Probability Functions for Waiting Times in Single-Channel Queues, with Emphasis on Simple Approximations

One approach to the waiting-time problem is through the integral equation, whose solution is the cumulative probability function—the method derived by Lindley. The equation is of the type for which a general solution was developed by Wiener and Hopf, and an adaptation of their method is presented in this paper. A quite wide range of arrival and service-time distributions can be accommodated, and approximations for the tail of the density curve are readily found. This is a convenient property, because the useful information about waiting times is fully provided by the mean value and the tail of the curve. The approximation process occurs within the solution, giving a reduction in the total mathematical labor compared with the forming of an approximation from an exact final formula.