Distribution and Moments of the Idle Period and Interarrival Time in the G/M/1 Queueing System

The Idle period is relevant in a wide variety of systems and has been extensively studied in the literature. This paper derives the distribution and moments of the idle period for different interrarrival time distributions in the G/M/1 queueing system. Specifically, Log-Normal (LN), Weibull, arbitrary-order Hyper-Exponential (HE), arbitrary order Erlang, and Deterministic distribution are considered. It is demonstrated that when the interarrival time follows HE (Erlang) distribution, the idle period follows a corresponding HE (Coxian) distribution. The effects of the distribution and coefficient of variation of the interarrival time, and the traffic load on the distribution and first four standardized moments of the idle period are numerically evaluated. Additionally, the accuracy of the derived distribution and moments of the idle period when the LN interarrival time is approximated by HE distributions of different orders (using the Expectation-Maximization algorithm) is investigated. Numerical results show a good fit accuracy between idle period distributions obtained under the LN and the m-th order HE interarrival time models. It is observed that the fitting accuracy (in terms of the Kolmogorov-Smirnov distance) improves as the order of the HE distribution increases. Finally, the standardized moments of the idle period are compared when the interarrival time follows either LN or Weibull distributions with identical first two moments. Numerical results indicate that the values of the standardized moments of the idle period are higher when the interarrival time follows a LN distribution, due to its heavier tail compared to the Weibull distribution.