 Solution

For the busy-period branching equation, during the first service of a busy period, every arriving customer initiates a subtree of further work. The independence and stationary increments of the Poisson process, together with independent services, give the distributional recursionwhere conditional on $S_k=s$, $N(S_k)$ is Poisson with mean $\lambda s$, and the descendant busy periods are independent copies of $B_k$. One may explore these subtrees in order; the server's total workload is unchanged by that ordering. Consequently, wherever the moment-generating functions are finite,For $\lambda <\mu$, this identity in particular holds in a sufficiently small neighbourhood of zero.