Stochastic Analysis of Complex System with Auto Changeover Switch and Advert Environment Employing Copula Approach

This paper studies the reliability measures of a system consisting of two subsystems in a series configuration for different types of failure and two types of repair. The subsystem-1 has four identical units in a parallel configuration operating under 3-out-of-4: G policy and this has connected to subsystem-2 which has three identical units arranged in a parallel configuration and working under 2-outof-3: F, scheme. The units of subsystem-1 are controlled by a controller for preventing failure effect and safety purposes. It is assumed that units of each subsystem have different types of failure and repair rates. The unsuitability of the environmental conditions such as overheating as a well-known natural cause of failure of any system and also weather conditions like heavy rain, thunderstorm, and catastrophic shakeups, etc. have treated as a complete failure of the system. This study considers the environmental causes of failure in the proposed repairable system as complete failure by which the system stops functioning. Human failure in the system is trickled as complete failure and repair employing copula (GumbelHougaard family copula distribution) like another complete failed states of subsystems. To analyze the proposed system, the supplementary variable technique are used and some measures of system reliability like availability, reliability; MTTF and incurred profit function for different values of parameters are derived. Some particular cases for different values of failure rates that have explicit are also studied.


/ :
The failure rates of units of the subsystem-1/ failure rate of subsystem-2.

/ :
Failure rates of system due to environmental failure/ human failure.

:
Failure rate of the controller of the subsystem-1.

( ):
State transition probabilities of the system in state S 0 .

P i (x, t):
State transition probability that the system is in state S i , i=1, 2, 3…..,8 with state transition probability P i (x, t) in which the system is under repair with repair variable x, t.
C θ (u 1, The expression for joint probability distribution (failed state S i to perfect state

Introduction
Industrial systems have become very compact and more complicated due to the excessive use of automation and miniaturization. Therefore, detecting and repairing faults in various units or components used in the system, sometimes, becomes imperatively challenging and time consuming as well. Technology advancement coupled with the complexity of networks is a great option for proper performance of system. In other words, it is crucial to keep the system failure-free. Hence, scientists and engineers have developed systems configurations that can perform better than conventional systems. The quality of the performance of the systems is tested by employing probability rules and distributions for failure and repair policies. Redundancy is another suitable technique which has widely recognised to improve the reliability of the system. In the case of redundancy of the system, some additional units or paths are created together with the central unit to support the complex industrial system to improve the reliability of the system. The k-out-of-n configuration structure is another proper structure which is widely applied to improve industrial systems performance. The k-out-of-n system configuration works, if and only if at least k of the n units/ components works. Thus, the k-out-of-n system plays a significant role in operations of industries system, which have attracted the attention of researchers.
Researchers working in the field of reliability and related fields have extensively developed and investigated systems particularly redundant systems, and evaluated the performances based on reliability measures of with a significant degree of satisfaction.
Refereeing to a few works wise, M. Ram et al. (2013) examined the stochastic analysis of a redundant standby system with waiting for repair strategy. The study demonstrated the effect of waiting time to repair the system to restore to operations mode. Rekha et al. (2013) has studied different reliability parameters for a complex redundant system under head-of-line repair. Ibrahim et al. (2012) studied a redundant system with three types of failure and emphasized on the comparative analysis of different situations. In this work, they observed that the preventive maintenance of a system is far better than without preventive maintenance of the system. Singh et al. (2013) discussed a system having two units in a series configuration with a controller for availability, MTTF and cost analysis. In continuation of the study of repairable systems,  examined the reliability characteristics of the complex system consisting of two subsystems in the series configuration under human and controller failure. Singh and Ram (2014) studied the operational behaviour of a, multi-state-state k-out-of-n: G; system and analysed for 2-out-of-3: G; the system as an exceptional case for computations. Dalah and Singh (2014) examined a two-unit standby system with the concept of switch failure. Eryilmaz et al. (2011) studied signaturebased analysis of m-Consecutive-k-out-of-n: F system with exchangeable components.
The human failure plays a major role in the evaluation of repairable systems performance during installation, production, and maintenance of the complex system. A slight negligence during operations of a complex system may have a cause of significant damage, which can destroy the whole system and might be rendering for a substantial loss in the sense of safety of human life. In the context of human failure, Surbhi et al. (2013) studied the operational behaviour of primary part assembly system of an automobile incorporating human error in maintenance and also intensive on environmental failure. Occasionally ecological failure in the system can damage the whole system and stop functioning of the system instantly. Unsuitability of the environmental condition may be one of the leading causes of failure of a system.  studied availability, MTTF and cost analysis of the complex system under pre-emptive resume repair policy using copula distribution approach. Suitability of environment is essential for proper operations of a complex system. In contrast to the study of human failure in the repairable systems, Dhillon et al. (1993), Vanderperre (1990) and Ram et al. (2010) studied reliability features of the complex system with common cause failure and reliability of duplex standby system by supplementary variable technique and Laplace transform. Rawal et al. (2014) studied the functioning of internet data centre with redundant server together with the primary mail server for different failure and repair facility using copula. Singh and Rawal. (2015) have studied a complex system consisting three units as superiority, priority, and ordinary under primitive resume repair policy using two types of repair. Jyoti Gulati et al. (2016) have examined performance of a system having three subsystems in series configuration employing copula linguistic approach and have concluded that copula repair is more beneficial over general repair. Lado et al. (2018, 2019) have studied a two units series system configuration under general repair policy and copula approach and conclude that copula repair improves performance of the system. In a competitive business world warranty on the newly launched products have played a significant role. The warranty for the replacement/repair of a product during a specific period for any equipment is an important policy factor, which frequently attracts the attention of customers. During the warranty period, the particular inferior part of the system is either replaced or repaired without any extra charge, and after the expiry of the warranty period, it would be charged for repair or replacement. In this context, Ram Niwas and M. S. Kadyan (2015) studied reliability characteristics of the maintained system with warranty and degradation using supplementary variable technique D. R Cox (1995 Though the various types of copulas available in the literature, in this paper, we have applied the Gumbel-Hougaard copula for the study of the analytical part which couples two kinds of distribution functions, namely, general distribution and exponential distribution.
The present paper studies a system, which consists of two subsystems (subsystem-1 & subsystem-2) in a series configuration. In addition, essential class of faults, i.e., the environmental failure which many a time is also the cause of damage to the system is considered beside other types of failures. Hence, a mathematical model which consists of two subsystems (subsystem-1 and subsystem-2) in a series configuration is devised. In subsystem-1, four identical units are arranged in a parallel configuration which is connected with subsystem-2 which has three identical units in parallel configuration. Initially, in state S 0, both the subsystems are in good operational condition, i.e., the system is in perfect state. During operations, if any one unit of subsystem-1 fails then, the system approaches to state S 1 . Further failure of any unit in the subsystem-1 the system approaches to state S 2 and further failing any unit in the subsystem-1 it approaches to state S 3 which is a complete failed state. Failing any one unit in the subsystem-2 it will be in state S 4 and again failing second unit of the subsystem-2, the system will approach to complete failed state S 5 as per policy 2-out-of-3: F. Failure due to the adverse effect of environmental conditions, i.e., ecological failure and human failure is indicated by the states S 6 and S 7 respectively. The system is repaired by employing general time distribution in the degraded status and using copula distribution in the complete fail state. The state S 8 is a complete failed state which brings the system in un-operational state. The system is analyzed using the supplementary variable technique and various measures of reliability especially availability, MTTF, cost analysis through profit evaluations' and some other particular cases are investigated to highlight the results.
The paper has organized in the following manner as prescribed format as Nomenclature, Introduction, State description, Assumptions, and state transition diagram of the model. The mathematical modelling and solution with computational analysis as Availability, Reliability, mean time to system failure (MTTF) and profit analysis of the present model have investigated. Results discussion and concussions is last section of this analysis.

State Description of The Model
State: State Description S 0: In the state S 0, both subsystems are in good condition.
S 1: The state S 1 represents the state due to failing one unit in subsystem-1, which is a minor partial failed situation in the subsystem-1.
State S 2 represents a major partial failed state in subsystem-1.
State S 3 is a complete failed state in system due to which system stop working and it need urgent repair. A copula distribution has employed for repair the failed system.
The state represents manifestation failure in subsystem-2. The system is in degraded but working state with minor partial failure. The system is under general repair and elapsed repair time is (x, t).

S 5:
The system is the complete failed state due to failure in subsystem-2 as 2-out-of-3: F scheme. The state S 7 represents the presence of environmental failure in the system due to unfavorable environmental conditions. The system is the complete failed state. The system is in repair and elapses repair time is (x, t).

S 8:
The state S 8 represents the advent of controller failure in the system. The system is the complete failed state. The system is in repair and elapses repair time is (x, t).

Assumptions
The following assumptions have been made throughout the discussion of the model.
1. Initially, the system is inthe perfect state S 0 , and both subsystems are in good working conditions. 2. The subsystem-1 works successfully until at least 3-units of its are in good condition. 3. The subsystem-2 work successfully when two units are in good condition. When more than two units fail, it approached to a complete failed state. 4. Only one transition is allowed at a time between two adjacent transition states. 5. Both human and environmental failures bring the system in a complete failed state. 6. Controller failure of the system brings the entire system in complete failed state. 7. The partially failed/ degraded state in the system is repaired using general time distribution. 8. The complete failed states need fast repairing, and hence these states are repaired-using Gumbel-Hougaard family copula. 9. The repaired system is assumed to work like a new one and repair did not damage anything.

Profit analysis
When the system is in operational mode and the manufacture is being done and let the profit of per unit item is K 1 and the production cost per unit item is K 2 in the interval [0, t) than the net profit for the system at any time in interval [0, t) can be obtain by equation (46) Tables. (4a  & 4b) which shows the expected profit on operation of system when the system follows copula repair and general repair. Conclusively the copula repair is more beneficial and profitable over general repair.    Fig.5(b). Expected profit in interval [0, t) when general distribution
If the revenue cost per unit time K1fixed at 1, service cost K 2 = 0.50, 0.40, 0.30, 0.20, 0.10, profit has been calculated and results obtained demonstrated in Fig. 5. One can easily conclude that as the service cost decreases profit increases.