On the exact distribution and mean value function of a geometric process with exponential interarrival times (Q2438513)

Let \(X,X_{1},X_{2},\dots\) denote i.i.d.\ random variables and suppose that \(X\) has an exponential distribution with mean \(\theta \). Let \(0<a<1\) and for \( n\geq 1\), let \(S_{n}=X_{1}+a^{-1}X_{2}+a^{-2}X_{3}+\dots+a^{1-n}X_{n}\). A geometric renewal counting process is defined as \(N(t)=\sup \left\{ n:S_{n}\leq t\right\} \). The corresponding renewal function is defined as \( M(t)=\operatorname{E}N(t)\). In this paper, the authors find analytic expressions for \(\operatorname{P}(N(t)=k)\) and \(M(t)\).