Transient analysis of interval availability for repairable systems modelled by finite semi-Markov processes

Let the semi-Markov process Y = (Y_t: t [0, )) be the modelof a repairable system, and let its finite state space 5 bepartitioned into the set of'up' states 11 and the set of 'down'states D; i.e.S = U D. For an interval [t₁,t₂] [0, x), theinterval availability A(t₁,₂) is defined as the fraction oftime spent by the system in U. In this paper, a system of integralequations is established for the interval availability. To demonstratethe practical utility of the integral equations, we computethe cumulative distribution function of the interval availabilityfor the semi-Markov model of a two-unit system with sequentialpreventive maintenance. The specific method devised for thenumerical solution of the resulting system of integral equationsis based on the two-point trapezoidal rule. The system of implementationis the matrix-computation package MATLAB on the Apple MacintoshQuadra 610. The numerical results are compared with those fromsimulation.     

Some new aspects of the transient analysis of discrete-parameter Markov models with an application to the evaluation of repair events in power transmission

In Markov reliability modelling, a partitioned state space isused to describe the behaviour of a system each state of whichis associated with the system either being functional or underrepair. Such a system alternates between working and repairperiods indefinitely. Recent research results on the distributionof the sequences of the lengths of working and repair periodsafford the reliability analyst a set of system characteristicswhich can be used in addition to the traditional ones (reliability,point availability, etc.) to describe the system‘s transientbehaviour. In this paper, we present a concise derivation ofclosed-form expressions for the probability mass function andthe factorial moments of the total cumulative ’time‘spent in a subset of the state space by an irreducible or absorbingdiscrete-parameter Markov chain during the first n time instances.This result is then applied to analyse the sequence of repairevents categorized as ’minor‘ and ’major‘of a Markov model of a power transmission system. The numericalimplementation using the Macintosh version of MatLab is alsodiscussed.