A Note on the Time to Failure of a Two-Unit Parallel Redundant System with Deterioration on a Lattice

In this paper, we consider a two-unit parallel redundant system with deterioration on a lattice, where each unit has multistage deterioration levels, say, n levels. The transition from one deterioration level to the subsequent level occurs following the well-known Marshall-Olkin bivariate exponential distribution. We derive the closed form of the Laplace transform of the time to system failure in the two-unit parallel redundant system with deterioration on n × n lattice without repair and simultaneous failure, as well as the simple system on 3 × 3 lattice. KeywordsTwo-unit parallel redundant system, Marshall-Olkin bivariate exponential distribution, Time to system failure, Laplace transform, Deterioration on a lattice.


Introduction
Two-unit parallel redundant systems play a central role to design actual redundant component systems in terms of fault-tolerance. In fact, a number of authors have paid their attentions to investigate the stochastic behaviour of redundant systems composed of two independent components (Borgerson and Freitas, 1975;Osaki and Nishio, 1980). On the other hand, a two-unit parallel redundant system with bivariate exponential lifetimes was first analyzed by Harris (1968), Osaki (1980). They applied the bivariate exponential distribution proposed by Marshall and Olkin (1967) to characterize the two-unit system with correlations. Sugasawa and Kaji (1981) and Goel et al. (1985) also took account of maintenance and inspection to the two-unit parallel system, respectively, for the two-unit parallel redundant systems. Sinha and Chaudhuri (1998) surveyed several results on the two-unit series systems based on the different definitions of bivariate exponential distributions. While the two-unit parallel redundant system is a rather standard system configuration, many authors (Ram and Singh, 2008;El-Said and El-Sherbeny, 2010;Ram et al., 2013;Singh et al., 2013;Ota et al., 2015) concerned to analyze different stochastic models as well. levels, say levels, and the transition from one deterioration level to the subsequent one occurs following an exponential distribution. More precisely, the deterioration time of the two-unit parallel system obeys a bivariate exponential distribution in the sense of Marshall and Olkin (1967), and the system failure time corresponds to the first passage time from the initial state ( , ) = (0, 0) to an absorbing state ( , ) = ( − 1, − 1) for respective units, where , = 0, 1, . . . , − 1 and ≥ 2. The present model under consideration is quite different from Harris (1968), Osaki (1980), and can be regarded as an abstract model for redundant structural systems in actual design.
The main purposes of this paper are to derive analytically (i) the Laplace transform of the time to failure in a simple two-unit parallel redundant system with deterioration on 3 × 3 lattice, and (ii) the Laplace transform of the time to failure in the two-unit parallel redundant system with deterioration on × lattice without repair and simultaneous failure. A numerical illustration is given to investigate the temporal behavior of the mean time to failure and its variance.

Two-Unit Parallel Redundant System
Let us consider a two-unit parallel redundant system with deterioration on a lattice. In the two-unit hot-standby parallel redundant system, each unit has multi-stage deterioration levels, say (= 2, 3, . . . ) levels, and the transition from one deterioration level to the subsequent one occurs following an exponential distribution. More precisely, let 1 and 2 be the independent deterioration times for Unit 1 and Unit 2, respectively, and denote the non-negative random variables having the marginal distribution functions: When both units may deteriorate simultaneously, on the other hand, let 12 denote the simultaneous deterioration time for them having 12 ( ) = Pr{ 12 ≤ } = 1 − exp{− 12 }, > 0, 12 > 0, In this case, the deterioration times for Unit 1 and Unit 2 taking account of the simultaneous deterioration are given by = min( 1 , 12 ), = min( 2 , 12 ), and the deterioration time distribution is described by the Marshall-Olkin bivariate exponential distribution (Marshall and Olkin, 1967): From Eq. (4) it is evident that 1 , 2 and 12 are independent exponentially distributed random variables. When one of units fails, the repair starts immediately and the repaired unit becomes as good as new. The repair times for Unit 1 and Unit 2 are independent from each other, and obey the common distribution functions 1 ( ) and 2 ( ), respectively. Hence, even if Unit 1 (2) fails, the system is operative during the period until an opponent Unit 2 (1) is operating, so that the state in which one of two units fails is a regeneration point, and the repair operation always starts at this point. On the other hand, the state in which both units fail is the non-regeneration point in this system (Osaki and Nishio, 1980). Figure 1 illustrates the configuration of the two-unit system on × lattice. In this system, each state is denoted by the pair ( − ), where (= 1, 2) is the unit number and (= 0, 1, . . . , − 1) is the deterioration level. The system failure time is defined by the first passage time from the initial state ( , ) = (0, 0) to an absorbing state ( , ) = ( − 1, − 1) for respective units, with , = 0, 1, . . . , − 1 and ≥ 2.  From the well-known Markovian analysis, it is straightforward to see that Taking the Laplace transform of both sides in Eq. (6), we have The problem is reduced to derive the Laplace transform of the time to failure, 0 ( ). Since ( ), , = 0,1, . . . , 2 − 1, are the exponential transition probabilities, it would be possible to calculate ( ) numerically even for relatively large . In the following sections, we focus on the analytically tractable cases and derive some useful results.

Special Case with =
As one of the simplest cases, we consider a two-unit parallel redundant system with deterioration on 3 × 3 lattice. This simple situation frequently occurs in many examples. For instance, Huang et al. (1995) and Garg et al. (1995) considered an AT & T billing application and described the software degradation behavior called software aging by a continuous-time Markov chain and a Markov regenerative process, respectively. In their modeling, it is assumed that the system failure occurs through one degradation level called failure probable state. So, if the software system is a one-unit system, the time to failure distribution is given by the Stieltjes convolution of two nonidentical exponential distributions, which is the hypo-exponential distribution. Rinsaka and Dohi (2005) performed the behavioral analysis of a two-unit fault-tolerant software system with rejuvenation and formulated the system availability, where the number of degradation levels for each unit is two. In this way, we note that the environmental diversity in software systems is based on the redundant configuration by either an identical replica of software or a non-identical hot standby system (Trivedi and Bobbio, 2017).
It can be found that the regeneration points are States 2, 5, 6 and 7, and that State 8 is a nonregeneration point.
Next, we derive the probability distribution function of the time to system failure.

Corollary 4.2
In the two-unit parallel redundant system deteriorating on × ( − 1) lattice without repair and simultaneous failure, the Laplace transform of the time to failure is 0 ( )/(2 ( )), where 0 ( ) is given in Eq. (61).

Corollary 4.3
In the two-unit parallel redundant system deteriorating on × lattice, the mean and variance of the time to failure are given by respectively.

Corollary 4.4
In the two-unit parallel redundant system deteriorating on × ( − 1) lattice, the mean and variance of the time to failure are given by respectively.
Corollary 4.2 and Corollary 4.4 are the direct applications of Theorem 4.1 and Corollary 4.3, respectively, and are useful to analyze an asymmetric lattice structure on deterioration.
We investigate numerically the temporal behavior of the mean time to failure and its variance. In Figures 3 and 4, we plot the mean time to failure and the variance with respect to the number of deterioration levels on × lattice. For an arbitrary 1 (> 0), the mean time to failure increases gradually as increases, where the increment between − 1 and approaches to 1/ 1 from a larger value. The increasing rate of the variance is higher than the mean time to failure but the increment is somewhat milder.

Conclusions
In this paper, we have considered a two-unit parallel redundant system with deterioration on a lattice, where each unit has multi-stage deterioration levels under the assumption that the transition from one deterioration level to the subsequent level occurred following the well-known Marshall-Olkin bivariate exponential distribution. We have derived the closed form of the Laplace transform of the time to failure in the two-unit parallel redundant system with deterioration on × lattice without repair and simultaneous failure, as well as the simple system on 3 × 3 lattice. Our explicit results in this paper can be easily applied to a general case where the degradation time of both units follows a Marshall-Olkin type bivariate distribution with the common marginal distributions.
Once the Laplace transform was derived, the Laplace inversion technique is applied to calculate the time to failure distribution numerically. However, the Laplace inversion transform is still instable in computation, and does not always provide feasible solutions. In future, we will propose any approximation scheme to compute the time to system failure distribution without the inversion technique.