Approximate Formula of Delay-Time Variance in Renewal-Input General-Service-Time Single-Server Queueing System

Approximate formulas of the variance of the waiting-time (also called as delay-time variance) in a renewal-input general-service-time single-server (GI/GI/1) system play an important role in practical applications of the queueing theory. However, there exists almost no literature on the approximate formulas of the delay-time variance in the GI/GI/1 system. The goal of this paper is to present an approximate formula for the delay-time variance. Our approach is based on the combination of a higher-moment relationship between the unfinished work and the waiting time, and the diffusion process approximation for the unfinished work. To derive the former relationship, we apply Miyazawa’s rate conservation law for the stationary point process. Our approximate formula is shown to converge to the exact result for the Poisson-input system as traffic intensity goes to the unity. The accuracy of our approximation is validated by simulation results.