An Approach to a Class of Queuing Problems

In a variety of situations, the amount of service activity wanted by a queuer is a function of the length of time during which he has had to wait for it. Additional want of service may emerge as a consequence of his own activity. New events may induce him to change the information, which he wants to communicate to somebody else. The latter case may also be regarded as a two-phase queuing problem. This paper considers the queuing problem characterized by Poisson streams of calls for service from the customers, the intensity of the individual rate of calls, and service times, being dependent on the position of the customer. An approximate solution is given for the general case, and special cases are examined with the aid of Monte-Carlo models. The solution covers a range of “machine-interference” problems and provides a pendant, in the range of finite numbers of customers, to the classical Erlangian solution of the infinite population delay system case. The model also accounts for the well-known fact that statistics frequently indicate a longer remaining duration of a telephone conversation, the longer it has been in progress. It remains to study, how the distribution of service times affects the accuracy of the solution.