= Solution
The service-period arrival count has <probability generating function>
$$
\boxed{g(s)=\sum_{j\ge0}a_js^j=\int e^{-\lambda(1-s)t}\,dG(t),\qquad |s|\le1,}
$$
with $g(1)=1$ and $g'(1)=\rho$. Let $\Pi(s)=\sum_{j\ge0}\pi_js^j$. In stationarity the recurrence and independence give
$$
\Pi(s)=g(s)\left[\pi_0+\frac{\Pi(s)-\pi_0}{s}\right],
$$
initially for $0<s<1$. Rearrangement gives $\Pi(s)=\pi_0(s-1)g(s)/(s-g(s))$. Letting $s\uparrow1$ and using $\Pi(1)=1$ shows $\pi_0=1-\rho$. Hence
$$
\boxed{\Pi(s)=\frac{(1-\rho)(s-1)g(s)}{s-g(s)}.}
$$
Analytic continuation gives the identity inside the unit disc; boundary values follow by convergence of the probability series, with removable limiting values wherever the displayed quotient is indeterminate, particularly at $s=1$.
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