Gamma-approximation for the waiting time distribution function of the M/G/1-∞ queue

In this paper we propose to approximate the waiting time distribution function (DF) of the waiting customers in a M/G/I-infinity queuing system by a Gamma-distribution whose parameters a and P are set by means of the first and second moment of the waiting time that are obtained from the Takaes recursion formula. Discrete-time analysis (DTA) is another approximation approach for the same objective. We show first for analytically feasible special cases that DTA is very accurate and then we use it to validate the accuracy of the new Gamma-approximation for wide parameter ranges regarding the service time distribution. We show that the Gamma-approximation respects well even the third moment of the service time distribution. As the new approach is very simple and fast, it may be used by engineers with only little background in queuing theory to calculate quantiles; for real-time control loops in technical systems.