Finite Waiting Line Models

Many practical waiting line problems that occur have the characteristics of finite waiting line models. This is true whenever the population of machines, people or items that may arrive for service is limited to a relatively small, finite number. The result is that we must express arrivals in terms of a unit of the population rather than as an average arte. In the infinite waiting line case, the average length of the waiting line is effectively independent of the number in the arriving population, but in the finite case, the number in the waiting line may represent a significant proportion of the arriving population and therefore the probabilities associated with arrivals are affected. For example, suppose there are 10 machines being serviced by a mechanic and 1 breaks down (arrives for service) .Then , there are only 9 that could possibly break down because 1 has been eliminated from the arriving population until it is serviced.

The resulting mathematical formulations are somewhat more difficult computationally than those for the infinite waiting line case. Fortunately, however, finite queuing tables (Peck and Hazelwood, 1958) are available that make the problem solution very simple. Although there is no- definite number that we can point to as a dividing line between finite and infinite applications, the finite queuing tables have data for populations from 4 up to 250, and these data may be taken as a general guide. We have reproduced these tables for population of 5, 10, 20 and 30 in Table of the Appendix to illustrate their use in the solution of finite waiting line problems. The tables are based on a finite model for negative exponential times between arrivals and negative exponential service times and a first come first served queue discipline.

Use of the Finite Queuing Tables

The tables are indexed first by N, the size of the population. For each population size, data are classified by X, the service factor (comparable to the utilization factor in infinite waiting line models),and by M, the number of parallel servers. The service factor X is compared from the following formula.

X = Service factor = ג / ג + µ

Where µ is the service rate as before, but ג is the mean arrival arte per population unit. For example, if our time unit is hours and each population unit arrives for service every 4 hours on the average, then ג = 1 / 4 = 0.25 per hour.

For a given N, X and M three factors are listed in the tables: D (the probability of a delay; that if a unit calls for services, the probability that it will have to wait), F (an efficiency factor, used to calculate other important data),and L q (the mean number in the waiting line) To summarize, we define the following factors:

W q =mean waiting time = 1 / µ X ( 1 – F / F )

L = Mean number waiting and being served = L q + FNX

W = Mean time in system = W q + 1 / µ

H = mean number of units being serviced FNX = L – ,L q

J = Mean number of units not being served = FN ( 1 – X)

M – H =Mean number of servers idle.

The procedures for a given case is as follows:

1) Determine the mean service rate µ ,and the mean arrival rate ג per population unit, based on data or measurements of the system being analyzed.
2) Compute the service factor X = ג / ( ג + µ )
3) Locate the section of the tables listing data for the population size N.
4) Locate the service factor calculated in 2 for the given population.
5) Read the values of D, F, and L q for the number of servers M, interpolating between values of X when necessary.
6) Compute the values for W q , H and J as required by the nature of the problem