# Queues, Theory of

## Queues, Theory of

a branch of queuing theory. The theory of queues studies systems in which customers who find a system to be busy do not go away but wait for the system to be free and are then served in some order—often with priority being granted to certain categories of customers. The results of the theory of queues are used for the efficient planning of mass servicing systems. From the mathematical point of view, the problems of the theory of queues can be included in the theory of stochastic processes; the answers are often expressed in terms of Laplace transforms of the unknown characteristics. The methods of the theory of queues must be used even in very simple cases in order to understand correctly the statistical regularities that occur in mass servicing systems.

** Example.** Let us assume there is a service facility that receives a random flow of customers. If the facility is free when a customer arrives, it immediately begins to serve him. Otherwise, the customer takes a place in line, and the facility serves the customers one after the other in the order of their arrival. Let us also assume that

*a*< 1, where

*a*is the expected number of customers who arrive in the time needed to serve one customer. We shall denote the length of a busy period—that is, the time interval from the moment when the facility has begun to serve a customer who has found it free to the first moment when the facility is again completely free—by

*T.*The theory of queues shows that, if natural assumptions are made, the mathematical expectation of

*T*is equal to m = 1/(1 —

*a*) and the variance is equal to (1 +

*a)m*For example, when

^{3}.*a*= 0.8, the corresponding values of

*m*and the variance are equal to 5 and 225. Thus, for a “well loaded” service facility—that is, for

*a*close to 1—the average value

*m*of the random variable

*T*is an extremely unreliable measure of

*T.*

### REFERENCES

Gnedenko, B. V., and I. N. Kovalenko.*Vvedenie ν teoriiu massowgo obsluzhivaniia.*Moscow, 1966.

*Prioritetnye sistemy obsluzhivaniia.*Moscow, 1973.

IU. V. PROKHOROV