On a Congestion Problem in an Aircraft Factory

Abstract

This is a study of the optimum number of clerks to be placed behind service counters. It involves queuing theory and was solved in two parts: First, we considered the counter from the customers' side, the “obverse” queue. Observational data showed that arrivals at the counter occur at random, the order of serving was random, and the distribution of the service times was exponential, so that Erlang's “C” formula for the probability that a customer has to wait applied. A formula is derived and graphed that shows the optimum number of clerks to be used behind the counter and the over-all minimum cost caused by waiting on the part of the customers and the clerks as a function of the queue input and the ratio of the cost of the customers' time to that of the clerks'. Second, the queue from the clerks' side was considered, that is, from the “reverse” queue. Clerks' time spent waiting for customers to appear is not altogether lost, if they can do work behind the counter. What happens to the clerks is governed by the characteristics of the “obverse” queue. From that is derived the probability distribution of the clerks' waiting time, considering the clerks to form a queue, as customers do. With this we calculate the saving in clerks' time that can be effected subject to the limitations of the shortest free period and the total length of time that can be used productively. This saving can then be subtracted from the cost computed in the first part of the study. The results obtained are applicable to any queuing operation which has the same characteristics. Operations Research, ISSN 0030-364X, was published as Journal of the Operations Research Society of America from 1952 to 1955 under ISSN 0096-3984.