This paper describes how studying the transient behaviour of the M|G|∞ system, starting at the beginning of a busy period, it is possible to determine a collection of service distributions for which the length of the busy period is exponential, with an atom at the origin. We determine also the distribution of the busy cycle length for these systems and the mean number of busy periods that begin in [0,t].

keywords: M|G|∞, busy period, busy cycle, Ricatti’s equation, exponential distribution