An importance measure for multi-component systems with Semi- Markov dynamics

We consider semi-Markov reliability models of multi-component systems with a discrete state space, general enough to include systems with maintenance or repair. We assume that for all system states the functioning or failure of each component is specified. In this setup we propose a component importance measure which is close in spirit to the classical steady state Barlow–Proschan importance measure for repairable binary coherent systems. We discuss our importance measure to some extent, highlighting the relation to the classical Barlow– Proschan measure, and present formulas expressing it in terms of quantities easily obtained from the building blocks of the semi-Markov process. Finally an example of a two-component cold standby system with maintenance and repair is presented which illustrates how our importance measure can be used in practical applications. Hellmich Mario, Berg Heinz-Peter An importance measure for multi-component systems with Semi-Markov dynamics 148 which has recently been generalized to include systems with repairable and multistate components in [12] and [13]. However, in the state space approach the problem of constructing and studying importance measures remains largely open. In [16] importance measures for systems with Markov dynamics were studied and explicitly calculated using a perturbation approach, but a systematic study of importance measures is lacking. In the present paper it is our purpose to report on a component importance measure introduced in [8] for multi-component systems with maintenance and/or component repair, whose time evolution is given by a semi-Markov process. It is close in spirit to the steady state Barlow–Proschan measure for repairable binary coherent systems. We present results which express the importance measure in terms of quantities easily calculated numerically from the defining quantities of the semiMarkov process, such as transition rates, mean recurrence times, etc., for both the time dependent as well as the steady state case. Moreover, we provide a discussion and an interpretation of the importance measure, and we explain its relation to the classical steady state Barlow–Proschan importance measure for binary coherent systems with component repair. The underlying mathematical setup which is employed can be briefly described as follows: We consider a system with n components and a finite state space E . The state space is assumed to be partitioned in two disjoint subsets, corresponding to the states of system functioning or failure. Furthermore, we assume that for each system state and each component it is specified whether the component is up (functioning) or down (failed, under repair, on standby, undergoing maintenance, etc.). As already mentioned above, the time evolution of the system is assumed to be given by a homogeneous semi-Markov process with values in E . Thus, the states successively visited form a Markov chain, and the sojourn time in each state is random, following a distribution which depends on the present state and the state to be visited next. 2. Assumptions and notation This section introduces our mathematical setup and notation used throughout the paper. 2.1. Multi-component systems We consider a repairable or maintained system with state space } ,..., 1 { E d = . The system is assumed to consist of n components, and we denote the set of components by } ,..., 1 { n = C . We suppose that in each system state E i ∈ a component C ∈ a can either be up or down as determined by the maps } 1 , 0 { : → E ca , with 1 ) ( = i ca when a is up in i and 0 ) ( = i ca when it is down. Moreover, we suppose that the state space is partitioned in two disjoint subsets U and D , i.e. D U E ∪ = , where U contains all states in which the system is up and D those in which it is down. An example of a system which fits in this framework is given in Section 4. 2.2. Semi-Markov processes The time evolution of the system under consideration is assumed to be given by a homogeneous semiMarkov process 0 )} ( { ≥ = t t Z Z with values in E , defined on some underlying complete probability space ) , , ( P F Ω . The corresponding Markov renewal process (MRP) is denoted by N ∈ n n n S J )} , {( , which is a sequence of random variables in + × R E such that ... 0 1 0 ≤ ≤ = S S , and such that the following Markov property } ,..., , ,..., | , { 0 0 1 1 n n n n S S J J t S j J ≤ = + + P = ) ( } | , { 1 1 n j n J n n n S t Q J t S j J − = ≤ = + + P is satisfied, where ) (t Qij denotes the semi-Markov kernel of the MRP. Upon introducing the random variables 0 0 0 = = S X , 1 − − = n n n S S X and conditioning on } { i Jn = the last equation can be written as } | , { ) ( 1 i J t X j J t Q n n n ij = ≤ = = − P , (1) independently of n , i.e., the process is homogeneous. The connection between Z and the corresponding MRP is given by ) ( ) ( t N J t Z = , where the counting