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Exact formulas for the moments of the first passage time of reward processes. (English) Zbl 1139.60340

Summary: Let \(\{{\mathcal Z}_\rho(t), t\geq 0\}\) be a reward process based on a semi-Markov process \(\{{\mathcal J}(t), t\geq 0\}\) and a reward function \(\rho\). Let \(T_z\) be the first passage time of \(\{{\mathcal Z}_p(t), t\geq 0\}\) from \({\mathcal Z}_p(0)= 0\) 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 \(ET_z\), \(ET^2_z\) and \(\text{var}(T_z)\). Formulas for certain type I counter models are given.

MSC:

60K15 Markov renewal processes, semi-Markov processes
60E10 Characteristic functions; other transforms
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