Moments and Blackwell convergence for repairable systems with heavy tailed lifetimes
type: Item
instance of: scholarly article
zbMATH Open document ID: 0995.60083
author: Ryszard Szekli
author: Günter Last
publication date: 2002-10-23T00:00:00Z
Mathematics Subject Classification ID: 60K10
Mathematics Subject Classification ID: 62N05
Mathematics Subject Classification ID: 60G55
zbMATH DE Number: 1679855
zbMATH Keywords: marked point process
zbMATH Keywords: imperfect repair
zbMATH Keywords: virtual age
zbMATH Keywords: Markov process
zbMATH Keywords: Lyapunov function
MaRDI profile type: Publication
title: Moments and Blackwell convergence for repairable systems with heavy tailed lifetimes
published in: Markov Processes and Related Fields
review text: 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.
reviewed by: Hans Daduna
description: scientific article; zbMATH DE number 1679855