Past exam of the mathematics course of the University of Cambridge / 2011 / ii / Paper 4 / 26J / iii / Solution

The service-period arrival count has probability generating functionwith $g(1)=1$ and $g^{\prime }(1)=\rho$. Let $\Pi (s)={\sum }_{j\ge 0}{\pi }_j{s}^j$. In stationarity the recurrence and independence giveinitially for $0<s<1$. Rearrangement gives $\Pi (s)={\pi }_0(s-1)g(s)/(s-g(s))$. Letting $s\uparrow 1$ and using $\Pi (1)=1$ shows ${\pi }_0=1-\rho$. HenceAnalytic 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 $s=1$.