Moments and Blackwell convergence for repairable systems with heavy tailed lifetimes
The authors consider a marked point process \(((T_n, V_n, Z_n): n\in\mathbb{N})\) describing an unreliable system by failure/repair-process. \(T_n\) are the occurrence times of the failures with immediate repair, \(1-Z_n\) is the grade of repair which determines the decrease of the actual virtual age of the system just before the failure to the virtual age \(V_n\) just after the repair. The interpoint distances \(T_{n+1}- T_n\) are distributed according to the residual life time distribution of a common life time distribution \(F\) given the virtual age \(V_n\). The associated triple process \(((T_t, V_t, Z_t): n\in \mathbb{N})\) in continuous time is assumed to be Markovian and the convergence to stationarity is studied with special emphasis on the case of \(F\) having heavy tails. Different modes of convergence are studied, e.g. implying total variation convergence and convergence of \(E(V^\beta_t)\) depending on the existence of suitable moments of \(F\). Several results on the speed of convergence are proved.
scientific article; zbMATH DE number 1679855