PERFORMANCE ANALYSIS OF A COMPLEX REPAIRABLE SYSTEM WITH TWO SUBSYSTEMS IN SERIES CONFIGURATION WITH AN IMPERFECT SWITCH

This paper presents the study of reliability measures of a complex system consisting of two subsystems, subsystem-1, and subsystem-2, in a series configuration with switching device. The subsystem-1 has five units that are working under 2-out-of-5: G policy and the subsystem-2 has two units that are working under 1-out-of-2: G policy. Moreover, the switching device in the system is unreliable, and as long as the switch fails, the whole system fails immediately. Failure rates of units of subsystems are constant and assumed to follow the exponential distribution. Still, their repair supports two types of distribution, namely general distribution and Gumbel-Hougaard family copula distribution. Using the supplementary variable technique, Laplace transformations, and copula approach differential equations developed. Important reliability characteristics such as availability of the system, reliability of the system, MTTF, profit analysis, and sensitivity analysis for MTTF have computed for fixed values of failure and repair rates. Particular cases corresponding to the switching device have also considered. Graphs demonstrate results, and consequently, conclusions have done.


INTRODUCTION
Complicated systems such as computers, automobiles industry, telephone networks, and various electronic networks are becoming a prevalent feature and essential requirements of our society.
The systems are built with multiple components/ parts to perform specified tasks adequately. It is often difficult to assure that the systems will perform particular tasks efficiently for which they designed. Due to various causes, it is difficult to anticipate the failure of a component and sometimes impossible to prevent the failure of the entire system. Reliability is a vital need for proper uses and repair of any engineering system. Achieving a high or required level of reliability and availability of the system is often an essential requisite based on system designed structure.
The importance and utility of a system depend on its successful performance, and its performance depends on its design. The availability and reliability of an industrial system may be enhancing using a highly reliable structural design of the system or subsystem of higher reliability. The best way to improve system reliability is to add redundant components in the design. A constructive and common form of redundancy is a k-out-of-n configuration. Many researchers have brought their attention to the study of k-out-of-n: G systems and k-out-of-n: F systems. The k-out-of-n: G system is good if and only if at least k of its n components is good, while k-out-of-n: F system fails if and only if at least k of its n components fails. For example, an airplane with four engines can be modeled as a 3-out-of-4: G system. Furthermore, consider a large truck with ten tires is an example of 6-out-of-10: G system. Although the system performance may be degraded if less than ten tires are operational, rearrangement of the tire configuration will result in adequate performance as long as at least six tires are operational. In nuclear power plant system 2-out-of-4: G; system can perform adequate power supply. Conclusively a k-out-of-n system plays a very crucial role in system reliability theory to the proper operation of the system.
In the past decade, many researchers have focused on k-out-of-n-type systems mainly because such systems are more general than pure parallel or pure series systems, and they frequently come across in practice. There is an extensive literature available for reliability analysis of k-out-of-n-type systems under various situation such as [10], repairable systems with different failure modes [25], three-unit series system under warm standby [15], consecutive k-out-of-n system using standby with multiple working vacations [19], generalized block replacement policy with respect to a threshold number of failed components and risk costs [13], non-identical components subject to repair priorities [1] and non-identical components considering shut-off rules using quasi-birth-death process [18] among others. There are some real systems such as 361 PERFORMANCE ANALYSIS OF A COMPLEX REPAIRABLE SYSTEM satellites, transmission systems, or computer systems where some new equipment groups need to add because of the requirement for better output of the system. Realizing this fact, authors like Alka and Singh [3] analyzed reliability analysis of a complex repairable system composed of two 2-out-of-3: G subsystems connected in a parallel configuration. They analyzed the system by using the supplementary variable technique and obtain various measures such as mean time to failure, steady-state probability, availability, and cost analysis. Yusuf et al. [7] focus on the comparative study of 2-out-of-3: G system for the different situations under the concept of general repair analyzed using Kolmogorov's forward equations method. The objective of this study is to see the effect of preventive maintenance and system design of 2-out-of-3. In addition, Yusuf et al. [8,9] developed an explicit expression for mean time to system failure for a 3-out-of-5 warm standby system involving common cause failure and ensured the maximum overall MTSF of the system.
Considering one type of repair/general repair to a totally failed system may cause a massive loss due to the non-operation of the system, and the industry/organization may drop its market reputation. Several authors, including El-Said and EL-Sherbeny [11], Bulama et al. [12], Gupta et al. [16,17] and Malik et al. [20] examined the reliability characteristics under the presumption that the failed unit can be repaired by employing only one type of repair. There are many situations in real life where more than one repair is possible between two adjacent transition states for quick repair of the failed system. When such type of possibility exists, the system is repaired using the Gumbel-Hougaard family copula; it couples the two distributions, namely general distribution and exponential distribution. Therefore, in contrast to this, authors have considered models in which they tried to address a problem where two different repair facilities are available between adjacent states, i.e., the initial state and totally failed state. Ram and Singh [14] have studied availability and cost analysis of a parallel redundant complex system with two types of failure under preemptiveresume repair discipline using the Gumbel-Hougaard family copula in repair. Singh et al. [21] have studied cost analysis of an engineering system involving two subsystems in a series configuration with controllers and human failure under the concept of k-out-of-n: G policy using Gumbel-Hougaard family copula distribution. Also, in [22,23], Singh et al. have studied the performance analysis of the complex system in the series configuration under different failure and repair disciplines using copula and controllers. Bona et al. [5] have discussed the reliability allocation based integrated factor method (IFM) approach to the aerospace system. The consequence of the study of reliability allocation method Di, Bona, Forcina, A, and Silvestri, A, [6] have proposed a new reliability allocation method as a critical flow method (CFM) for the 362 VIJAY VIR SINGH, PRAVEEN KUMAR POONIA AND AMEER HASSAN ADBULLAHI thermonuclear system. Recently Lado et al. [2] analyzed two subsystems connected in a series configuration and operated by a human operator. In this study, they concluded that copula repair is more reliable compared to general repair. Also, Babu et al. [4] studied a δ-shock maintenance model for a deteriorating system with an imperfect delayed repair under partial process. In addition, Singh and Poonia [24] studied two units parallel system with correlated lifetime under inspection using regenerative point technique.
Authors who studied k-out-of-n systems have put attention toward the operation of units in parallel/series configuration or in a circular arrangement with catastrophic failure and preventive maintenance but did not consider transfer switch and its failure. Therefore, realizing the fact and necessity of such type of configuration, we in the present analyzing a complex system having two subsystems viz. subsystem-1 and subsystem-2 under k-out-of-n: G configuration. Both subsystems connected in series, and each linked with a switching device for the proper functioning of the system, which may be perfect or imperfect at the time of need. The subsystem-1 follows 2-out-of-5: good configuration, and subsystem-2 follows 1-out-of-2: good configuration. All the units in both the subsystems are in a parallel configuration. The system has three possible transition states: Good, partially failed and complete failed. The system may move to the failed state as per the following options: (iii) The switching device of the subsystem-1 / subsystem-2 fails.
In addition to this, the system will be in a partially failed state in the following situations: To carry out performance analysis, the authors have evaluated the expressions for the availability of the system, reliability of the system, MTTF, profit analysis, and sensitivity analysis corresponding to MTTF using the supplementary variable technique. All the results analyzed in the model using mapple17. This paper is planned in various sections. Section 1 describes the brief introduction of the paper, which focuses on the relevant literature reviewed for the study of 363 PERFORMANCE ANALYSIS OF A COMPLEX REPAIRABLE SYSTEM the proposed design. Section 2 to 6 covers the state description, assumptions, nomenclature of notation used for the study of a mathematical model, and transition diagram. Section 7 and 8 cover the analytical part of the paper in which some particular cases are taken for discussion and elaboration. Section 8 describes the conclusion of the study with results.

STATE DESCRIPTION
The description of the various possible state of the model after failing the units in both the subsystems, including transfer switch failure, is given in Table 1. The states {S0, S1, S2, S3, and S5} are operative states, and {S4, S6, S7, and S8} are inoperative states of the system. This is a perfect state, and all units of subsystem-1 and subsystem-2 are in good working condition.

S1
The indicated state represents that the system is degraded but is in operational mode after the failure of any one unit in subsystem-1, but both units of subsystem-2 are in a good operational state. The system is under repair.

S2
The indicated state represents that the system is degraded but is in operational mode after the failure of any two units in subsystem-1, but both units of subsystem-2 are in a good operational state. The system is under repair.

S3
The indicated state represents that the system is degraded but is in operational mode after the failure of any three units in subsystem-1. Still, both units of subsystem-2 are in a good operational state. The system is under repair.

S5
The indicated state represents that the system is degraded but is in operational mode after the failure of anyone unit in subsystem-2, but all the units of subsystem-1 are in a good operational state. The system is under repair.

S4
The states represent that the system is in totally failed mode after failing more than three units in the subsystem 1. The system is under repair using the Gumbel-Hougaard family copula distribution.

S6
The states represent that the system is in a complete failed state after failing both units in subsystem 2. The system is under repair using employing copula distribution.

S7
It is a complete failed state due to switch failure in the subsystem-1.

S8
The state S8 represents a complete failed state by failing switch device in subsystem-2.

ASSUMPTIONS
The following assumptions have been made throughout the study of the model: 1. Initially, the system is in the state 0 S , and all the units of subsystem-1 and subsystem-2 are in good working conditions.
2. The subsystem-1 works successfully until three or more than three units are in good working condition, i.e., 2-out-of-5:G policy.
3. The subsystem-2 works successfully if one or both units are in good working condition, i.e., 1-out-of-2:G policy.
4. Both the subsystems having switching devices, which may be unreliable at the time as long as the switch fails, the whole system fails immediately.
5. The units in both the subsystems are in parallel mode and hot standby and ready to start within a negligible time after the failure of any unit in the subsystems.
6. Repairperson is available to full time with the system and maybe called as soon as the system reaches to partially or totally failed state.
7. All failure rates are constant and follow the exponential distribution.
8. The complete failed system needs repair immediately. For this, copula can be employed to restore the system. 9. No damage reported due to the repair of the system. 10. As soon as the failed unit repaired, it is ready to perform the task as good as new.
The state transition probability that the system is in state i S at an instant 0 i = .

Ps
Laplace transformation of the state transition probability ( ) The probability that the system is in the state i S for 1 to 8 i = and the system is under repair with elapsed repair time is , xt.
x is repaired variable and t is time variable.

SYSTEM CONFIGURATION AND STATE TRANSITION DIAGRAM
System configuration is shown in Fig 1 (a) while the transition diagram in Fig 1 (b). In transition diagram, S0 is perfect state, S1, S2, S3, and S5 partial failed/degraded and S4, S6, S7, and S8 are complete failed states. Due to failure in any unit in the subsystem-1 and in subsystem-2, the transitions approach to partially failed states S1, S2, S3, and S5, respectively. The state S4 and S6 are complete failed states due to the failure of units in both the subsystems. The states S7 and S8 are complete failed states due to transfer switch failure.

FORMULATION OF THE MODEL
By a probability of considerations and continuity arguments, we can obtain the following set of difference-differential equations associated with the present mathematical model:

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

Reliability Analysis
In order to obtain system reliability, consider repair rates equal to zero. Like availability, the same three cases are discussed here.

Mean Time to Failure (MTTF)
Taking all repair rate to zero and the limit as s tends to zero in (53) for the exponential distribution; we can obtain the MTTF as:

Cost Analysis
Let the service facility be always available, then expected profit during For the same set of parameters defined in (53), one can obtain (63). Therefore,

Sensitivity Analysis corresponding to MTTF
The sensitivity in MTTF of the system can be studied through the partial differentiation of MTTF with respect to the failure rates of the system. By applying the set of parameters as

CONCLUSION
In this paper, the reliability analysis of a complex system consisting of two subsystems, subsystem-1 and subsystem-2 in a series configuration with the switching device, is studied. The subsystems have five and two units, respectively. Furthermore, the switching device in the system is unreliable, and the function of the switch is: "as long as the switch fails, the whole system fails immediately." Using the supplementary variable technique and the Laplace transform various measures like availability of the system, reliability of the system, MTTF, profit analysis, and sensitivity analysis for MTTF are derived in this model.    When revenue cost per unit time fixed at 1 1 K = and service costs at 2 0.6,0.5,0.4,0.3, K = 0.2 and 0.1, the expected profit has been calculated (Table 5), and the results are demonstrated by the graph (Figure 5). It reveals that expected profit increased as time increased for lower values of 2 K while it is decreased for higher values of service cost. Thus, for low service costs,