Applications Of Semi-markov Processes In Reliability

F.Grabski Applications of semi-Markov processes in reliability - RTA # 3-4, 2007, December - Special Issue Grabski Franciszek Naval University, Gdynia, Poland Applications of semi-Markov processes in reliability Keywords semi-Markov processes, reliability, random failure rate, cold standby system with repair Abstract The basic definitions and theorems from the semi-Markov processes theory are discussed in the paper. The semiMarkov processes theory allows us to construct the models of the reliability systems evolution within the time frame. Applications of semi-Markov processes in reliability are considered. Semi-Markov model of the cold standby system with repair, semi-Markov process as the reliability model of the operation with perturbations and semi-Markov process as a failure rate are presented in the paper. 1. Introduction 2. P{ξ 0 = io , ϑ 0 = 0} = P{ξ 0 = i0 } = pio The semi-Markov processes were introduced independently and almost simultaneously by P. Levy, W.L. Smith, and L.Takacs in 1954-55. The essential developments of semi-Markov processes theory were proposed by Cinlar [3], Koroluk & Turbin [13], Limnios & Oprisan [14]. We would apply only semiMarkov processes with a finite or countable state space. The semi-Markov processes are connected to the Markov renewal processes. The semi-Markov processes theory allows us to construct many models of the reliability systems evolution through the time frame. hold. From the above definition it follows that a Markov renewal chain is a homogeneous two-dimensional Markov chain such that the transition probabilities do not depend on the second component. It is easy to notice that a random sequence {ξ n : n = 0,1,2,...} is a homogeneous one-dimensional Markov chain with the transition probabilities p ij = P{ξ n +! = j | ξ n = i} = lim Qij (t ) . The matrix [ (4) Is called a Markov renewal kernel. Both Markov renewal kernel and the initial distribution define the Markov renewal chain. This fact allows us to construct a semi-Markov process. Let ξ n , ϑn , n = 0,1,2,... are the random variables defined on a joint probabilistic space ( Ω , F, P ) with values on S and R + respectively. A two-dimensional random sequence {(ξ n , ϑ n ), n = 0,1,2,...} is called a Markov renewal chain if for all i0 ,...., i n−1 , i ∈ S , t 0 ,..., t n ∈ R+ , n ∈ N 0 : = P{ξ n +1 = j , ϑ n +1 ≤ t | ξ n = i} = Qij (t ), ] Q(t ) = Qij (t ) : i, j ∈ S , Let S be a discrete (finite or countable) state space and let R+ = [0, ∞ ) , N 0 = {0,1,2,...} . Suppose, that { (3) t →∞ 2. Definition of semi-Markov processes with a discrete state space 1. P ξ n+1 = j, ϑn+1 ≤ t | ξ n = i, ϑn = t n ,...,ξ 0 = i0 , ϑ0 = t 0 (2) τ 0 = ϑ0 = 0, τ n = ϑ1 + ... + ϑ n , τ ∞ = sup{τ n : n ∈ N 0 } } A stochastic process following relation (1) - 60 - {X (t ) : t ≥ 0} given by the F.Grabski Applications of semi-Markov processes in reliability - RTA # 3-4, 2007, December - Special Issue X (t ) = ξ n for t ∈ [τ n , τ n+1 ) It means that the process {X (t ) : t ≥ 0} has th