= Solution
An <M G 1 queue> has Poisson arrivals of rate $\lambda$, independent identically distributed service times with a general law of mean $m$, and one server. During the first service time $S$, a mean $\lambda\mathbb ES=\lambda m$ customers arrive, and each initiates an independent descendant busy-period contribution. Thus
$$
\mathbb EB=m+\lambda m\,\mathbb EB,
$$
so for traffic intensity $\rho=\lambda m<1$,
$$
\mathbb EB=\frac{m}{1-\rho}.
$$
For $\rho\geq1$ the expected busy period is infinite.
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