# New PDF release: Abstract inference

By Grenander U.

ISBN-10: 0471082678

ISBN-13: 9780471082675

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Example text

Etc. This gives us in principle the properties of the system at the particular sequence of time instants under consideration. While this does not answer all questions that might be asked about the system, it is enough for many purposes. As we would expect from our treatment of the continuous-time processes in earlier sections, if the traffic intensity, p =@I < 1, a stationary probability distribution {1Tk} exists, that is e" lim prob(q" = k) = 1Tk' ,,~ao The probability 1Tk also has the interpretation that it approximates the proportion of a large number of successive customers who leave behind them a queue of size k.

For the original system agree reasonably with observation. 30 Queues Quite often a combination of simulation and mathematical analysis is effective, the simulation being used,for example, to check on simplifying assumptions made in the mathematical analysis. (v) Formulation o/practical recommendation This will involve choosing from among the practicable modifications examined in (iv) the one that satisfies some optimality criterion. The criterion may either be expressed solely in terms of cost, or may be that a cost is to be minimized subject to some such condition as that the probability that a customer's waiting-time exceeds, say wo, is not to exceed, say (lo.

Suppose then that at the initial moment t =0 there are qo customers in the queue. • qo from the head of the queue, and suppose that service is just about to commence for customer number 1. As further customers arrive and join the queue, number them qo + 1, qo + 2, ... and so on, in an obvious manner. We shall then write qn for the number of people in the queue at the moment the service of customer number n terminates, and ~ n for the number of customers that arrive during the servicetime of customer number n.