Copula based Measures of Repairable Parallel System with Fault Coverage

The main objective of this study is to analyse the reliability behaviour of parallel systems with three types of failure, namely unit failure, human failure and major failure. For this purpose, we apply three different statistical techniques, namely copula, coverage and copula-coverage. More precisely, this study presents a stochastic model for analysing the behaviour of a multi-state system consisting of two non-identical units by incorporating the concept of coverage factor and two types of repair facilities between failed state to a normal state. The system could be characterized as being in a failed state due to unit failures, human failure and major failures, such as catastrophic and environmental failure. All failure rates are constant and it is assumed that these are exponentially distributed whereas, repair rates follow the Gumbel-Hougaard copula distribution. The entire system is modelled as a finite-state Markov process. Time-dependent reliability measures like availability, reliability and mean time to failure (MTTF) are obtained by supplementary variable techniques and Laplace transformations. The present study provides a comparative analysis for reliability measures among the aforementioned techniques, while a discussion referring to which technique makes the system more reliable is also developed. Furthermore, numerical simulations are presented to validate the analytical results. KeywordsPerfect coverage, Gumbel-Hougaard copula, Parallel system, Human failure, MTTF, Markov process.


Introduction
A manufacturer designs the system keeping in mind some objectives to deliver good services. However, after a certain time or due to environmental conditions the system may not perform as well as the manufacturer expected and consequently, the desired result may not be achieved and the system could be then treated as unreliable. Therefore, a reliable system is one that performs failure rate and imperfect coverage. Ram and Goyal (2018) studied on flexible manufacturing system under a combinatorial copula-coverage approach and have shown that the mixed copulacoverage technique improves the reliability of the system.
In the past, many researchers have studied the Gumbel-Hougaard copula, the fault coverage, and both techniques, but the copula-coverage technique with perfect coverage has not been covered yet. Therefore, in the present study three techniques namely copula, coverage and copula-coverage with perfect coverage have been discussed. The proposed model consists of two dissimilar units in a parallel configuration. The system gets degraded states due to failure of any one of the units and failed states due to both unit's failure, human failure, and major failure. Further, the effect of these techniques on reliability measures has been analysed. In the context of the copula, the Gumbel-Hougaard family has been used. The main objective of this study is to present a comparative analysis to identify which technique is better to improve system's reliability.
The present paper is organized as follows: Section 2 gives a brief introduction of copula and coverage techniques. Section 3 presents assumptions associated with the model, mathematical model details containing notations and structure of the model. Section 4 describes the formation and solution of the model. Numerical calculations such as availability, reliability, and MTTF have been obtained in Section 5. Section 6 discussed the result of this work. Conclusion of the present paper and the direction of future work is presented at length in Section 7.

Brief Introduction of Copula and Coverage
The term copula was first introduced by Sklar (1973). Copulas are the functions that connect multivariate distributions functions to their one-dimensional marginal distribution functions (Nelsen, 2007). If F is m-dimensional cumulative distribution function with one-dimensional margins F1, F2, ………, Fm then there exists an m-dimensional copula C such that F(y1,……,ym)= C (F1(y1),……..Fm(ym)) where, An m-dimensional copula is a function such that The copula approach requires the specification of the marginal distribution of each random variable along with a function that connects them. Patton (2009) has given the idea of conditional copula and applied it on the time-series based model. Copulas are enabled to structure flexible multivariate distributions appearing rich patterns in tail behaviour, different kinds of asymmetry and varying from tail independence to tail dependence (Rodriguez, 2007;Trivedi and Zimmer, 2007).

Gumbel-Hougaard family copula
Gumbel-Hougaard copula is defined as Gumbel-Hougaard copula does not allow negative dependence. If u1 = e x and u2 = μ(x) then by Gumbel-Hougaard copula the joint probability expression is given by Further, the concept of coverage has introduced by Arnold (1973). Coverage is a technique used to improve the reliability of the system. Coverage is defined as System recover from fault coverage factor = Total fault occur in thesystem .
Coverage is the probability of automatic recovery of the system from the fault that occurred during system operation. Coverage provides a more accurate picture of system behaviour and gave more persuasion in reliability estimates. If the value of the coverage factor is equal to one, it is called perfect coverage. Perfect coverage means that the system is able to detect and remove the failure of the redundant system (Manglik and Ram, 2014). Pham (1992) studied a high voltage system with imperfect coverage and constant fault coverage. Levitin and Amari (2007) evaluated reliability characteristics of fault tolerance systems with multi fault coverage using the universal generating function technique. Reibman and Zaretsky (1990) presented a reliability model for network reconfiguration system using fault coverage in a high-reliability model and improve the network's reconfiguration capabilities.

Assumptions, Notations and System Description
This study provides a comparative analysis among three techniques namely copula, coverage and copula-coverage technique. The assumptions of the proposed model are as follows: (i) The system consists of two non-identical units in a parallel configuration. (ii) Initially, the units and the system are in good condition. (iii) The system contains three states of functioning, namely good, degraded and failed. (iv) The system could fail due to both unit failure, human failure and major failures, such as a catastrophic failure and environmental failure.
All failure rates are assumed to be constant and follow the Exponential distribution. (vi) The repair rates follow the Gumbel-Hougaard copula distribution. (vii) Only a complete state has been repaired and in completely failed states, the system has been repaired by two repair facilities i.e., the Gumbel-Hougaard family of the copula. (viii) Once the system is repaired, it works like a new one.
The system contains two non-identical units in a parallel configuration. At the beginning, the system is supposed to be in good working condition, say state S0. On failure of any one of the two units, another unit takes the load and the system goes to degraded state say state S1 and S2. The system is said to be in a completely failed state (S3, S4, S5, S6) on the failure of both the units, human failure and major failures like a catastrophic failure, electronic failure and environmental failure in the system. The system has two types of repair facilities between completely failed states (S3, S4, S5, S6) and original state S0. In such a system, detected faults have been perfectly covered with coverage factor c. All failure rates are taken to be constant and follow the exponential distribution. However, repair rates from the state (S3, S4, S5, S6) to state S0 follow the Gumbel-Hougaard family of the copula. The whole system is analysed under three different technique namely copula, coverage and copula-coverage to improve the reliability of the system. The description of the transition states associated with the model are given as S0: The system is in good working condition as both the units operate well. S1: The first unit fails, the second unit takes the load of the first unit, the system is in a degraded state. S2: The second unit fails, the system works with the first unit, degraded state of the system. S3: Completely failed state due to failure of the second unit after first unit failure. S4: Completely failed state due to failure of the first unit after second unit failure. S5: Completely failed state due to human failure. S6: Completely failed state due to major failure in the system. Furthermore, some notations used throughout the lines of the present paper are displayed in Table  1. Given the abovementioned assumptions of the proposed model, a transition diagram is developed as shown in Figure 1.

Mathematical Modelling of the Proposed Method
This section gives the states transition probabilities associated with the model, these probabilities are further used to evaluate the reliability characteristics such as availability, reliability and MTTF. From the state transition diagram, the following differential equations have been derived.   After solving the Equations (10) -(13) by the aid of (14) -(17), one can get the state transition probabilities as Up and downstate probability are as follow: 3,4, ,

Numerical Calculations
In this section, some numerical computations have been accomplished by the aid of the abovementioned differential equations.

Availability Analysis
Generally speaking, availability simply means that the system performs its work in a specified time t. Availability is one of the most important reliability measures, since it is associated with both the failure and repair rate of the system. It is always a time function. In this subsection, the availability of the underlying system with different techniques is presented.

Availability of the System without Copula-Coverage.
In this case, availability expression is given by By setting the values of λ1 = 0.05, λ2 = 0.10, λh = 0.15, λm = 0.20 and 1 μ = in Equation (27) and then taking inverse Laplace transform, we may produce several numerical results. Table 2 displays the availability of the system for time t from 0 to 25.

Availability of the System with Copula (When Coverage is Ignored).
In this case, availability of the system is given by putting By replacing λ1 = 0.05, λ2 = 0.10, λh = 0.15, λm = 0.20 in the above availability expression and then taking inverse Laplace transform of Equation (28), we may produce several numerical results. Table 3 displays the availability of the underlying system for time t from 0 to 25.

Availability of the System with Coverage (When Copula is Ignored).
In this case, availability expression is given by putting ( ) 1 x =  in Equation (25).
Further, by putting λ1 = 0.05, λ2 = 0.10, λh = 0.15, λm = 0.20 and then taking the inverse Laplace transform of Equation (29), we may produce several numerical results. Table 4 presents the availability of the system for time t from 0 to 25 at different values of coverage factor.

Reliability Analysis
Reliability is a probability that any system works well without any damage within a specific time under the stated condition. The reliability of the system can be found in the case that repair facilities not available. Hence, the reliability of the system can be calculated by putting a repair rate equal to zero.

Reliability of the System without Copula-Coverage
Reliability of the system without copula-coverage is given by We next replace λ1 = 0.05, λ2 = 0.10, λh = 0.15, λm = 0.20 and then take inverse Laplace transform of Equation (31). Table 6 displays the reliability of the system for time t from 0 to 25.  Figure 6. Reliability of the system without copula -coverage.

Reliability of the System with Copula
Ignored coverage factor and put repair rate is equal to zero in Equation (25) to get the reliability expression of the system with copula, given by as follow We first set λ1 = 0.05, λ2 = 0.10, λh = 0.15, λm = 0.20 and then take inverse Laplace transform of Equation (32). Table 7 displays the reliability of the system for time t from 0 to 25.

Reliability of the System with Coverage
In this case, the reliability of the system is given by We first set the values of failure rates λ1 = 0.05, λ2 = 0.10, λh = 0.15, λm = 0.20 and then take inverse Laplace transform of Equation (33). Further, by putting c = 0.2, 0.4, 0.6, 0.8 and varying t = 0 to 25, we get the following table.

Mean Time to Failure (MTTF) Analysis
MTTF is the mean time to failure i.e., the time that the system is available or not failed. Taking s tends to zero in reliability expression to get the MTTF of the system.

MTTF of the System without Copula-Coverage
Taking all repair rate equal to zero and s tends to zero, we get following MTTF expression without copula-coverage Now setting λ1 = 0.05, λ2 = 0.10, λh = 0.15, λm = 0.20 and varying λ1, λ2, λh, λm one by one from 0.1 to 0.9, we get the table of MTTF as Table 10 for λ1, λ2, λh, λm Table 10. MTTF of the system without copula-coverage. Figure 10. MTTF of the system without copula-coverage.

Result Discussion
From Figures 2-5, it is clearly shown that the availability of the system decreases slowly and it becomes constant after a long time, that is to say, that the availability of the system decreases as an increment in the time variable. During one type of repair between two transition states of the system (i.e., repair rate follows exponential distribution) and when no coverage technique applied, the availability of the system is achieved 0.786 as time varies from 0 to 25 units of time, which is shown by the Figure 2. While the availability of the system arising from Figure 3 is 0.909 as of time vary from 0 to 25, Which is the result of availability when there is no coverage technique is applied and repair rate follows Gumbel-Hougaard family of copula distribution. Figure 4 shows the availability of the system at different values of coverage factor and when the repair rate follows an exponential distribution. With a close look of Figure 4, it is clear that availability of the system decreases with the passage of time and the highest value of availability is obtained at coverage value of 0.2 and lowest value of availability is obtained at coverage value of 0.8, which is even better from the figure 2 and 3's results. Further, Figure 5 shows the availability of the system when both copula-coverage techniques applied together. With time availability of the system decreases up to 0.976 at coverage value of 0.2 and 0.92 at coverage value of 0.8. From Figure 2  easily seen that the availability is improved by using the mentioned techniques namely copula, coverage and copula-coverage. Among these techniques, the copula-coverage technique is the better one. The order of these techniques, to improve the availability of the system is copula-coverage> coverage> copula> without copula-coverage.
The reliability's results of the system using the mentioned techniques are shown by figures 6-9. Figure 6 shows the reliability of the system when the repair rate follows exponential distribution and when the coverage technique is not applied. It is seen that as time increases the reliability of the system decreases. Figure 7 shows the reliability under the copula technique. As time increases, the reliability of the system decreases and it is up to 0.03 as time varies from 0 to 25. Figure 8 demonstrates the reliability of the system using the coverage technique. At different values of coverage factor, the reliability of the system decreases with the increment in the time variable. Reliability is highest in case of coverage value 0.2 and lowest in case of coverage value 0.8. Figure  9 shows the reliability when the copula and coverage technique applied together. The reliability of the system decreases as time increases at different values of coverage and it decreases up to 0.09 for the coverage value 0.8. It may be seen from Figures 6-9, the values of reliability are different under the copula, coverage and copula-coverage techniques. So, the copula-coverage technique must give the best value of reliability.
Table 10-13 gives the value of mean time to failure of the system and correspondingly Figures 10-13 show the MTTF behaviour of the system. In all cases, MTTF variation is low for unit failure and high for human failure which indicates that human failure is more responsible for the proper functioning of the system. Also, it is to be noted that the coverage and copula-coverage techniques give the same values of MTTF. It means that there is no effect of copula technique on MTTF because MTTF does not depends on repair rate, either a single repair facility or more than one repair facility is available for the system.

Conclusions and Future Work
In this work, the authors proposed an approach for analysing the reliability measures of a parallel system consisting of two non-identical units, using Gumbel-Hougaard copula distribution and perfect coverage. A non-identical unit parallel system is often found in aircraft, medical equipment, power generating station, etc. Three different techniques namely copula, coverage and copulacoverage for improving the reliability of the system are discussed. The effect of copula, coverage, and copula-coverage, as well as the failure component of the system on to the behaviour of the system, is investigated by varying the different parameters of the system. Based on the results illustrated throughout this work, it is concluded that the availability and reliability of the system tend to become higher when copula and coverage techniques have been used together. The performance of the system, using these techniques is evaluated in descent order as follows: copulacoverage technique is better than the coverage technique, which is better than the copula technique. The comparative study of this work leads to the conclusion that, in order to improve system's reliability, there is a need to simultaneously incorporate copula and coverage techniques. Also, by preventing the occurrence of human failure, the system becomes more reliable. The area of this research is wide open in which one may investigate the behaviour of k-out-of-n systems using these techniques. Also, this study can be more interesting if one may think about other types of copulas instead of the Gumbel-Hougaard family of copula. In future work, our research will look at repairing the degrade states and evaluating the sensitivity of availability, sensitivity of reliability and sensitivity of MTTF.