M/M/∞ busy period and busy cycle distribution function bounds

Abstract

The busy period length distribution function knowledge is important for any queue system, and in particular for the M/M/∞ queue. But the mathematical expressions are in general very complicated, with a few exceptions, involving usually infinite sums and multiple convolutions. So, in this work are deduced some bounds for the M/M/∞ system busy period length distribution function, meaning the second M exponential service time, which analytic expressions are simpler than the exact one. As a consequence, also some bounds for the M/M/∞ system busy cycle length distribution function are presented