaves the k-th phase, the next customer may enter the l-st, but not before. The reader will observe that this is not the same thing ask serial service-channels, because only one customer at a time may be receiving service in the present case, whereas up to k customers could be involved in some aspect of the service-mechanism in the other.