Characterizing The Idle Time Distribution in A Nonexponential Server System

Abstract

Understanding the behavior of an idle time of a limited resource is the key to increase productivity in service operations. When the system consists of nonexponential properties of time distributions it becomes difficult to provide results for the general case. We derive the MacLaurin series for the moments of the idle time with respect to the parameters in the service time and interarrival time distributions for a G I/G/1 queue. The light traffic derivatives are obtained to investigate the quality of a well-known MacLaurin series. The expected error bound under this approach is identified. The coefficients in these series are expressed in terms of the derivatives of the interarrival time density function evaluated at zero and the moments of the service time distribution, which can be easily calculated through a simple recursive procedure. The result for the idle period is easily taken as input to the calculation of other performance measures of the system, e.g., cycle time or interdeparture time distributions. Numerical examples are given to illustrate these results.