Exact formulas for the moments of the first passage time of reward processes

Abstract

Let {Zρ(t), t ≥ 0} be a reward process based on a semi-Markov process {J (t), t ≥ 0} and a reward function ρ. Let Tz be the first passage time of {Zρ(t), t ≥ 0} from Zρ(0) = 0 to a prespecified level z. In this article we provide the Laplace transform of the E[T k z ] and obtain the exact formulas for ETz, ET 2 z and var(Tz). Formulas for certain type I counter models are given.