UBC Theses and Dissertations
Steady state single channel queues Low, Siew Nghee
This thesis extends the application of waiting line theory to situations where both arrival rate and service rate distributions are arbitrary or non-random. It does so only for single channel, single phase, steady state, infinite queues with no feed-back. Previous work by A.K. Erlang had shown that queuing characteristics could be predicted for one case of an arbitrary service rate distribution, the constant service time. Also, F. Pollaczek had shown that, where arrival rates are random, queue lengths and waiting times were independent of the form of the service rate distribution, being functions of the coefficient of variance squared. But all of the works assumed random arrivals around a stable mean arrival rate and, except for the constant service time case, most applications were limited to cases where both arrival and service rates were random. This restriction has limited applications severely and has required that most analysis of queuing characteristics be done by simulation. This study develops and proves by inference the hypothesis that system length is dependent on these factors only: the square of the coefficient of variance of the interarrival time distribution, C²a, the square of the coefficient of variance of the service time distribution, C²s, and the ratio of mean arrival rate to mean service rate, p. Through a combination of calculation and simulation a set of curves has been developed covering values, C²a from 0 to 6, C²s from 0 to 6 and of p from 0.1 to 0.9. These curves permit the prediction of system length, and then of average queue length and waiting time, for any case where only the mean and variance of the arrival and service time distributions are known, even though nothing is known about the form of the distributions. In the usage of the set of graphs (figures 10-29), the following steps are all that is required to obtain the necessary characteristics: a) Calculate the average interarrival time, [formula omitted] (Total time of observation/Total number of Arrivals). b) Calculate the variance for interarrival times, [formula omitted] Total number of arrivals. c) Calculate the fractional coefficient of variance squared for interarrival time distribution, [formula omitted]. d) Calculate the average service time, [formula omitted] (Total time service facility is in operation/ Total number serviced). e) Calculate the variance for service times, [formula omitted] Total number serviced. f) Calculate the fractional coefficient of variance squared for service time distribution, [formula omitted]. g) Calculate the utilization factor, p = (Average service time/Average interarrival time). h) With the values p, C²a, and C²s, read from the set of graphs (figures 10-29) the verticle axis, L. i)Compute Lq, W and Wq.
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