Probabilistic analysis of a system consisting of two subsystems in the series configuration under copula repair approach

Probabilistic analysis of a system consisting of two subsystems in the series configuration under copula repair approach

1. V. Singh

Department of Mathematics,

Yusuf Maitama Sule University, Kano State, Nigeria

Corresponding author: singh_vijayvir@yahoo.com

1. K. Poonia

Department of General Requirement

Sur College of Applied Sciences, Sur, Oman

Email: pkpmrt@gmail.com

Monika Gahlot

Department of Mathematics

Mewar University Chittorgarh, Rajasthan, India

Email: monikagahlot12@gmail.com

Abstract

Redundancy commonly employed to improve the system reliability of the repairable systems. In most situations, components in the standby configurations assumed to be statistically similar but independent. In many realistic models, all standby parts are not treated as identical as they have different failure possibilities. The system’s operational structure has subsystem-1 with five identical components working under 2-out-of-5: G; policy, and the subsystem-2 has two units and functioning under 1-out-of-2: G; strategy. Failure rates of units of subsystems are constant and assumed to follow an exponential distribution. The system examined using the supplementary variable and the Laplace transform.

Keywords: Repairable systems; k-out-of-n: G system; availability; MTTF; cost analysis; Gumbel-Hougaard family copula distribution.

1. Introduction

Many deteriorating repairable systems, such as aircraft, space shuttles, hydraulic control systems, nuclear plant systems, satellite projection systems, electric power generating systems, and communication systems, suffer from unavoidable failures due to complex degradation processes and environmental conditions. In practice, we come across several complex systems where an unpredicted failure of any of the parts results in the reduction of system efficiency. Redundancy is a technique extensively used to improve system performance and minimize failure effects. The identical components are aliens in such a way when any component fails, and the others will keep the system functioning. Also, redundancy is highly cost-optimistic in achieving a certain reliability level from the system. To enhance reliability and adequate performance of the k-out-of-n system type structure consisting of at least k components out of n; functioning for the system to be operational state plays a vibrant role.  In a communications system with three transmitters, the average message load can manage by at least two transmitters to be system operational at all times. Thus, the functioning of the transmission system is an example of the 2-out-of-3: G system.  Drive of  Eight-cylinder V stoke configuration engine car, only firing of four cylinders enable to move car appropriately is a case of 4-out-of- 8: G configuration. Also, k-out-of-n: G/F, repairable systems have been studied extensively in the context of computing the reliability, indices of the system.

In the past decades, several articles on reliability and other related measures of complex systems have been published by many researchers and presented significant work to improve the reliability of real-life and industrial systems. To investigate the system performance, a study of reliability measures such as availability and mean time to system failure is required. More than a few authors, including Moustafa [3], Kullstam [4], and Liang et al. [5] examined the reliability characteristics using k-out-of-n repairable systems with different failure modes. Park and Pham [6] studied the block replacement policy for the k-out-of-n system on threshold numbers of fail components and the system’s risk cost. Among them, Kumar and Gupta [7], Vanderperre [8], Mokaddis et al. [9], Kumar [10], Dhillon [11] have studied the reliability measures of considered systems by taking different types of failure and general repair employing the k-out-of-n: operation policy. The type of repair approach predicts the performance of repairable systems. The cited literature from [1] to [11] has treated the system repair can be employing general repair between two transition states. In a realistic situation, more than one repair is possible between two adjacent transition states. Such types of possibility insist the researchers repair the completely failed state employing the copula approach. B. Nelson [12]. To cite a few of them, Alka and Singh [13] analyzed the reliability analysis of a complex repairable system composes of two subsystems in a series configuration using Gumbel- Hougaard family copula repair strategy and computation have made for a structure 2-out-of-3: G subsystem. Goyal et al. [14] studied a three-unit series system under k-out-of-n redundancy and analyzed the system’s sensitivity analysis. Singh et al. [15, 16] have studied the performance analysis of the complex system in the series configuration under different failure and repair disciplines using copula concept. Lado et al. [17] have evaluated the reliability measures (Availability, reliability, and MTTF, sensitivity, and profit analysis) of the repairable complex system with two subsystems connected in a series configuration using the supplementary variable and Laplace transforms. Anuj Kumar et al. [18] present a novel method for availability analysis of an engineering system involving subsystems in series configuration incorporating waiting time to repair. Sharma and Kumar [19] analyzed availability improvement for the successive k-out-of-n machining system with multiple working vacations. The repairman can choose multiple working vacations of random length during its dormant time. El-Damcese et al. [20] analyzed availability and reliability for the r-out-of-m: G system with three types of failures using the Markov model. Ram and Singh [21] have studied availability and cost analysis of a parallel redundant complex system with two types of failure under preemptive- resume repair discipline using the Gumbel-Hougaard family copula in repair. Singh et al. [22] 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 repair approach. Xinzhuo Bao [23] examined reliability characteristics for the series Markovian repairable system by considering the system failure’s repair time as too short and long and tried to delay the failure effect. Zheng et al. [24] studied a single unit Moskov reparable system with neglecting the repair time. Authors Kumar and Ram [25] have studied sensitivity analysis of coal handling thermal power plant with two subsystems (Wagon Tripler and Conveyor) in a series configuration with one standby unit in both subsystems with different failure rates and general repair concept. Sensitivity assessment of air and refrigeration systems with four equipment (Compressor, condenser, expansion device, and evaporator) have premediated

by Nupur Goyal et al. [26]. M. El- Damcese et al. [27] have illustrated reliability and MTTF for the three-element system in series and parallel configuration employing Fuzzy failure rates. M. A. El- Damcese and N. H. El- Sodany, [28] have studied reliability and sensitivity analysis of the k-out-of-n: G warm standby parallel repairable system with replacement at common cause employing Markov model.

2 Model Description and Notations

2.1 System Description

The numerous models with the standby unit have widely studied in explore literature. Moreover, the configuration of k-out-of-n: G/F have also studied by various researchers but the structure of type k-out- of- n together with series and parallel configurations have not much attended by investigator due of the complexity of configuration. Because of our prediction, the proposed model of a complex system consisting of two subsystems 1 and 2, connected in a series configuration, has studied with a copula repair approach. Under significances, 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. In the model, each subsystem has three states: perfect operation, partial failure, and complete failure. Both the subsystems are connected by a switching device, which may or may not reliable at the time of need. The switch’s function is: as long as the switch fails, the whole system fails immediately. Failure rates of units of subsystems are constant and assumed to follow an exponential distribution. Still, their repair supports two types of distribution, namely general distribution and Gumbel-Hougaard family copula distribution. Two types of repair (general repair and copula repair) have been employed for partially failed and completely failed states. The repair rate of each unit in subsystem-1 is the same, whereas, in subsystem-2, it is different for each unit. We used the supplementary variable technique (Cox, 1; Oliveira et al., 2) and Laplace transformation technique to evaluate various characteristics like transition state probabilities, availability, reliability, MTTF, and profit analysis. Some particular cases have also been discussed for different values of failure rates. Graphs demonstrate the results, and conclusions have been drawn. The paper is organized as follows:

Section-2 introduces system description with assumptions and notations.

Sections-3 discuss the description of the states of the model.

Section-4 presents the mathematical formulation of the model, and Section-5 gives the analytical study of the model that includes availability, reliability, MTTF, and profit analysis.

Conclusions of the proposed analysis are given in Section-6.

Finally, some exceptional cases of the complex system are taken to highlight the system’s reliability characteristics. These are as follows:

• Both the subsystems have a switching device.
• Only subsystem-2 has the switching device.
• No subsystems have switching devices.

The state description of the considered system is given in Table 1, and the transition state diagram of the investigated system is shown in Figure 1.

2.2. Assumptions

Following assumptions have been considered for the study of model:

1. Initially, the system is in state S₀, and all the units are in good working conditions.
2. The subsystem-1 having five identical units and works successfully under the policy .
3. The subsystem-2 having two non-identical units and works successfully under the policy .
4. Both the subsystems are connected via a switching device, which in the system may be unreliable at the time of need. Moreover, if the switch fails, the whole system fails immediately.”
5. The units in both the subsystems are in parallel mode and warm standby and ready to start within a negligible time after the failure of any unit in the subsystems.
6. The repairman is available to full time with the system and maybe called as soon as the system reaches a wholly or partially failed state.
7. The failure rate of all the units in subsystem-1 is the same, while the units’ failure rates in subsystem-2 are different.
8. Both the subsystems, including switching devices, have constant failure rates and follow an exponential distribution.
9. The complete failed system needs repair immediately. For this, copula repair can be used to restore the system. No damage has been reported due to the repair of the system.
10. As soon as the failed unit repaired, it is ready to perform the task and new.

2.3. Notations

Time scale

Laplace transform variable

The failure rate of each unit in subsystem-1.

The failure rate of both the units A and B in subsystem-2.

The failure rate of switching devices between units for subsystem-1/subsystem-2.

Repair rate of each unit in subsystem-1.

Repair rate of each unit A and B in subsystem-2.

The state transition probability that the system is in state at an instant.

Laplace transformation of the state transition probability.

The probability that the system is in the state for and the system is under repair with elapsed repair time is .  is repaired variable and is time variable.

Expected profit in the interval.

Revenue generated and service cost per unit time, respectively.

According to the Gumbel-Hougaard family copula, an expression of the joint probability from failed state S_i to good state S₀ is given as where. Here is the parameter.

3. System configuration and state transition diagram

The state description of the model shown in table 1 that highlights S₀ is a perfect state where both the subsystems are in good working condition. S₁, S₂, S₃, S₅, and S₆ are the states where the system is in degraded mode, and general repair is employed, states S₄, S₇, S₈ and S₉ are the states where the system is in the total failure mode and repair is being applied using Gumbel-Hougaard family copula distribution. System configuration is shown in Fig 1 (a) while the state transition diagram in Fig 1 (b).

Table 1 State Description

Figure 1(a) System configuration

Figure 1 (b) State transition diagram of the model

4. Formulation of the mathematical model

By a probability of considerations and continuity arguments, we can obtain the following set of difference-differential equations:

; (1)

(2)

(3)

(4)

(5)

(6)

; (7)

Boundary conditions

(8)

(9)

(10)

(11)

(12)

(13)

(14)

(15)

(16)

Initials conditions

, and other state probabilities are zero at (17)

Solution of the model

Taking Laplace transformation of equations (1) to (16) and using equation (17), we obtain

;   (18)

(19)

(20

(21

(22

(23

; (24)

Boundary conditions

(25)

(26)

(27)

(28)

(29)

(30)

(31)

(32)

(33)

Solving all the above equations with the implications of boundary conditions, and   than with one may get Laplace transform of state transition probabilities as:

(34)

(35)

(36)

(37)

(38)

(39)

(40)

(41)

(42)

(43)

Where
The up and downstate probabilities of the system are given by

(44)

where and

(45)

5. Analytical Study

5.1 Availability Analysis

When repair follows two types of distributions as general and Gumbel-Hougaard family copula distribution, then we have

Let us choose the values of parameters as in (44), then taking inverse Laplace transform. We obtain the system’s availability as per the following three cases on switching device:

1. When both the subsystems have switching device, we get,

(46)

1. When subsystem-2 does not have a switching device, i.e., we obtain,
2. When both subsystems 1 and 2 do not have a switching device, i.e., we obtain,

We can write similar expressions for availability in case (b) and (c) using Maple. Different values of time-variable units of time may get different values, as shown in Table-2 and the corresponding Figure-2.

Table 2 Variation of availability with respect to time in various cases

Figure 2 Availability as a function of time

5.2 Reliability of the system

Taking all repair rates equal to zero and obtaining inverse Laplace transform, we get an expression for the reliability of the system after taking the failure rates (44).  Now consider the same cases as availability, we have

1. When both the subsystems have switching device, we obtain,

(47)

1. When subsystem-2 does not have a switching device, i.e., we obtain
2. When both subsystem-1 and 2 do not have a switching device, i.e., we obtain,

We can write similar expressions for reliability in case (b) and (c) using Maple. By taking different values of time-variable units of time, one may get reliability with the help of as shown in table-3 and the corresponding figure-3.

Table 3 Computed values of reliability corresponding to the different cases

Figure 3 Reliability as a function of time

5.3 Mean Time to Failure (MTTF)

Taking all repair rate and Laplace parameter s to zero in (44) for the exponential distribution, we can obtain the meantime to failure as:

(48)

where and

Now taking the values of different parameters as and varying one by one respectively as in (48), the variation in MTTF, concerning failure rates, can be obtained as per table 4 and figure 4.

Table 4 Computation of MTTF corresponding to the failure rates

Figure 4 MTTF as a function of failure rates

5.4 Cost Analysis

If the service facility is always available, then expected profit during the interval is

(49)

For the same set of parameters defined in (44), one can obtain (50). Therefore,

(50)

Settingandrespectively and varying units of time, the results for expected profit can be seen in table-5 and figure-5. 

Table 5 Profit computation for different values of time

Figure 5 Expected profit as a function of time

6. Conclusion

Warm-standby redundancy has been used as an effective technique for improving the availability and reliability of the system while attaining the equilibrium between fast repair and low process cost. Depending on the level of operation promptness of standby units, there can exist many standby modes, each categorized by altered standby maintenance and startup costs. Therefore, in this paper, we considered two subsystems in a series configuration with a switching device. The subsystems-1 have five identical units, while subsystem-2 have two non-identical units, and both are connected via switching device. The system’s availability can be seen from table-2 and figure-2 when failure rates are fixed at . It is moving down as the value of t increases in all the three cases considered based on switching devices and eventually becomes stable after a sufficiently long interval.

On the other hand, the reliability experiences a steep fall in all three cases for the same failure rates, as it is evident from table-3 and figure-3. Besides, from table-2 and table-3, it is clear that resultant values of availability are more significant than the values of reliability, highlighting the necessity of regular repair for repairable systems. Furthermore, the study for availability and reliability reveals that switching devices for both the subsystems have a significant effect on the output.

Table-4 and Figure-4 yield the MTTF of the system concerning variation in failure ratesrespectively when other parameters are fixed. We observe that MTTF on average based on the failure rate  is maximum with little variation, while a similar variation for subsystem-1 and switching device. Thus, the failure rates are more responsible for the system’s effective operation. The expected profit can reveal from table-5 and figure-5. Profit is maximum for K₂= 0.2 and minimum at K₂=0.6. Conclusively, we can observe that as service cost decreases, profit increase with a variation of time. The model given in this paper is suitable to be applied to several real systems such as power plant and transmission system, server design for the network, and so on.

Conflict of Interest

The work is original and has not been submitted anywhere for publication.

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