By Dr. Gordon F. Newell (auth.)

For many stochastic carrier structures, provider capacities sufficiently big to serve a few given shopper call for is accomplished just by offering a number of servers of low potential; for instance, toll plazas have many toll creditors, banks have many t- lers, bus traces have many buses, and so on. If queueing exists and the common queue dimension is big in comparison with the quantity n of servers, all servers are stored busy as a rule and the carrier behaves like a few "effective" unmarried server wit:l suggest se.- vice time lin instances that of a precise server. The habit of the queueing method could be defined, a minimum of nearly, through use of recognized effects from the a lot studied single-channel queueing method. For n» 1 , even though, (we are considering p- ticularlyof circumstances during which n ~ 10), the procedure should be relatively congested and relatively delicate to adaptations widespread even if the typical queue is small in comparison with n. The habit of one of these procedure will, as a rule, range fairly considerably from any "equivalent" single-server procedure. the next research offers with what, within the usual class of queueing structures, is named the G/G/n method; n servers in parallel with self sustaining s- vice occasions serving a reasonably basic kind of patron arrival strategy. rhe arrival cost of consumers will be time-dependent; specific cognizance is given to time - pendence standard of a "rush hour" within which the arriving expense has a unmarried greatest most likely exceeding the means of the service.

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This smoothing of the step function will cause a larger variance than for a S o on the early side of the step, but a smaller variance than for aS = 0 on the late side of the step. The latter effect is again analogous to the competition -52between fluctuations in servers and arrivals that exist for I > 1. transition, when queueing starts, introduces a large variance in N(t) The first over a short time, much as would be created by a batch of customers arriving at a predictable time but with a random batch size.

4 shows a discontinuity in the slope of from Ac(t) to As(t). D(t) at to or D(t) after N(t) , will depend tl - to in Fig. 4. where it switches Since all customers have the same service time, the rate at which customers enter the service at time t , when there is a queue, is the same as the rate at which servers become free, which, in turn, is the rate at which customers entered the service at time will be repeated a time s t - s Any discontinuity in the slope of later, if there is still a queue.

E{N(t) + n} the relative fluctuation in If is of order E{N(t)} + n aN/n is ~ (I/n)l/~ which we interpret as being small compared with 1. To evaluate fluctuations in E{N(t)} + n , it is not necessary, however, to subtract away the Ac(t - s) by choosing this as a new origin for counting. 5) by the same graphical in Fig. 4 from any realization Ac(t) , re- gardless of the magnitude of the fluctuations, as long as they do not cause a queue of customers. The above is not completely trivial. time of a server from the curve an expectation of kth A (t) c E{A (t)} for fixed server to enter service c t • One cannot determine the expected idle because the notation implies that this is If one wants the expected idle time of the Ws(k) , one must take expectations for fixed k -27Under the same conditions as above By analogous arguments, The curve for E{W (k)} s could be obtained from a graph of E{A-1(k)} is not the same as the inverse of the curve for c E{A (t)}, c although in most practical applications, the difference between these should not be enough to make much difference.

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