Nt1310 Unit 1 Application Paper

Assume also $S_{n}=\sum_{i=1}^{n}\zeta_{i}$ and let $\Psi(\tau)=Sup\{n:S_{n}\leq \tau\}$.\\ Therefore $\Psi(\tau)$ is considered as the complete collection of cycles computed in the given interval of $[0,\tau].$ $\Phi(\tau)=E[\Psi(\tau)]$ which is known as renewal function connected with variables $\{\zeta_{n}\},$ which has been studied in [\cite{Hellen2016nonparametric, frees1986warranty, harel1995asymptotics}] \\
It is known that the average of instant system availability $\Lambda_{a}(t)$ is considered as portion of time in assigned interval $(0,\tau]$ such that the system component is available for being used as discussed in \cite{hagenimana2016computation}. It characterizes the mean value of the instant system availability function throughout a period of time $(0,\tau]$ where it is illustrated as the following expression,

For $\tau$, we hold $S_{\Psi(\tau)}\leq \tau\leq S_{\Psi(\tau)+1},$ then finally $\Phi(\tau)\nu_{\zeta}\leq \tau\leq (\Phi(\tau)+1)\nu_{\zeta}.$ Suppose that, the system operation at any given time $\tau=0$, $\overline{\beta}(\tau),$ the average availability when the system is up, is defined in the respected interval $[0,\tau]$ which can be expressed as
\begin{equation*}