ANALYSIS OF SERIES–PARALLEL SYSTEM’S SENSITIVITY IN CONTEXT OF COMPONENTS FAILURES

. This research examines the reliability characteristics of a broad series–parallel system. Complex system, having elements in series and parallel configuration is considered here to compute the reliability. Due to the failure of component, the system can either breakdown or can work with reduced frequency. To analyze the effect of component failure, sensitivity of the system with respect to reliability parameters is analyzed. In this study, generally distributed and exponentially distributed repair rates are taken into consideration from failed state to the working state of the system. The mathematical model of the designed structure is developed using Markov process and supplementary variable technique. The comparative study of the system availability and its sensitivity with respect to general distribution and general as well as exponential distribution has been examined. The Gumbel– Hougaard family of copula is used to analyze the effect of both the distributions together. For a better explanation of this work, a numerical example has been provided and shown the results graphically.


Introduction
In today's modern environment, everyone depends on the continuous operation of complicated systems/equipment. When purchasing equipment or working on a system, the first thing a person wants to know is how reliable it is, or how well it works or acts in the way that person expects it to. Reliability is therefore most prevalent crucial attributes of components, goods, and complicated systems. Leitch [17] defined "reliability" as a transient but desirable characteristic of a product or service that is typically assessed quite subjectively in everyday life. Being able to quantify reliability is essential because for an engineer, it has important financial and potentially even safety implications. Regarding both qualitative and quantitative research, Roberts and Priests [26] have shown validity and reliability. Both the integrity of research methodologies and the dependability of research outcomes are demonstrated and communicated using the terms reliability and validity. According to Mohajan [22], the reliability of a measuring device relates to one's confidence in the data obtained using the device or the degree to which any measuring equipment corrects for random mistake. The validity of a measuring device is concerned with what it measures and how well it does so.
Concerns about issue characteristics and methods of solution continue to emphasize on the reliability of a series-parallel system. Also, in pursuance of improving these systems, several researches have been carried out. Juang et al. [13] explained a series-parallel system comprehensively. In their opinion, a series-parallel system includes subsystems that have many components connected in series first, then in parallel, or subsystems that have many components connected in series first, then in parallel. A system and its functions can be defined primarily by its components, the logical connections between those components (series, parallel, etc.), and its functions. According to Schorr [27], there are primarily two categories of systems: those that cannot be repaired and those that can. A system is considered unrepairable if it cannot be repaired to carry out its specialized duties after failing. This does not always indicate that it can never be fixed. It simply means that a system failure when it is in use has consequences that are essentially irreversible. In contrast, a system is deemed repairable if, after malfunctioning, it can be rectified by swapping out the damaged components for new ones or being restored in some other way. The definition of a repairable system, according to Tyagi et al. [29], is one that can be updated to keep performing its duties.
A reliability allocation challenge was given by Yalaoui et al. [33] in a series-parallel system taking into account the Tillman and Truelove functions and examined the performances using different cost functions. In order to examine a repairable parallel-series multi-state system, Bisht and Singh [1] introduced an interval universal generating function (IUGF) technique and employed the Markov process to assess the probability of various components. Tian et al. [28] and Levitin et al. [19] also examined reliability and its performance for multi-state series-parallel systems. An issue of a series-parallel system's bi-objective reliability and cost was identified by Garg [4] using a collaborative approach. He then used the PSO (Particle Swarm Optimization) technique to resolve the issue, and then the genetic approach was used. PSO is based on communication and interaction, where individuals (particles) within the population (swarm) share information with one another. This technique attempts to improve a solution iteratively with respect to a specified metric of quality in order to optimize the problem. Beginning with a swarm of particles whose positions are initial solutions and whose velocities are randomly given in the search area, PSO describes the process. Nourelfath and Nahas [23] have effectively addressed a problem of series-parallel structural redundancy optimization utilizing multi-state models. Kumar and Kumar [14] used the Laplace transformation to resolve the system's Markov model, and looked at reliability metrics of the two parallel-operating tripod turnstile machines. The supplementary variable approach and the Laplace transformation were used by Yuan and Meng [36], Manglik and Ram [21], and Ram et al. [25] to overcome challenges in their individual investigations on reliability characteristics and steady-state system indices utilizing the Markov process. A comparison of the Universal Generating Function (UGF) & the recursive methods, C-K theorem and differential equations was performed by Guilani et al. [8] for the evaluation of the reliability of non-repairable systems using Markov model and found that the beta distribution is the same as the distribution of time to failure. Tyagi et al. [30] taken into account sensitivity analysis and reliability modelling of an Internet of Things (IoT) based Flood Alerting System (FAS) and used Markov process to obtain its state transition probabilities which were solved using Laplace transformation. By merging the structural functions of Markov processes and s-coherent multi-state system, Xue and Yang [32] were able to determine the parameters for two-state reliability and examine the reliability of multi-state systems. By using two Markov models with standby, low coverage, reboots, and common cause failure, Jain and Kumar [11] projected the development of a repairable fault-tolerant system. Model II, which included certain realistic elements, was used to study a two-unit system, whereas Model I in this work was made up of a functioning unit and a backup unit. They finally arrived at the explicit words for the various indices after some time. Additionally, Yang and Tsao [34] took into account a matrix-analytic method to determine the availability and reliability of standby systems with working vacations. They conducted a sensitivity analysis using Laplace transformation to evaluate the MTTF and the reliability function, and the results show that increasing the number of spare parts and maintenance rates can increase system reliability. Gopalan and Venkatachalam [5] computed the reliability and availability of a two-unit, two-server system that has undergone preventative maintenance and repair. They also derived explicit Laplace transforms of mean downtime and mean time to failure of the system. Additionally, a precise equation for the steady-state availability is found. For a repairable system with several vacations and incomplete fault coverage, Jain and Gupta [10] devised the ideal replacement strategy and used the Laplace transformation and supplementary variable technique to calculate the reliability indices.
The desire to come up with fresh, effective solutions to the issues presented is what gave rise to the current work. The primary goal of this work is to demonstrate how to assess the reliability of complex systems. This is how the paper is organized: Section 2 briefly describes the model's specifics, including the system description, state description and notations. The formulation of mathematical model is covered in Section 3. Some significant measures, including availability, MTTF, reliability and cost analysis are discussed in Section 4. In Section 5, the sensitivity analysis of certain significant indices, such as availability, MTTF and reliability is presented. In Sections 6 and 7, respectively, the results discussion and conclusion are offered.

System description
An examination of a general series-parallel system's reliability is presented in this work. The system is made up of two subsystems, and , which are linked in series, sub-system having two components 1 and 2 that arranged in parallel combination while sub-system having two components, 1 and 2 connected in series. Further, they both are connected to one another component, 3 in parallel. Home solar power systems and water treatment plants are real-world examples of this type of system. The system configuration is shown in the Figure 1. With the aid of the Markov process and the supplementary variable technique, the entire system is mathematically modelled. For complex systems, it is quite complicated to portray them in the right way. According to Ram [24], the working of any complex system is done in three states: good state, partially failed (degraded) state, and completely failed state. It is assumed that a system's failure can be either partial or complete. A partial failure can result from an internal component failing, which renders the system less effective, and a complete failure can cause the breakdown of the whole system [12,31].

Assumptions
(i) Initially we assumed that the system is in good working condition. (ii) After repair, the system works like a new one. (iii) It is assumed that components 1 and 2 have same failure rate ( ) and in the same way components  (v) The repair rate follows two types of distribution i.e. general distribution and exponential distribution. (vi) 100% revenue is available.

State description
The various states as shown in Figure 2 are described in Table 1.

Notations
The notations concerned with the work are listed in Table 2.

Mathematical model formulation
A variable whose significance is entirely determined by its current state, without reference to any past behavior, can have its value predicted using Markov analysis. According to Russian mathematician Andrei Andreyevich Markov, the state transitions are exponentially distributed, which means that the transition states are constant. The Markov process shatters and transforms into a non-Markovian process when either the failure time or the repair time are time dependent. The non-Markovian process develops into Markovian by adding a supplementary variable [2]. The hazard rate function, and essential boundary and initial conditions are required when the elapsed service time is used as a supplementary variable.
A copula known as the Gumbel-Hougaard family copula permits any particular degree of (upper) tail dependency between individual variables. It is an asymmetric Archimedean copula that exhibits more dependence in the positive tail than in the negative tail. It is given as, The Gumbel-Hougaard copula models independence at = 1, and it converges to comonotonicity at → ∞.
As a result of employing the Markov process and the mathematical model that is described, the following differential equations are produced: The necessary boundary conditions are given as, 0 (0) = 1 is the beginning condition, and at time = 0, all other state probabilities are zero.
Once we apply the Laplace transformation to equations (1) through (12) and use equation (13), we obtain, and boundary conditions, Now, solving equations (14)-(21) using equations (22)-(25), we get the transition state probabilities, Then, this is how the system's up-state probability underwent Laplace transformation: And, this is how the system's down-state probability is transformed using Laplace transformation:

Availability analysis
The availability of a system is determined by the number of failures that occur and how quickly they got rectified [15]. By setting the failure rates as = 0.03 and = 0.04 and repair rate as = 1 in (34) and using the inverse Laplace, the system's availability is given as, Also, by setting the failure rates as = 0.03 and = 0.04 and repair rate as = 2.71828 in (34) and using the inverse Laplace, the system's availability with copula is given as,  (36) and (36a), we obtain numerous availability metrics for the system in numerical form, presented in Table 3 and its graphical illustration is shown in Figure 3.

Mean time to failure (MTTF) analysis
MTTF calculates the mean time expected until the first failure of a non-repairable system. So, putting the repair rate = 0 and taking the limit tends to zero in (34), we obtain MTTF as,    Table 4, and it is graphically depicted in Figure 4

Cost analysis
The expected profit, taking into account service costs for the range [0, ], is as follows if the service facility is considered to be accessible at all times, Using equation (34) in equation (40) and by putting 1 = 1 and 2 = 0.1, 0.2, 0.3, 0.4, 0.5 respectively we obtained Table 6 and Figure 6 in that order.

Sensitivity analysis
Sensitivity analysis, in general, is a technique for determining how various independent variable values influence a certain dependent variable under a specified set of assumptions [3,7]. According to Goyal and Ram [6], a function's partial derivative with respect to a certain factor gives an indication of how sensitive the function is to that component. Here we have determined the sensitivity of availability, sensitivity of MTTF and sensitivity of reliability in relation to the assumed failure rates.

Sensitivity of availability
The inverse Laplace equation (34) is differentiated to determine the sensitivity of availability (with and without copula), and then is changed from 0 to 30 with respect to and one by one respectively, displayed in Table 7, and Figure 7 displays a graphical depiction of them.

Sensitivity of MTTF
For calculating the sensitivity of MTTF, differentiating (37) with respect to and one by one and varying them as 0.10, 0.15, 0.20, 0.25, 0.30, 0.35, 0.40, 0.45, 0.50, then we get the values for sensitivity of MTTF in relation to both the failure rates, shown in Table 8 and Figure 8 depicts it graphically.

Sensitivity of reliability
For calculating the sensitivity of reliability, differentiating (38) one by one with respect to and respectively and varying from 0 to 30 we obtained numerical figures of sensitivity of reliability displayed in Table 9 and Figure 9 shows its graphical illustration.

Result discussion
In the present work, authors have examined the reliability indices and sensitivity for a series-parallel system comprising component failures. After performing our considered methodology on the designed system, some key outcomes are obtained those are discussed as  -With the aid of Table 3 and corresponding Figure 3, authors are able to demonstrate that if the failure rates are set at varying levels, the system's availability varies with the passage of time . For the fixed values of failure rates as = 0.03 and = 0.04, the system's availability reduces with time and its probability of failure rises. After a long run it becomes constant. Figure 3 also evidently shows that the availability of the system exceeded highly after employing Gumbel-Hougaard family of copula approach for repairs. -The MTTF of the system is determined by considering variations in and as described in Table 4. The graphs of MTTF shown in Figure 4 demonstrate that the system's MTTF decreases gradually in context of both the failure rates, and .   Figure 7. Sensitivity of availability.
- Table 5 reveals that the change in time causes the change in system's reliability, and its behavior is illustrated in Figure 5. From the graph of reliability, one can see that the system's reliability decreases uniformly as time grows and after a long run, it tends to zero. - Figure 6 illustrates the analysis of expected profit with respect to service cost as well as time graphically.
From the critical examination of graphs, one can see that the system gives the profit increasingly as time passes but if service cost increases then the profit attained by the system is decreases. - Table 7 shows the sensitivity of availability of the system with and without implementing Gumbel-Hougaard family of copula approach with respect to both the failure rates i.e. and . From the critical analysis of graph one can observe that the sensitivity availability of the system with respect failure rate initially decreases till = 15 units and after that it becomes constant. Similarly, sensitivity availability with respect to failure rate first decreases till = 10 units. Then from = 10 to 25 units it increases very slightly and after that it becomes constant. Again, after employing copula, the sensitivity availability of the system with respect to both the failure rates and it initially decreases slightly and after = 10 it becomes constant.  -Critical analysis of the graphs demonstrated in Figure 8 gives that when failure rates vary from 0.1 to 0.2, the sensitivity of the MTTF with respect to and has some difference but after the value of failure rate 0.25, the sensitivity of MTTF is equal with respect to both. Also, as the failure rate increases, the sensitivity of MTTF first increases straightly and after that it increases smoothly. -The graphs of system's sensitivity of reliability with regard to both the failure rates and is shown in Figure 9. It reveals that the sensitivity of reliability decreases very smoothly as time increases. Initially, the reliability sensitivity is approximately same with respect to both the failure rates but after a short period it has a significant difference with respect to and .

Conclusion
This work carried out the analysis of reliability measures and their sensitivities of a series-parallel system consisting of two sub-systems with different configuration incorporating component failures. Laplace transformation, supplementary variable approach and Gumbel-Hougaard family of copula approach have been used in this study. A wide literature is available on series-parallel system ( [16,18,20], etc.) but they did not consider two types of repair facilities. The results of this study enable us to draw the conclusion that the system's availability and reliability deteriorate over time and increased failure rates lead to a reduction in the system's MTTF. However, comparative study shows that system's availability increases eminently after utilizing copula approach.
Through the overall study, it is concluded that the system is highly sensitive with respect to failure of subsystem . Hence, it stands to reason that lowering the system's failure rates makes it less sensitive and controlling the service cost, makes the system more profitable. The results obtained in this paper after employing copula approach helps the designers and engineers to make a highly reliable and more profitable system. In future, authors can develop mathematical model to maximize reliability and minimize cost and sensitivity for cost effective systems. The model can be developed with the aid of reliability function to optimize reliability and other characteristics of the system by using meta-heuristics.