A model for a continuous state system with (s,S) repair policy

Abstract

A model for a system whose state changes continuously with time is introduced. It is assumed that the system is modeled by a Brownian motion with negative drift and an absorbing barrier at the origin. A repairman arrives according to a Poisson process and repairs the system according to an $(s,S)$ policy, i.e., he increases the state of the system up to $S$ if and only if the state is below $s$. A partial differential equation is derived for the distribution function of $X(t)$, the state of the system at time $t$, and the Laplace-Stieltjes transform of the distribution function is obtained by solving the partial differential equation. For the stationary case the explicit expression is deduced for the distribution function of the stationary state of the system