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In this paper, a generalized class of run shock models associated with a bivariate sequence {(Xi, Yi)}i≥1 of correlated random variables is defined and studied. For a system that is subject to shocks of random magnitudes X1, X2, ... over time, let the random variables Y1, Y2, ... denote times between arrivals of successive shocks. The lifetime of the system under this class is defined through a compound random variable T = ∑Nt=1 Yt , where N is a stopping time for the sequence {Xi}i≤1 and represents the number of shocks that causes failure of the system. Another random variable of interest is the maximum shock size up to N, i.e. M = max {Xi, 1≤i≤ N}. Distributions of T and M are investigated when N has a phase-type distribution.

Phase-type distributions have been found to be very useful in reliability, queueing theory, and some other operational research applications. In this work, we de_ne a new mixed shock model and study lifetime properties of the corresponding system using phase-type distributions. The system under concern fails upon the occurrence of k consecutive shocks of size at least d1 or a single large shock of size at least d2, d1 < d2. In particular, using closure properties of phase-type distributions, we obtain survival function and mean time to failure of the system. Optimal replacement time is also studied.

WOS: 000444437300015 delta-shock model is one of the widely studied shock models in reliability. Under this model, the system fails when the time between two consecutive shocks falls below a fixed threshold . In this paper, the survival function and the mean time to failure of the system are obtained when the times between successive shocks follow proportional hazard rate model.

This paper is concerned with the Birnbaum importance measure of a component in a binary coherent system. A representation for the Birnbaum importance of a component is obtained when the system consists of exchangeable dependent components. The results are closely related to the concept of the signature of a coherent system. Some examples are presented to illustrate the results.

Consider a system with m elements which is used to fulfill tasks. Each task is sent to one element which fulfills a task and the outcome is either fulfillment of the task (''1'') or the failure of the element (''0''). Initially, m tasks are sent to the system. At the second step, a complex of length m"1 is formed and sent to the system, where m"1 is the number of tasks fulfilled at the first step, and so on. The process continues until all elements fail and the corresponding waiting time defines the lifetime of the binary sequence which consists of ''1'' or ''0''. We obtain a recursive equation for the expected value of this waiting time random variable.

Recently, Grabner et al. [Combinatorics of geometrically distributed random variables: run statistics, Theoret. Comput. Sci. 297 (2003) 261–270] and Louchard and Prodinger [Ascending runs of sequences of geometrically distributed random variables: a probabilistic analysis, Theoret. Comput. Sci. 304 (2003) 59–86] considered the run statistics of geometrically distributed independent random variables. They investigated the asymptotic properties of the number of runs and the longest run using the corresponding probability generating functions and a Markov chain approach. In this note, we reconsider the asymptotic properties of such statistics using another approach. Our approach of finding the asymptotic distributions is based on the construction of runs in a sequence of m-dependent random variables. This approach enables us to find the asymptotic distributions of many run statistics via the theorems established for m-dependent sequence of random variables. We also provide the asymptotic distribution of the total number of non-decreasing runs and the longest non-decreasing run.

Consider a system which has n independent components (or subsystems) each consisting of m dependent elements. Let X"i=(X"i^1,X"i^2,...,X"i^m), i=1,2,...,n denote the random strength vector of the ith component, where X"i^j denotes the random strength of the jth element of the ith component. The elements of the components are subjected to a common random stress Y=(Y"1,Y"2,...,Y"m) over time. In this paper, we setup a multivariate stress-strength model based on the conditional ordering between X"is and Y and evaluate the reliability of coherent structures in this setup.

In this article, we study a shock model in which the shocks occur according to a binomial process, i.e. the interarrival times between successive shocks follow a geometric distribution with mean 1/p. According to the model, the system fails when the time between two consecutive shocks is less than a prespecified level. This is the discrete time version of the so-called δ-shock model which has been previously studied for the continuous case. We obtain the probability mass function and probability generating function of the system’s lifetime. We also present an extension of the results to the case where the shock occurrences are dependent in a Markovian fashion.