Busy period ($B$)

A busy period is the interval from an arrival that finds the system empty until the next departure leaving it empty. It excludes the preceding idle waiting time. In a stable M/G/1 queue with arrival rate $\lambda$ and mean service time $m$, its mean is $m/(1-\lambda m)$.

In an M/G/1 queue, arrivals during the first service generate independent subtrees of work. Conditional on service time $S=s$, the number of descendants is Poisson with mean $\lambda s$. Thus $B\stackrel{d}{=}S+{\sum }_{j=1}^{N(S)}{B}_j$, with independent descendant busy periods. Conditioning and the Poisson generating function give the displayed transform equation wherever its moment-generating functions are finite. Taking expectations yields $\mathbb{E}B=\mathbb{E}S+\lambda \mathbb{E}S\mathbb{E}B$ in the stable regime.