Stavros Papastavridis

Runs on a circle

We study the number of occurrences of non-overlapping success runs of length k in a sequence of (not necessarily identical) Bernoulli trials arranged on a circle. An exact formula is given for the probability function, along with some sharp bounds which turn out to be very useful in establishing limiting (Poisson convergence) results. Certain applications to statistical run tests and reliability theory are also discussed.

CHAPTER 6 - Consecutive-k-out-of-n Systems

A consecutive-k-out-of-n:F system fails if and only if at least k successive components of the system are simultaneously down. Since its introduction, a decade ago, there has been an exponential growth of the literature on this subject and several modifications, extensions and generalizations have been proposed. Due to the fact that most of them are of special interest in the study of real life engineering systems, it is necessary to take stock of the state-of-art. The present chapter conducts a survey of this rapidly expanding area, focussing mainly on the most recent results.

Consecutive k out of n: F systems with Cycle k

An r consecutive k out of n: F system is a system of n linearly arranged components which fails if r non-overlapping sequences of k components fail. When r=1 we have the classic consecutive k out of n: F system about which there is an extensive literature. In this research we study the situation where there are k distinct components with failure probabilities qi for i=1,...,k and where the failure probability of the jth component (j=mk+i (1[less-than-or-equals, slant]i[less-than-or-equals, slant]k)) is qi. We call such a system an r consecutive k out of n: F system with cycle (or period) k. We obtain exact expressions for the failure probability of an r consecutive k out of n: F system and in particular show that it is independent of the order of the k components if n is a multiple of k. Interesting applications are given for the arrangement of sport competitions and inspection procedures in quality control.

Consecutive k-out-of-n systems with maintenance

A consective k-out-of-n system consists of n linearly or cycliccally ordered components such that the system fails if and only if at least k consecutive components fail. In this paper we consider a maintained system where each component is repaired independently of the others according to an exponential distribution. Assuming general lifetime distributions for systems components we prove a limit theorem for the time to first failure of both linear and circular systems.

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.

A reliability bound for 2-dimensional consecutive k -out-of- n :F systems

The 2-dimensional consecutive-k-out-of-,?:F system was introduced by Salvia & Lasher [I] by generalizing the notion of the consecutive-k -out-of-11 :F system [53]. It consists of a square grid of size n (containing )I2 components) and fails if and only if there is at least one square grid of size k (1 < k I W) whose components arc failed. This system has rcccntly rcccivcd extensive research interest, a fact mainly due to its applicability in various areas such as safety monitoring systems, design of electronic devices, discasc diagnosis and pattern detection. For a coherent system whose minimal cut sets have been specified, the classical lower bound is the one obtained by Esary & Proschan [4] (see also Barlow & Proshan 151). In a 2-dimensional consecutive-k -outof-n:F system the minimal cut sets consist of k2 components placed on a rectangular kxk grid. Therefore, denoting by p, (c/ii = l-p,,) the survival (failure) probability of system’s components we

Bounds for coherent reliability structures

Esary and Proschan (1963) proposed a lower bound (resp. upper bound) for the reliability of a general coherent system, which was expressed via systems minimal cut (resp. path) sets. In this article we provide a lower bound based on the minimal path sets and an upper bound based on the minimal cut sets. These bounds, which include as a special case the bounds derived recently by Fu and Koutras (1994b), are subsequently used for the efficient reliability approximation of various coherent structures.

Note: Pairwise rearrangements in reliability structures

Boland, Proschan, and Tong [2] used the notion of criticality of nodes in a coherent system to study the optimal component arrangement of reliability structures. They also provided a sufficient minimal cut (path) based criterion for verifying the criticality ordering of two nodes. We develop a necessary and sufficient condition for two nodes to be comparable and provide specific examples illustrating our result’s applicability. As a corollary, certain optimal arrangement properties of well-known systems are derived. 0 1994 John Wiley & Sons, Inc.

Formulae and Recursions for the Joint Distribution of Success Runs of Several Lengths

Consider a sequence of n independent Bernoulli trials with the j-th trial having probability pj of success, 1 ≤ j ≤ n. Let M(n,K) and N(n, K) denote, respectively, the r-dimensional random variables (M(n, k1),..., M(n,kr) and (N(n,k1), ..., N(n, kr)), where K = (k1, k2, ..., kr) and M(n, s) [N(n, s)] represents the number of overlapping [non-overlapping] success runs of length s. We obtain exact formulae and recursions for the probability distributions of M(n, K) and N(n, K). The techniques of proof employed include the inclusion-exclusion principle and generating function methodology. Our results have potential applications to statistical tests for randomness.

CONTEXT ORIENTATED TEACHING

The CONTEXT ORIENTATED TEACHING (COT) is a model for teaching mathematics and applications, based on Problem Solving and Modelling ideas, where the proper selection of the Context used to introduced, discus and achieve the didactical goals, is the central issue. Related to COT is the idea of Preformal Proving.