91Ó°ÊÓ

Show that \(\lim _{t \rightarrow \infty} V(t) / t=\sigma^{2} / \mu^{3}\), where \(V(t)\) is the variance of a renewal process \(N(t)\) and \(\mu\) and \(\sigma^{2}<\infty\) are the mean and variance, respectively, of the interarrival distribution.

Consider a renewal process with underlying distribution function \(F(x)\). Let \(\mathbb{W}\) be the time when the interval duration from the preceding renewal event first exceeds \(\xi>0\) (a fixed constant). Determine an integral equation satisfied by $$ V(t)=\operatorname{Pr}\\{W \leq t\\} $$ Calculate \(E[W] .\) (Assume an event occurs at time \(t=0 .\) )

Consider a renewal process \(N(t)\) with associated distribution function \(F(x)\). Define \(m_{k}(t)=E\left[N(t)^{h}\right]\). Show that \(m_{k}(t)\) satisfies the renewal equation $$ m_{k}(t)=z_{k}(t)+\int_{0}^{t} m_{k}(t-\tau) d F(\tau), \quad k=1,2, \ldots $$ where $$ z_{k}(t)=\int_{0}^{t} \sum_{j=0}^{k-1}\left(\begin{array}{l} k \\ j \end{array}\right) m_{j}(t-\tau) d F(\tau) $$ Ilint: Use the renewal argument.