= Busy-period branching equation
{title2=$\phi_B(\theta)=\phi_S(\theta+\lambda(\phi_B(\theta)-1))$}
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 $B\overset d=S+\sum_{j=1}^{N(S)}B_j$, 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 $\mathbb EB=\mathbb ES+\lambda\mathbb ES\,\mathbb EB$ in the stable regime.
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