Ourania Chryssaphinou

Multi-State Reliability Systems Under Discrete Time Semi-Markovian Hypothesis

We consider repairable Multi-state reliability systems with components, the lifetimes and the repair times of which are -independent. The -th component can be either in the complete failure state 0, in the perfect state , or in one of the degradation states . The sojourn time in any of these states is a random variable following a discrete distribution. Thus, the time behavior of each component is described by a discrete-time semi-Markov chain, and the time behavior of the whole system is described by the vector of paired processes of the semi-Markov chain and the corresponding backward recurrence time process. Using recently obtained results concerning the discrete-time semi-Markov chains, we derive basic reliability measures. Finally, we present some numerical results of our proposed approach in specific reliability systems, namely series, parallel, k-out-of-n:F, and consecutive-k-out-of-n:F systems.

On Discrete Time Semi-Markov Chains and Applications in Words Occurrences

Let a discrete time semi-Markov process {Z γ;γ ∈ ℕ} with finite state space an alphabet Ω. Defining the process {U γ; γ ∈ ℕ} to be the backward recurrence time of the process {Z γ; γ ∈ ℕ}, we study the Markov process {(Z γ, U γ); γ ∈ ℕ}. We give its transition probabilities of first and higher order, the limiting distribution, and the stationary distribution. Using this Markov process we construct a k-dimensional process and we study its basic properties. As an application we consider a finite set of words W = {w 1, w 2,…, w ν} of equal length k which are produced under the semi-Markovian hypothesis and we focus on the waiting time for the first word occurrence from the set W. The corresponding probability distribution, the generating function, as well as the mean waiting time and variance are obtained.

Compound Poisson approximation in reliability theory

The use of the compound Poisson local formulation of Steins method to obtain reliability bounds is discussed. In the context of the consecutive-k-out-of-n model, the approximations obtained are comparable with the best known for the system, which rely heavily on its special structure. In contrast, Steins method for the compound Poisson approach can be applied equally effectively in many more-complicated models. >

Compound poisson approximation in systems reliability

The compound Poisson “local” formulation of the Stein-Chen method is applied to problems in reliability theory. Bounds for the accuracy of the approximation of the reliability by an appropriate compound Poisson distribution are derived under fairly general conditions, and are applied to consecutive-2 and connected-s systems, and the 2-dimensional consecutive-k-out-ofn system, together with a pipeline model. The approximations are usually better than the Poisson “local” approach would give.

Poisson Approximation for the Non-Overlapping Appearances of Several Words in Markov Chains

Let X1, …, Xn be a sequence of r.v.s produced by a stationary Markov chain with state space an alphabet Ω = {ω1, …, ωq}, q ≥ 2. We consider a set of words {A1, …, Ar}, r ≥ 2, with letters from the alphabet Ω. We allow the words to have self-overlaps as well as overlaps between them. Let E denote the event of the appearance of a word from the set {A1, …, Ar} at a given position. Moreover, define by N the number of non-overlapping (competing renewal) appearances of E in the sequence X1, …, Xn. We derive a bound on the total variation distance between the distribution of N and a Poisson distribution with parameter EN. The Stein–Chen method and combinatorial arguments concerning the structure of words are employed. As a corollary, we obtain an analogous result for the i.i.d. case. Furthermore, we prove that, under quite general conditions, the r.v. N converges in distribution to a Poisson r.v. A numerical example is presented to illustrate the performance of the bound in the Markov case.

Compound Poisson Approximation for Multiple Runs in a Markov Chain

We consider a sequence X1, ..., Xn of r.v.s generated by a stationary Markov chain with state space A = {0, 1, ..., r}, r ≥ 1. We study the overlapping appearances of runs of ki consecutive is, for all i = 1, ..., r, in the sequence X1,..., Xn. We prove that the number of overlapping appearances of the above multiple runs can be approximated by a Compound Poisson r.v. with compounding distribution a mixture of geometric distributions. As an application of the previous result, we introduce a specific Multiple-failure mode reliability system with Markov dependent components, and provide lower and upper bounds for the reliability of the system.

On the Number of Appearances of a Word in a Sequence of I.I.D. Trials

Let X1,...,Xn be a sequence of i.i.d. random variables taking values in an alphabet Ω=ω1,...,ωq,q ≥ 2, with probabilities P(Xa=ωi)=pi,a=1,...,n,i=1,...,q. We consider a fixed h-letter word W=w1...wh which is produced under the above scheme. We define by R(W) the number of appearances of W as Renewal (which is equal with the maximum number of non-overlapping appearances) and by N(W) the number of total appearances of W (overlapping ones) in the sequence Xa1≤a1≤n under the i.i.d. hypothesis. We derive a bound on the total variation distance between the distribution ℒ(R(W)) of the r.v. R(W) and that of a Poisson with parameter E(R(W)). We use the Stein-Chen method and related results from Barbour et al. (1992), as well as, combinatorial results from Schbath (1995b) concerning the periodic structure of the word W. Analogous results are obtained for the total variation distance between the distribution of the r.v. N(W) and that of an appropriate Compound Poisson r.v. Related limit theorems are obtained and via numerical computations our bounds are presented in tables.

On a Conjecture by Eriksson Concerning Overlap in Strings

Consider a finite alphabet Ω and strings consisting of elements from Ω. For a given string w, let cor(w) denote the autocorrelation, which can be seen as a measure of the amount of overlap in w. Furthermore, let aw(n) be the number of strings of length n that do not contain w as a substring. Eriksson [4] stated the following conjecture: if cor(w)>cor(w′), then aw(n)>aw′(n) from the first n where equality no longer holds. We prove that this is true if ∣Ω∣≥3, by giving a lower bound for aw(n)−aw′(n).

A generalized multi-standby multi-failure mode system with various repair facilities

Abstract We consider a system of ( m + 1) non-identical units—one functioning and m standby. Each unit of the system has the following states: normal, N types of partial failures and corresponding to them N types of total failures. There are k distinct major repair facilities and one on the spot repairman. One unit can pass from one state to another with known probability and then the time of staying in this state has a general distribution. The system starts to work at t = 0 and fails when the ( m + 1)th unit after a total failure is finally rejected. Using semi-Markov techniques we obtain Laplace transforms of transition probabilities. Considering particular cases we derive known results for systems which have been defined in the past.

A general repairable system with N failure modes and k standby units

Abstract A repairable system with N failure modes and k standby units is studied under quite general conditions. Laplace transforms of the transition probabilities of the system are obtained by using semi-Markov techniques. A particular case is discussed.