|
Article Excerpt
1. Introduction
In the literature on reliability the considered distribution functions are frequently continuous. Lifetime models are then selected from several distributions that have proved useful in practical applications, such as the exponential, Erlang, Weibull and others. When the model is constructed from observations, sometimes it is difficult to select one particular distribution, because none of the known ones closely fits the dataset. The continuous case solves this problem by approaching the distribution or the dataset by a continuous phase-type distribution; this is made possible by the density of the continuous class in the family of continuous distribution functions defined in the positive real line. Many stochastic models involve, in one way or another, phase-type probability distributions. The family of phase-type distributions (PH) was introduced by Neuts (1981) as a tool for unifying a variety of stochastic models and for constructing new models that would lead to algorithmic analysis.
Markovian processes have been used for modeling reliability systems. Liu and Kapur (2006) developed reliability measures for dynamic multi-state systems that have discrete states of working efficiency. Maillart (2006) studied the problem of adaptively scheduling perfect and imperfect observations and preventive maintenance actions for a multi-state Markovian deterioration system with obvious failures. Diamantidis et al. (2007) studied serial flows lines and their presented model is used as a decomposition block for solving larger lines. The decomposition block is solved via exact Markovian analysis. Few papers concern systems involving continuous phase-type distributions. In the following list, we cite some references in recent years. Perez-Ocon and Montoro-Cazorla (2006) studied a repairable warm standby n-unit system and showed that this system is governed by a level-dependent Quasi-Birth-and-Death (QBD) process. They use the methodology given in Naoumov (1997) for calculating the stationary distribution of the model. Neuts et al. (2000) studied a unit system submitted to several types of failures with the operational and repair times being phase-type distributed. Perez-Ocon and Montoro-Cazorla (2004) considered a system comprised of units arrayed in cold standby. This system is governed by a continuous QBD process. Barata et al. (2002) solved maintenance problems through Monte Carlo simulations for deteriorating systems. A multi-component G-out-of-M system was analyzed by Perez-Ocon et al. (2006) through a Markov process with vectorial states incorporating geometrical processes. Two general repairable models with two independent units involving PH distributions were developed in Perez-Ocon and Ruiz-Castro (2004). Recently, Ruiz-Castro et al. (2008b) modeled a complex redundant system involving discrete phase-type distributions. They showed that the process that governs the system is a QBD one. Also, Ruiz-Castro et al. (2008a) developed a model for studying the behavior of a discrete multi-state cold standby system with loss of units.
Nevertheless, all the systems cannot be continuously monitored, and they must be observed at certain epochs. Due to several causes, such as the impossibility of a continuous observation, or the inner structure of the system, it can occur that the systems can be observed only at certain discrete times. Thus, some reliability systems, such as digital computer systems, have a discrete behavior by time. On the other hand, a system can be submitted to periodic inspections. In this case, it is assumed that the state of the system can only be known by inspections, in discrete times.
Then, discrete distributions can be used for describing the evolution of systems. When general distributions are used for modeling a reliability system, phase-type distributions can be taken into account in a natural way, given that any non-negative discrete distribution of the probability is a phase-type distribution. As stated above, any non-negative continuous distribution of probability can be approximated through PH distributions. In the discrete case, the approximation is equivalence. Neuts (1975) pointed out that all discrete distributions with finite support can be represented by discrete-phase (PH) distributions. Alfa and Neuts (1995) showed the elapsed-time representation for such distributions. Shi et al. (1996) pointed out that for every discrete random variable a Markov chain can be constructed to generalize the result. Shi and Liu (1998) established the representation for the elapsed time. Meanwhile, Alfa and Castro (2002) considered these results for modeling a system that involved general discrete distributions. Thus, any discrete distribution is phase-type distributed. Given this analysis, a general discrete renewal process can be expressed by considering the time among arrivals that are PH distributed. Alfa (2004) formalized the notation of this result for the general renewal-type distribution and applied it for modeling a queuing system.
The system we study is a warm standby n-system, with a repair channel. There is an online unit and the others are in warm standby or in repair. The system is up when there is a unit online, but otherwise it is down. If the online unit fails, it goes to repair, and instantaneously the standby unit becomes an online unit. Any warm standby unit can become an online unit with a certain probability. After repair, the repaired units go to standby if there is an operating online unit, or become the online unit if the system is down.
The present paper differs in several ways from others previously published. For this system the Markov process that governs the system is constructed, and we show that the above mentioned multiple n-system, in which phasetype distributions are involved, is governed by a discrete Markov process of the M/G/1 type (Neuts, 1981). The resultant Markov process is identified by constructing its transition probability matrix. The stationary distribution is calculated using matrix-analytic methods for the discrete case, applying methods suitable for this case that are not a direct consequence of the continuous case. The availability and the conditional probability of failure in stationary and transient regimes are calculated. The distribution function of the up period is determined. The rewards associated with the different operations involved in the system are studied. The model is directly applicable to a system involving general discrete probability distributions, without considering any approach procedure, as in the continuous case. Throughout the paper, it can be seen that the methodology, the calculations and the structure in the discrete case are not deduced from the continuous case in a trivial way. A numerical application illustrates the calculations.
Phase-type distributions occupy a central role in the paper. We give the following definition and notation. They are studied in detail in Neuts (1981).
Definition 1. A probability distribution is a phase-type distribution if and only if it is the distribution of the time until absorption in a finite Markov process with one absorbent state and the other state being transient.
Therefore, a density {[p.sub.k]} of the non-negative integers is of phase-type if and only if there exists a finite Markov chain with a transition probability matrix P of order m + 1 of the form:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
and initial probability vector ([alpha], [[alpha].sub.m+1]), where [alpha] is a row m-vector. Here, T is a sub-stochastic matrix such that Te + [T.sup.0] = e, (I-T) is a non-singular matrix and e a column vector of ones of order m. Throughout this paper e will denote a column vector with all components equal to one for which the order is determined by the context. The order of the matrix T (and the one of the vector [alpha]) is the order of the phase-type distribution. The matrix T contains the transition probabilities among transient states and [T.sup.0] is a column...
|
