Cost Benefit Analysis of a k-out-of-n: G Type Warm Standby Series System Under Catastrophic Failure Using Copula Linguistics

This paper deals with the study of reliability measures of a complex engineering system consisting three subsystems namely L, M, and N in series configuration. The subsystem-L has three units working under 1-out-of-3: G; policy, the subsystem-M has two units working under 1-out-of-2: G policy and the subsystem-N has one unit working under 1-out-of-1: G; policy. Moreover, the system may face catastrophic failure at any time t. The failure rates of units of all subsystems are constant and assumed to follow the exponential distribution however, their repair supports two types of distribution namely general distribution and Gumbel-Hougaard family copula distribution. The system is analyzed by using the supplementary variable technique, Laplace transformation and Gumbel-Hougaard family of copula to derive the differential equations and to obtain important reliability characteristics such as availability of the system, reliability of the system, MTTF, and profit analysis. The numerical results for reliability, availability, MTTF, and profit function are obtained by taking particular values of various parameters and repair cost using maple. Tables and figures demonstrate the computed results and conclude that copula repair is more effective repair policy for better performance of repairable systems. It gives a new aspect to scientific community to adopt multi-dimension repair in form of copula. Furthermore, the results of the model are beneficial for system engineers and designers, reliability and maintenance managers. Keyword: K-out-of-n, G system, Availability, MTTF, Catastrophic failure, Gumbel-Hougaard family copula distribution. Nomenclature s , t Laplace transform / Time scale variable   1 1 / x   Failure rate / Repair rate of each unit in subsystem-L.   2 2 / x   Failure rate / Repair rate of each unit in subsystem-M. 3  Failure rate of the unit in subsystem-N. E  Deliberate failure rate when two units in subsystem-L and one unit in subsystem-M failed. C  Failure rate related to catastrophic failure mode.   0 P t The state transition probability that the system is in Si state at an instant for 0 i  .   P s Laplace transformation of the state transition probability   P t .   , i P x t The Probability that the system is in state i S for 1 to 9, E, C i  and the system is under repair with elapsed repair time is , x t . x is repaired variable and t is time variable. 1. Corresponding Author Email: pkpmrt@email.com   p E t Expected profit in the interval  0, t  . 1 2 , K K Revenue generated and service cost per unit time respectively.   0 x  An expression of the joint probability from failed state Si to good state S0 according to Gumbel-Hougaard family copula is given as         0 1 2 , x C u x u x        1 exp log x x             where     1 u x x   and   2 x u x e  . Here  is the parameter1     . Introduction Determining accurate reliability and availability of an existing structure or product is a crucial task in the reliability engineering. In case of failure, money and time will be wasted and even disaster may occur. In order to achieve reliable system functioning, components are designed to be highly reliable in the sense that they rarely suffer from sudden failures. Nevertheless, components might degrade gradually with usage. Redundant strategy is often used by engineers to 36 / IJRRS / Vol. 3/ Issue 1/ 2020 P. K. Poonia and A. Sirohi ensure the reliability and availability of the systems and/or to improve these characteristics of the systems. Thus, variety of standby systems have been designed and analyzed during the last few decades. The main objective of these studies is to develop methods and tools for evaluation and demonstration of the reliability, availability, and cost analysis. Redundant systems, which have been widely used in practice, such as space shuttles, communication satellites, a hybrid car, Nuclear reactors, or a fighter plane, are frequently discussed in research literature. Initially, redundant parts are designed to improve the reliability of the system, meaning that some additional paths are created or identical components connected in such a way that when one component fails the others will keep the system functioning. It is a technique commonly used to improve system reliability and availability. Redundancies can be categorized into the following: (i) Cold standby in which the standby unit is only called upon when the primary or operating unit fails. These inactive components have a zero failure rate and cannot fail while in standby state; (ii) Hot standby in which the standby unit has the same failure rate as when it is run with the operating unit; (iii) Warm standby in which the standby unit runs in the background of operating unit. It can fail in this state but its failure rate is less than that of the operating unit. Moreover, redundancy is highly cost effective in achieving a certain reliability level of the system. Therefore, in order to enhance reliability k-out-of-n system structure in which at least k components of n must be functioned. In order to improve the reliability of k-out-of-n systems, numerous researches have presented their works and contributions by constructing different types of complex repairable systems under the different types of failure and repair distributions. For instance, authors consider warm standby system by She and Pecht [1], generalized multi state system by Huang et al. [2], repairable consecutive systems with r repairman by Wu and Guan [3], two-stage weighted systems with components in common by Chen and Yang [4], main unit with helping unit by Kumar and Gupta [5], Markov repairable system with neglected or delayed failures by Bao and Cui [6], evaluated exact reliability formula for consecutive repairable systems by Liang et al. [7], general system with non-identical components considering shut-off rules using quasi-birth-death process by Moghaddass et al. [8], and generalized block replacement policy with respect to a threshold number of failed components and risk costs by Park and Pham [9]. The occurrence of failure in any complex repairable or non-repairable engineering system is a natural phenomenon, which arises due to the different working conditions. The k-out-of-n effective policy plays a crucial role in maintaining the reliability of repairable systems. The researchers have focused on evaluating reliability and availability of the redundant repairable systems like k-out-of-n in series configuration. In particular, Singh et al. [10] analyzed an engineering system, which consists of two subsystems, viz. subsystem-1 and subsystem-2 with controllers in series. Subsystem-1 works under the k-out-ofn: good policy. Subsystem-2 consists of three identical units in parallel configuration. In this case, controllers control the working of both subsystems. Authors evaluated reliability characteristics using supplementary variable technique. Ram et al. [11] investigated the reliability of a standby system incorporating waiting time to repair. In this case, system consists of two units’ namely main unit and standby unit. Whenever the main unit fails, the whole load is transferred to the standby unit instantaneously by a switching-over device. As regards to the repairing of the main unit, it has to wait for repair whenever it fails due to unavailability of repair facility. Munjal and Singh [12] analyzed a complex repairable system composed of two 2-out-of-3: G subsystems connected in parallel. Jia et al. [13] studied repairable multistate two-unit series systems when repair time can be neglected. Goyal et al. [14] studied the sensitivity analysis of a three-unit series system under k-out-of-n redundancy. Singh et al. [15] developed a model of a complex repairable system having two subsystems in series configuration. Both subsystems includes two units in parallel, and it is assumed to work till at least one unit of both the subsystems are in good operative condition. Gahlot et al. [16] assessed a repairable system in series configuration under different types of failure and repair policies using copula linguistics. Singh and Poonia [17] assessed 1-out-of-2: G system with correlated lifetimes under inspection. Some specific papers related to this paper are as follows. Lado and Singh [18] analyzed an engineering system, which consists of two subsystems in series configuration operated by a human operator. Both the subsystems have two units in parallel. In this papers authors proved that copula repair is more reliable than general repair. Pundir and Patawa [19] studied repairable two dissimilar units’ cold standby system waiting for repair facility after failure of system units. They stimulated exponential failures, arbitrary waiting and arbitrary repair rate. Singh et al. [20] studied two subsystems in series configuration with imperfect switch connected with both subsystems. Recently, Zhao et al. [21] and Singh et al. [22] studied some real system problems related to our study. System Description Researchers around the world have presented their research works on reliability analysis of complex repairable system however they have not focused on the study of the system consisting of three subsystems connected in series configuration with catastrophic failure. Catastrophic failure is a complete, sudden, often unexpected breakdown in the entire system. Such a break down may occur due to animal related IJRRS: Vol. 3/ Issue 1/ 2020 / 37 Cost Benefit Analysis of a k-out-of-n: G Type Warm Standby Series System ... disruption or change in environment related conditions like Corona virus nowadays. Sometimes a single component in a critical location fails; resulting in downtime for the entire system also comes under catastrophic failure. The term catastrophic failure is most commonly used for organizational failures, but has often been extended to many other disciplines in which total and irrecoverable loss occurs. Treating the above realities in the present study, the model consisting three subsystems in series configuration considering catastrophic failure. The subsystem-L has three identical units, subsystem-M has two identical units and subsystem-N has one unit only. The subsystem-L is working under 1-out-of-3: G; scheme, the subsystem-M is working under 1-out-of-2: G; scheme, however, the subsystem-N works under 1-outof-1: G; scheme. The catastrophic failure is treated as a complete failing state. During operation, the system will be in any of the three states: perfect operation, partial failure, and complete failure. The failure rates of units of subsystems are constant and assumed to follow the exponential distribution, but their repair supports two types of distribution namely general distribution and Gumbel-Hougaard family copula distribution. Then, based on the behavior of the whole system, all the system states can also be classified into three subsets as follows. Classification I: The system operates perfectly; in this situation, all the components in both subsystems are in the perfect functioning state. Classification II: The system is partially working; in this situation, at least one component in one or both subsystems is in the failure state, and the remainder is perfectly functioning. Classification III: The system is completely failed; in this situation, either subsystem L, M or N is in the complete failure state. Further, system may be completely failed due to catastrophic failure. Therefore, the system remains working until one of the subsystems is completely failed. Based on the above-mentioned assumptions, the system could be modeled by a continuous-time stochastic process. The present study accomplished two objectives using supplementary variable technique. First the expressions for the reliability of the system, availability of the system, mean time to failure and profit function are obtained. Second numerical simulation with respect to profit function is performed. Explicit expressions for reliability, availability, MTTF, and cost analysis functions are obtained with help of MAPLE (software). Tables and graphs present a comparative analysis of results. The system configuration and transition state diagram of the designed model are shown in fig. 1(a) and 1(b) respectively. Assumptions The following assumptions are made through this paper: 1. Initially the system is in state 0 S , and all the units of subsystem-L, M, and N are in proper working conditions. 2. The subsystem-L works successfully if minimum one unit is in proper working condition i.e. 1-out-of-3: G policy, the subsystem-M works successfully if minimum one unit is in proper working condition i.e. 1-out-of-2: G policy, and the subsystem-N works successfully if the lonely unit is in proper working condition i.e. 1-out-of-1: G policy. 3. As soon as repair of a unit in all of the three subsystems completed, it again becomes operational (as good as new). No damage reported due to repair of the system. 4. Whenever there is a failure in two units of subsystem-L and one unit in subsystem-M, the system goes to perilous state where system has to stop functioning deliberately to avoid further failures with emergency failure rate E  . 5. There may be unpredictable catastrophic failure to the system at any time (t). 6. One repairperson is available full time with the system and may be called as soon as the system reaches to partially or completely failed state. 7. All failure rates are constant and follows the exponential distribution. 8. The failure rate and repair rate in all the three subsystems is same unit wise, while different subsystem wise. 9. The complete failed system needs repair immediately. For this Gumbel-Hougaard, family of copula can be employed to restore the system. Copula A d-dimensional copula is a distribution function on [0, 1] with standard uniform marginal distributions. Let C(u) = C (u1, ...,ud) be the distribution functions which are copulas. Hence C is a mapping of the form C: [0, 1] → [0, 1], i.e. a mapping of the unit hypercube into the unit interval. The following three properties must hold: (i) C(u1, ..., ud) is increasing in each component ui. (ii) C(1, ...1, ui, 1, ..., 1) = ui for all i {1, ...d}, ui [0, 1]. (iii) For all (a1,..., ad), (b1,..., bd)  [0, 1] with ai ≤ bi we have: Where uj1 = aj and uj2 = bj for all j  {1, ..., d}. The copulas are multivariate distribution functions whose one-dimensional margins are uniform on the interval [0, 1]. The copula (joint probability distribution) approach is very natural when a complex system repaired in a couple of ways. For θ=1 the GumbelHougaard copula 1/ 1 2 1 2 ( , ) exp( (( log ) ( log ) ) , C u u u u          1     , θ=1 the GumbelHougaard copula models become independence, and for θ→∞ it converges monotonically. Although the different copulas have employed by 38 / IJRRS / Vol. 3/ Issue 1/ 2020 P. K. Poonia and A. Sirohi various researchers due to simplicity conventional purpose GumbelHougaard family copula have employed to assessing analytical cases of the paper. System Configuration and Transition Diagram System configuration shown in Fig 1 (a) while transition diagram in Fig 1 (b). HereS0 is perfect state, S1, S2, S3, S4 and S5 partial failed/degraded and S6, S7, S8, S9, SE and SC are complete failed states. Due to failure of unit (s) in the subsystem-L, M or/and N, the transitions approaches to partially failed states S1, S2, S3 S4 and S5. The state S6, S7, S8 and S9 are complete failed states due to failure of units in all the subsystems, while SE is completely failed state due to deliberate failure. The states SC is complete failed state due to catastrophic failure. Fig. 1(a) System configuration Fig. 1(b) State transition diagram of the model In the transition diagram above S0 is a state where all the subsystems are in good working condition. S1, S2, S3, S4 and S5 are the states where the system is in partially failure mode/ degraded, and the general repair is employed, states S6, S7, S8, S9, SE and SC are the states where the system is in the totally failure mode. Repair is being applied using Gumbel-Hougaard family copula distribution. Table 1. State Description State Description S0 This is a perfect state and all units of subsystem-L, M and N are in proper working condition. S1 The indicated state is degraded but is in operational State Description mode after the failure of the one unit in subsystemL. All units of subsystem-M and N are in the proper operational state. The system is under general repair. S2 The indicated state is degraded but is in operational mode after the failure of two units in subsystem-L. All units of subsystem-M and N are in the proper operational state. The system is under general repair. S3 The indicated state is degraded but is in operational mode after the failure of the one unit in subsystemM. All units of subsystem-L and N are in the proper operational state. The system is under general repair. S4 The indicated state is degraded but is in operational mode after the failure of the one unit in subsystemL and one unit in subsystem-M. All units of subsystem-N are in the proper operational state. The system is under general repair. S5 The indicated state is degraded but is in operational mode after the failure of two units in subsystem-L and one unit in subsystem-M. All units of subsystem-N are in the proper operational state. The system is under general repair. S6, S7 S8, S9 SE, SC The states represent that the system is in completely failure mode and the system is under repair using Gumbel-Hougaard family copula distribution. Formulation of mathematical model By probability of considerations and continuity arguments, we can obtain the following set of difference-differential equations associated with the present mathematical model (see Appendix-1):

  0 x  An expression of the joint probability from failed state S i to good state S 0 according to Gumbel-Hougaard family copula is given as

Introduction
Determining accurate reliability and availability of an existing structure or product is a crucial task in the reliability engineering. In case of failure, money and time will be wasted and even disaster may occur. In order to achieve reliable system functioning, components are designed to be highly reliable in the sense that they rarely suffer from sudden failures. Nevertheless, components might degrade gradually with usage. Redundant strategy is often used by engineers to ensure the reliability and availability of the systems and/or to improve these characteristics of the systems. Thus, variety of standby systems have been designed and analyzed during the last few decades. The main objective of these studies is to develop methods and tools for evaluation and demonstration of the reliability, availability, and cost analysis. Redundant systems, which have been widely used in practice, such as space shuttles, communication satellites, a hybrid car, Nuclear reactors, or a fighter plane, are frequently discussed in research literature. Initially, redundant parts are designed to improve the reliability of the system, meaning that some additional paths are created or identical components connected in such a way that when one component fails the others will keep the system functioning. It is a technique commonly used to improve system reliability and availability. Redundancies can be categorized into the following: (i) Cold standby in which the standby unit is only called upon when the primary or operating unit fails. These inactive components have a zero failure rate and cannot fail while in standby state; (ii) Hot standby in which the standby unit has the same failure rate as when it is run with the operating unit; (iii) Warm standby in which the standby unit runs in the background of operating unit. It can fail in this state but its failure rate is less than that of the operating unit. Moreover, redundancy is highly cost effective in achieving a certain reliability level of the system. Therefore, in order to enhance reliability k-out-of-n system structure in which at least k components of n must be functioned. In order to improve the reliability of k-out-of-n systems, numerous researches have presented their works and contributions by constructing different types of complex repairable systems under the different types of failure and repair distributions. For instance, authors consider warm standby system by She and Pecht [1], generalized multi state system by Huang et al. [2], repairable consecutive systems with r repairman by Wu and Guan [3], two-stage weighted systems with components in common by Chen and Yang [4], main unit with helping unit by Kumar and Gupta [5], Markov repairable system with neglected or delayed failures by Bao and Cui [6], evaluated exact reliability formula for consecutive repairable systems by Liang et al. [7], general system with non-identical components considering shut-off rules using quasi-birth-death process by Moghaddass et al. [8], and generalized block replacement policy with respect to a threshold number of failed components and risk costs by Park and Pham [9]. The occurrence of failure in any complex repairable or non-repairable engineering system is a natural phenomenon, which arises due to the different working conditions. The k-out-of-n effective policy plays a crucial role in maintaining the reliability of repairable systems. The researchers have focused on evaluating reliability and availability of the redundant repairable systems like k-out-of-n in series configuration. In particular, Singh et al. [10] analyzed an engineering system, which consists of two subsystems, viz. subsystem-1 and subsystem-2 with controllers in series. Subsystem-1 works under the k-out-of-n: good policy. Subsystem-2 consists of three identical units in parallel configuration. In this case, controllers control the working of both subsystems. Authors evaluated reliability characteristics using supplementary variable technique. Ram et al. [11] investigated the reliability of a standby system incorporating waiting time to repair. In this case, system consists of two units' namely main unit and standby unit. Whenever the main unit fails, the whole load is transferred to the standby unit instantaneously by a switching-over device. As regards to the repairing of the main unit, it has to wait for repair whenever it fails due to unavailability of repair facility. Munjal and Singh [12] analyzed a complex repairable system composed of two 2-out-of-3: G subsystems connected in parallel. Jia et al. [13] studied repairable multistate two-unit series systems when repair time can be neglected. Goyal et al. [14] studied the sensitivity analysis of a three-unit series system under k-out-of-n redundancy. Singh et al. [15] developed a model of a complex repairable system having two subsystems in series configuration. Both subsystems includes two units in parallel, and it is assumed to work till at least one unit of both the subsystems are in good operative condition. Gahlot et al. [16] assessed a repairable system in series configuration under different types of failure and repair policies using copula linguistics. Singh and Poonia [17] assessed 1-out-of-2: G system with correlated lifetimes under inspection.
Some specific papers related to this paper are as follows. Lado and Singh [18] analyzed an engineering system, which consists of two subsystems in series configuration operated by a human operator. Both the subsystems have two units in parallel. In this papers authors proved that copula repair is more reliable than general repair. Pundir and Patawa [19] studied repairable two dissimilar units' cold standby system waiting for repair facility after failure of system units. They stimulated exponential failures, arbitrary waiting and arbitrary repair rate. Singh et al. [20] studied two subsystems in series configuration with imperfect switch connected with both subsystems. Recently, Zhao et al. [21] and Singh et al. [22] studied some real system problems related to our study.

System Description
Researchers around the world have presented their research works on reliability analysis of complex repairable system however they have not focused on the study of the system consisting of three subsystems connected in series configuration with catastrophic failure. Catastrophic failure is a complete, sudden, often unexpected breakdown in the entire system. Such a break down may occur due to animal related disruption or change in environment related conditions like Corona virus nowadays. Sometimes a single component in a critical location fails; resulting in downtime for the entire system also comes under catastrophic failure. The term catastrophic failure is most commonly used for organizational failures, but has often been extended to many other disciplines in which total and irrecoverable loss occurs. Treating the above realities in the present study, the model consisting three subsystems in series configuration considering catastrophic failure. The subsystem-L has three identical units, subsystem-M has two identical units and subsystem-N has one unit only. The subsystem-L is working under 1-out-of-3: G; scheme, the subsystem-M is working under 1-out-of-2: G; scheme, however, the subsystem-N works under 1-outof-1: G; scheme. The catastrophic failure is treated as a complete failing state. During operation, the system will be in any of the three states: perfect operation, partial failure, and complete failure. The failure rates of units of subsystems are constant and assumed to follow the exponential distribution, but their repair supports two types of distribution namely general distribution and Gumbel-Hougaard family copula distribution. Then, based on the behavior of the whole system, all the system states can also be classified into three subsets as follows.
Classification I: The system operates perfectly; in this situation, all the components in both subsystems are in the perfect functioning state.
Classification II: The system is partially working; in this situation, at least one component in one or both subsystems is in the failure state, and the remainder is perfectly functioning.
Classification III: The system is completely failed; in this situation, either subsystem L, M or N is in the complete failure state. Further, system may be completely failed due to catastrophic failure. Therefore, the system remains working until one of the subsystems is completely failed. Based on the above-mentioned assumptions, the system could be modeled by a continuous-time stochastic process. The present study accomplished two objectives using supplementary variable technique. First the expressions for the reliability of the system, availability of the system, mean time to failure and profit function are obtained. Second numerical simulation with respect to profit function is performed. Explicit expressions for reliability, availability, MTTF, and cost analysis functions are obtained with help of MAPLE (software). Tables and graphs present a comparative analysis of results. The system configuration and transition state diagram of the designed model are shown in fig. 1(a) and 1(b) respectively.

Assumptions
The following assumptions are made through this paper: 5. There may be unpredictable catastrophic failure to the system at any time (t). 6. One repairperson is available full time with the system and may be called as soon as the system reaches to partially or completely failed state. 7. All failure rates are constant and follows the exponential distribution. 8. The failure rate and repair rate in all the three subsystems is same unit wise, while different subsystem wise. 9. The complete failed system needs repair immediately. For this Gumbel-Hougaard, family of copula can be employed to restore the system.

Copula
A d-dimensional copula is a distribution function on [0, 1] d with standard uniform marginal distributions. Let C(u) = C (u 1 , ...,u d ) be the distribution functions which are copulas. Hence C is a mapping of the form C: e. a mapping of the unit hypercube into the unit interval. The following three properties must hold: The copulas are multivariate distribution functions whose one-dimensional margins are uniform on the interval [0, 1]. The copula (joint probability distribution) approach is very natural when a complex system repaired in a couple of ways. For θ=1 the Gumbel-Hougaard copula the Gumbel-Hougaard copula models become independence, and for θ→∞ it converges monotonically. Although the different copulas have employed by various researchers due to simplicity conventional purpose Gumbel-Hougaard family copula have employed to assessing analytical cases of the paper.

System Configuration and Transition Diagram
System configuration shown in Fig 1 (a) while transition diagram in Fig 1 (b). HereS 0 is perfect state, S 1 , S 2 , S 3 , S 4 and S 5 partial failed/degraded and S 6 , S 7 , S 8 , S 9 , S E and S C are complete failed states. Due to failure of unit (s) in the subsystem-L, M or/and N, the transitions approaches to partially failed states S 1 , S 2 , S 3 S 4 and S 5 . The state S 6 , S 7 , S 8 and S 9 are complete failed states due to failure of units in all the subsystems, while S E is completely failed state due to deliberate failure. The states S C is complete failed state due to catastrophic failure. In the transition diagram above S 0 is a state where all the subsystems are in good working condition. S 1 , S 2 , S 3 , S 4 and S 5 are the states where the system is in partially failure mode/ degraded, and the general repair is employed, states S 6 , S 7 , S 8 , S 9 , S E and S C are the states where the system is in the totally failure mode. Repair is being applied using Gumbel-Hougaard family copula distribution.  The states represent that the system is in completely failure mode and the system is under repair using Gumbel-Hougaard family copula distribution.

Formulation of mathematical model
By probability of considerations and continuity arguments, we can obtain the following set of difference-differential equations associated with the present mathematical model (see Appendix-1):

Solution of the model
Taking Laplace transformation of equations (1) to (18) and using equation (19), we obtain  

Availability Analysis
When repair follows general and Gumbel-Hougaard family copula distribution, we have

Reliability Analysis
Taking all repair rates equal to zero and obtain inverse Laplace transform, we get an expression for the reliability of the system after taking the failure rates as Taking all repair rate to zero and the limit as s tends to zero in (54) for the exponential distribution; we can obtain the MTTF as:

Cost Analysis
For the assumed failure and repair rates in section 6.1 and corresponding to the state transition diagram, we have computed the incurred profit for two cases when the system follows copula repair and general repair in (62 a) & (62 b). Let the service facility be always available, then expected profit during the interval

Conclusion
This paper studies the reliability characteristics of a complex repairable standby system consisting of three subsystems in series configuration under catastrophic failure. First Subsystem-L is composed of three identical units in parallel configuration working under 1-out-of-3: G policy, second subsystem-M has two identical units working under 1-out-of-2: G: policy, while the third subsystem have one unit that working under a-out-of-a: G policy. Explicit expressions have been derived using supplementary variable technique. Warm standby redundancy has been used as an effective technique for improving the reliability of system design. Table 2 and Figure 2 give the analysis of availability of the system in three different possibilities. One can clearly observe that availability of the system initially decreases with respect to time and later on it seems to be constant as the time increases. Table-3 and  figure 3 give information for reliability of the system at different values of time. The graph showing a steep fall in reliability from the top to the lowermost in a very short period based on the failure rate of units. From table-2 and 3, one can observe that corresponding values of availability are greater than the values of reliability, which highlights the requirement of systematic repair for any complex systems for desirable performance. Additionally, availability is more in case (a) as compared to other cases that indicates that copula repair is far better than general repair. Table 4 and figure 4 yield the MTTF of the system with respect to variation in failure rate 1 2 3 E , , , and C      respectively, when other parameters were kept constant. MTTF of the system is decreasing concerning different failure rates. MTTF of the system is the highest for the failure rate of subsystem-3 and is the lowest concerning the catastrophic failure that indicates subsystem-3 is responsible for proper operation of the system. The MTTF in case of deliberate failure is almost the same for all E  . An acute examination from table-5 and 6 and figure-5 and 6 reveals that expected profit increases as service cost K 2 decreases, while the revenue cost per unit time is fixed at K 1 =1 in case of both copula and general repair. The calculated expected profit is maximum for K 2 = 0.1 and minimum for K 2 =0.6. We observe that as service cost decreases, profit increase with variation of time. In general, for low service cost, the expected profit is high in comparison to high service cost. Conclusively, copula repair is more effective repair policy for better performance of repairable systems.