Stochastics and StatisticsA reliability system under different types of shock governed by a Markovian arrival process and maintenance policy K
Introduction
Engineering systems sometimes are considered as unit ones when the interest is focused in the behavior of the system as a whole. In many cases the systems operate under environmental conditions producing shocks that cause damage and eventually the failure of them. These conditions can generate different types of shocks (vibrations, heat, strength) that usually cause deterioration to the system. Several probabilistic models can be applied to govern the different systems under shocks, such as the competitive risk model when the different shocks can be considered independent. In general, the independence among the interarrival time of the shocks is usually assumed.
We study a system under different types of shocks causing deterioration and eventually the failure. It will be assumed that all the shocks arise from a source and they can be of different types. The shocks can cause repairable or non-repairable damages, these last are fatal ones. There are shocks causing only a mild deterioration in the system in such a way that it continues operational without repair, but the number of these is limited and after a certain fixed number of them the system is replaced. After a non-repairable failure the system is also replaced. Repair is as good as new. This system is studied following the matrix-analytic methods.
The study of shock and wear models initiated with (Esary, Marshall, & Proschan, 1973), in it the arrival of shocks followed a Poisson process. From then many papers have been written related to shock models (see Finkelstein, 2007, Finkelstein, 2008, Tang, 2006, and references in them). The first version of this paper under matrix-analytic methods and incorporating an explicit expression of the reliability function of the system under shocks is due to Neuts and Bhattacharjee (1981). The matrix-analytic methods have proved to be useful in the study of reliability systems incorporating several types of failure and repair with a unified treatment from the methodological point of view. Neuts (1979) introduced the Markovian arrival processes as an arrival model of events to the positive real half line. These have the property of having dependent interarrival times, extend previous models and allow to consider more general models in the study of reliability systems under shocks. Neuts (1981) studied the phase-type distributions, these are being a versatile model for the study of reliability systems under general operational and repair random times since they are dense in the family of the continuous distribution functions defined on the positive half real line. In Asmussen (2000) a survey about phase-type distributions and Markovian arrival processes is performed showing applied aspects of them and indicating how to fit them to datasets. Other papers that have applied phase-type distributions to shocks and wear models are references (Frostig and Kenzin, 2009, Montoro-Cazorla et al., 2009). Papers considering phase-type distributions as a model for the occurrence of a wear failure and repair times, and Markovian arrival processes as a model for the arrival of shocks causing failure, each one independent from the other, are references (Montoro-Cazorla and Pérez-Ocón, 2006a, Montoro-Cazorla and Pérez-Ocón, 2006b, Montoro-Cazorla and Pérez-Ocón, 2011, Montoro-Cazorla and Pérez-Ocón, 2012a, Montoro-Cazorla and Pérez-Ocón, 2012b, Montoro-Cazorla et al., 2009, Pérez-Ocón and Segovia, 2009).
In the present paper there are several contributions to the study of reliability systems. (1) A single source for the failures is considered, governed by a Markovian arrival process. This structure allows us to include all the events associated to the different types of shock by introducing six parametric matrices in the Markovian arrival process, each one representing the rate of occurrence of the different types of shock. The behavior of the system is governed by a multidimensional Markov process. (2) Two types of failure are considered. This allows us to model the arrival of shocks to a system composed by two well-differentiated parts, as mechanical and electrical ones. (3) Every type of shock causes different damage in the system: minor, reparable, and fatal damage. (4) Policy K of replacement is included in the maintenance of the system and the performance measures are calculated in terms of K. (5) The performance measures calculated are: the availability, the reliability, and the rates of occurrence of the different shocks in transient and stationary regime. (6) A comparison among these measures for different values of K is performed to determine the system with optimal availability and rate of occurrence of failures. The expressions of these measures are given in algorithmic form to be computationally implemented and so to be applied in practical applications. The versatility of the Markovian arrival process to be a model for a very complex system and its algorithmic advantages are shown throughout the paper.
The paper is organized as follows. In Section 2 the multidimensional Markov model is constructed. In Section 3 the performance measures are calculated in transient regime. In Section 4 the stationary study of the system is performed. In Section 5 several special cases studied in the literature are included as particular and related ones to the present system. In Section 6 a numerical application is performed. Conclusions are included at the end of the paper.
The phase-type distributions, the Markovian arrival process, and the operations of Kronecker play an important role in the present paper. They are the basic elements in the application of the matrix-analytic methods. We define them formally for a better comprehension of the paper. For more details see (Bellman, 1997, Graham, 1981, Neuts, 1981). Definition 1 The distribution function defined on is a continuous phase-type distribution (PH-distribution) if it is given by
It is associated to a finite Markov process with one absorbent state. The initial vector of the process is . Matrix S is the submatrix of the generator of the process restricted to the transient states. Vector e is a column vector of 1s of appropriate order. The absorption vector is denoted by and it satisfies . The order of the vectors and matrices involved are the same and it is the order of the distribution. It is said that the distribution has representation and it is written .
A similar definition applicable to a Markov chain of discrete parameter is valid for defining discrete phase-type distributions. Definition 2 Let D be an irreducible infinitesimal generator of a Markov process. Let a sequence of matrices , be non-negative and the matrix with non-negative off-diagonal entries. The diagonal entries of are strictly negative and it is non-singular. It is satisfied
All the matrices are square and have the same order. Associated to this Markov process there is renewal Markov process performing an arrival process to real line operating as follows. Matrix governs the interarrival times and matrix governs the arrival of type . This is the Markovian arrival process (MAP) associated to the initial Markov process. The order of the MAP is the order of the involved matrices. Definition 3 If A and B are rectangular matrices of orders and , respectively, their Kronecker product is the matrix of order , written in compact form as . The Kronecker sum of the square matrices C and D of orders p and q, respectively, is defined by , where denotes the identity matrix of order k.
Section snippets
The model
Let a system be subject to shocks produced by a unique source. Two types of shocks are distinguished. Each one of these produce several types of failures depending on the damage produced in the system. The occurrence of all these types of failures is governed by an MAP that will be defined below. The behavior of the system is governed by the following assumptions. Assumption 1 The occurrence of the shocks is governed by an MAP. Assumption 2 Two types of shocks are distinguished, denoted by type I and type II, producing
Performance measures
Associated to the Markov process with generator (2.1) are the transition probability functions among the macro-states:
Denoting the matrix form of the forward Kolmogorov differential equations is under the initial condition . It is well-known that the solution of this equation is , and this can be solved by using computational programs.
The reliability measures of the system are
Stationary probability vector
The stationary study of the system is based in the stationary probability vector that we define now. Associated to the state-space of macro-states we define the stationary probability vector of the system aswhere denotes the probability of occupancy macro-state ; and denotes the occupancy probabilities of macro-state , all of them at any time. Every is a vector formed by the probabilities of occupancy of the
Special cases
We consider two special cases of the general model. The first one is the initial shock and wear model in Esary et al. (1973) under the matrix-analytic methodology. The second one considers the wear following a phase-type distribution and it has been published in Montoro-Cazorla and Pérez-Ocón (2006a) under some different assumptions. We present them here showing that the MAP allows us to extend previous works incorporating dependence among the interarrival times of the shocks.
Numerical application
We illustrate the calculations throughout the paper by means of a numerical application in stationary regime. The system we have studied depends on the value K and so all the performance measures. We present a system below and perform a comparison for the different performance measures in terms of the parameter K. Let a system be as the one defined in Section 2 subject to shocks arriving following an MAP whose parameter matrices are the following:
Conclusions
We have considered a system under different types of shocks governed by an MAP, a stochastic model whose interarrival times are dependent. Two types of shocks have been studied, and for them three different types of shocks have been included, depending on their effect on the system. This model can be applied straightaway to systems composed by two well-differenced elements, such as devices composed by mechanical and electrical parts. But many other systems composed by more than two parts and
Acknowledgements
We are very grateful to the two Reviewers for their comments that helped to improve the final version of the manuscript. This paper is partially supported by the project MTM2010-17996 of the Ministerio de Ciencia e Innovación, Spain.
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