MTBF evaluation for 2outof3 redundant repairable systems with common cause and cascade failures considering fuzzy rates for failures and repair: a case study of a centrifugal water pumping system
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Abstract
In many cases, redundant systems are beset by both independent and dependent failures. Ignoring dependent variables in MTBF evaluation of redundant systems hastens the occurrence of failure, causing it to take place before the expected time, hence decreasing safety and creating irreversible damages. Common cause failure (CCF) and cascading failure are two varieties of dependent failures, both leading to a considerable decrease in the MTBF of redundant systems. In this paper, the alphafactor model and the capacity flow model are combined so as to incorporate CCF and cascading failure in the evaluation of MTBF of a 2outof3 repairable redundant system. Then, using a transposed matrix, the MTBF function of the system is determined. Due to the fact that it is difficult to estimate the independent and dependent failure rates, industries are interested in considering uncertain failure rates. Therefore, fuzzy theory is used to incorporate uncertainty into the model presented in this study, and a nonlinear programming model is used to determine system’s MTBF. Finally, in order to validate the proposed model, evaluation of MTBF of the redundant system of a centrifugal water pumping system is presented as a practical example.
Keywords
Mean time between failures (MTBF) Redundant repairable systems Common cause failure (CCF) Cascading failure Fuzzy parametersIntroduction
Redundancy is a wellknown and widely used approach to enhancement of failuresensitive systems which are subject to both dependent and independent failures. In most reliability analyses and mean time between failures (MTBF) evaluation models for redundant systems, components are considered independent of one another with respect to failure. This results in an incorrect and inaccurate evaluation of system features. Therefore, it is highly crucial to identify and consider dependent failures in the evaluation of reliability and MTBF of systems. Common cause failure (CCF) is one of the most important dependent failures in redundant systems, in which an intrinsic factor leads to the propagation of failure in all components, resulting in the simultaneous failure of components in the redundant system (Kančev and Čepin 2012). Failing to incorporate CCF into MTBF of redundant systems leads to irreversible damage. In March 22, 1975, negligence of CCF led to a fire that occurred in a nuclear power plant located in the state of Alabama, USA (Mortazavi et al. 2016). After this event, in order to prevent the recurrence of such incidents, extensive research was conducted on CCF, resulting in the development of various standards (Mosleh et al. 1988, 1998). As another example, one may refer to the failure of all four engines of the Boeing 747 in fight BA 009 on 24 June 1982 over the Indian Ocean (Tootell 1985). The engineers estimated the likelihood of all four engines failing during the same flight to be negligible; however, a CCF of volcano ash proved otherwise. Other catastrophic incidents resulted by ignoring the likelihood of dependent failures in redundant systems have been reported, e.g., the hydraulic pumps failure due to engine explosion in United Airlines DC10 in July 1989 (Galison 2000).
Another variety of dependent failures in redundant systems is cascading failure (also known as load share); when a component in the redundant system fails, the intact components undergo greater load or the failure propagates through the system to other components; hence, their failure rates change. As an example, the costly collision of the space rocket Ariane 5 on 4 June 1996 was due to cascading failure (Gleick 1996). Ariane 5 and its precious cargo of four expensive satellites were destroyed due to an error in the rocket navigation computer that led to generation of a number which was too large for the system to calculate. This in turn resulted in handing over control to an identical redundant computer in which the same failure occurred. The outofcontrol rocket changed direction to compensate for an estimated error and was finally destroyed in its own turbulence. The significance of the influence of cascading failures can be recognized through many examples of catastrophic incidents reportedly stem from this type of dependent failure, e.g., the explosion of Boeing 707, Pan Am flight 214, on 8 December 1963 and Boeing 747, TWA 800, on 17 July 1999 (Negroni 2013).
Considering the significant impact of dependent failures (CCF and cascading failure) on reliability of redundant systems, these types of failures substantially affect MTBF of such systems. Neglecting the role that these types of failures play may result in significantly misleading MTBF evaluation models which in turn lead to incorrect MTBF value. Therefore, this paper addresses these two important types of failures and incorporates those into a proposed model for MTBF evaluation of a 2outof3 repairable redundant system.
The organization of the paper is as follows: In the next section, a review of the related literature is presented. In "Alpha factor model and capacity flow model" section, addresses the alpha factor model and capacity flow model. In "Formulating the model" section, the MTBF for 2outof3 redundant repairable system is computed with CCF and cascading failure based on alpha factor model and capacity flow model and, also the developed MTBF function together with fuzzy parameter is discussed, and the NLP method is used to determine the membership function. In "Case study" section, in order to validate the proposed model, a case study on the redundant system of centrifugal pumps is presented. In "Comparison of results" section, the results of the developed model are compared with those of a model offered in previous studies and an analysis is carried out. Finally, "Discussion and conclusion" section, concludes the paper and presents some guidelines for future works.
Literature review
Studied extensively in recent years, the fuzzy theory is an efficient tool for considering uncertainty in reliability analyses (Purba et al. 2014; Sharma and Sharma 2015; Gupta et al. 2016; Kumar and Goel 2017). Due to uncertainty and imprecision, it is not easy to estimate dependent and independent failure rates. Therefore, different industries in the real world are interested in considering these rates as intervals (minimum failure rate and maximum failure rate) using the fuzzy theory and fuzzy numbers. Wu and Tsai (2000) developed weightedbased fuzzy clustering procedure to estimate timetofailure distribution. The fuzzy reliability computation of single component is a fundamental problem. Jiang and Chen (2003) focused on the computation of the fuzzy reliability of a single component. The idea is that if the value of fuzzy reliability of a single component can be determined, it will be possible to compute the fuzzy reliability of the whole system by conventional methods. The general redundant system with the random fuzzy lifetime was considered by Zhao and Liu (2004). In their research, three system performances with random fuzzy lifetimes are studied. In addition, Liu et al. (2007) regarded component failure rate and lifetime as fuzzy variables and established mathematical models for nonrepairable series and parallel systems. In another research by Liu et al. (2010), component lifetime and repair time were modeled by random fuzzy exponential distribution. In addition, system MTBF and mean time to repair (MTTR) were calculated. Liu et al. (2011) used bivariate exponential distribution, assuming that components have interdependent lifetimes.
To estimate lifetime and repair time, Liu et al. (2014) used random fuzzy exponential distribution and evaluated the availability of a redundant system with nonidentical repairable components. GonzálezGonzález et al. (2016) used nonlinear regression to model a degradation process in order to predict mean time to failure (MTTF). To include uncertainty in the extrapolation process, they used trapezoidal fuzzy numbers to shape the timetofailure estimation. Aghili and HajianHoseinabadi (2017) introduced a flexible form of Markov process to evaluate the reliability of repairable systems. They then used fuzzy arithmetic calculations to incorporate uncertainty into their presented model. Some other studies focused on repairable systems with fuzzy repair rates. Chhoker and Nagar (2015) and Hu and Su (2016) developed frameworks for modeling, analyzing and predicting the reliability of redundant repairable systems with fuzzy parameters.
Redundant systems are in various types. Standby systems are one of the most widely used types of redundant systems, in which one or several components are put on standby mode. In case of failure of an operating component, the standby component replaces the failed component and prevents system breakdown. Ke et al. (2008) proposed a procedure to construct the MTTF membership function of a redundant repairable system with two primary components and one standby. They assumed component failures to be independent of one another. Lin et al. (2012) analyzed the reliability and MTBF of a redundant repairable system with one primary component, one standby component, and one unreliable service station. They assumed failure and repair rates to follow fuzzy exponential distributions. In the study by Huang et al. (2006), a parametric NLP approach is addressed to analyze the MTBF of a repairable system with switching failure and fuzzy parameters, assuming that components fail only due to their independent failure. Jahanbani Fard et al. (2017) applied the concepts of αcuts and fuzzy algorithm to a repairable system with two primary units in parallel as one active and one standby redundancy with imperfect coverage. They proposed a method to construct membership functions for MTBF and availability using paired NLP models.
Lethal shocks cause redundant system components to fail simultaneously (simultaneous failure). In some studies, such shocks are known as common cause shock failure. Only few studies have investigated dependent failure with fuzzy parameters. Huang et al. (2008) addressed the fuzzy availability and fuzzy MTTF of a system with two components in series and parallel configurations. They categorized system failures into two groups of individual failures and common cause shock failures assuming failure rates to be fuzzy numbers with trapezoidal membership functions. Jain et al. (2012) investigated a repairable redundant system with imperfect coverage, common cause shock failure, reboots, and recovery, and determined the fuzzified reliability, availability, and MTTF. Jain (2016) studied a repairable redundant system with warm standby components and repair facility. He took into account the real system conditions including repair, repair delay, switching failure, and CCF. Taking these into consideration helped developing appropriate availability functions for the redundant system.
In most of the aforementioned researches, dependent failures are ignored. A few studies have merely hinted at a single kind of dependent variables (e.g., CCF or common cause shock failure). However, there are many redundant systems which are exposed to cascading failure as well as CCF. Dependent failures and reparability are among the features that should be taken into consideration in evaluation of MTBF of redundant systems in order to obtain realistic results. Hence, it is important to develop a method which incorporates these features into the MTBF function. Furthermore, it was demonstrated in this paper how dependent failures affect and reduce redundant system MTBF. If dependent failures are not incorporated into reliability analyses, reliability parameters are not correctly evaluated creating misleading results about the redundant systems. Some other studies have been silent regarding fuzzy failure rates, whereas engineers designing redundant systems would prefer the failure rates expressed as linguistic terms that can be effectively modeled as fuzzy numbers. Therefore, this research presents a method to evaluate the MTBF of a 2outof3 redundant repairable system with independent failure, CCF, cascading failure, and fuzzy parameters.
Alpha factor model and capacity flow model
Notations
t  Time scale 

P _{ k }(t)  The probability of k components fail at time t (k = 0,1,2,3) 
A _{I}, B _{I}, C _{I}  Independent failures of components A, B, and C 
C _{ AB }, C _{ AC }, C _{ BC }  Failures of A&B, A&C, B&C due to CCF 
C _{ ABC }  Failure of A&B&C due to CCF 
Q _{1}  Independent failures rate 
Q _{2}  Simultaneous failure rate of two components caused by common cause 
Q _{3}  Simultaneous failure rate of three components caused by common cause 
Q _{ x } ^{*}  Failure rate of surviving components after suffering the failure of X component 
γ  Load factor 
ω  Repair rate 
\(\tilde{Q}_{1}\)  Fuzzy number for independent failure 
\(\tilde{Q}_{2}\)  Fuzzy number for CCF (Q _{2}) 
\(\tilde{Q}_{3}\)  Fuzzy number for CCF (Q _{3}) 
\(\tilde{\omega }\)  Fuzzy number for repair rate 
a, b, c  Triangular fuzzy number parameters 
f(t)  System probability density function at time t 
R(t)  System reliability function at time t 
In the above equation, x is the number of failed components in a redundant system and γ is the load factor. Load share exists in many redundant systems such as water pumps, electric generators, suspension bridge cables, and computer parts (e.g., CPUs, graphics cards, laptop RAMs).
Formulating the model

The system and the components have two states: they either work or fail.

Load share in centrifugal pumping 2outof3 redundant system is detected by the use of technical measures.

Initially, all the system components operate (they are not failed).

After each repair, the system restarts and operates exactly the same as a newly installed and started system.

Component failure is repaired immediately after detection.

Failure rates and repair rate of the components are constant (component lifetime has exponential distribution).

Components are repaired individually (there is one repairman).
Case study
Fuzzy failure and repair rates
Subject  Value 

Fuzzy number for independent failure  (0.003, 0.004, 0.005) 
Fuzzy number for CCF (Q2)  (0.0045, 0.0055, 0.0065) 
Fuzzy number for CCF (Q3)  (0.006, 0.007, 0.008) 
Fuzzy number for repair rate  (0.02, 0.035, 0.05) 
Load factor γ  0.25 
Figure 6 illustrates the membership function of failure rates and repair rate of centrifugal water pumps.
MTBF values obtained by applying different αcuts values
α  X _{1α } ^{ L }  X _{1α } ^{ U }  X _{2α } ^{ L }  X _{2α } ^{ U }  X _{3α } ^{ L }  X _{3α } ^{ U }  X _{4α } ^{ L }  X _{4α } ^{ U }  MTBF _{ α } ^{ L }  MTBF _{ α } ^{ U } 

0.00  0.003  0.005  0.0045  0.0065  0.006  0.008  0.066  0.074  35.63052  50.97627 
0.10  0.0031  0.0049  0.0046  0.0064  0.0061  0.0079  0.0664  0.0736  36.18262  49.91712 
0.20  0.0032  0.0048  0.0047  0.0063  0.0062  0.0078  0.0668  0.0732  36.75132  48.89942 
0.30  0.0033  0.0047  0.0048  0.0062  0.0063  0.0077  0.0672  0.0728  37.33738  47.92080 
0.40  0.0034  0.0046  0.0049  0.0061  0.0064  0.0076  0.0676  0.0724  36.77827  46.97908 
0.50  0.0035  0.0045  0.005  0.006  0.0065  0.0075  0.068  0.072  38.56482  46.07222 
0.60  0.0036  0.0044  0.0051  0.0059  0.0066  0.0074  0.0684  0.0716  39.20793  45.19835 
0.70  0.0037  0.0043  0.0052  0.0058  0.0067  0.0073  0.0688  0.0712  39.87190  44.35571 
0.80  0.0038  0.0042  0.0053  0.0057  0.0068  0.0072  0.0692  0.0708  40.55775  43.54268 
0.90  0.0039  0.0041  0.0054  0.0056  0.0069  0.0071  0.0696  0.0704  41.26655  42.75773 
1.00  0.004  0.004  0.0055  0.0055  0.007  0.007  0.07  0.07  41.99947  41.99947 
It is concluded from Table 3 that for α = 1, the MTBF value of the centrifugal pumping redundant system is 41.99947 days. For possibility level α = 0, the MTBF range of the centrifugal redundant pumping system is approximately [MTBF _{ α=0} ^{ L } = 35.63052, MTBF _{ α=0} ^{ U } = 50.97627]. If Eq. (21) is used for defuzzification, MTBF equals 41.99947 days. All three values of MTBF _{ α=0} ^{ L } , \({\text{MTBF}}_{\alpha = 0}^{U}\), and MTBF_{ α=1} can be beneficial for system designers and maintenance operators.
Comparison of results
Obviously, (Q _{2}) and (Q _{1} ^{*} ) rates are ignored in this function. In centrifugal water pumping systems, however, these two failure rates must be taken into consideration. In other words, it can be said that cascade failure and CCF both exist in these systems. If the redundant system consists of more than two components, in order to take CCF into consideration, models such as alphafactor and MGL should be used (Beckman 1995).
MTBF values obtained by applying different αcuts values ignoring Q _{1} ^{*} and Q _{2}
α  X _{1α } ^{ L }  X _{1α } ^{ U }  X _{3α } ^{ L }  X _{3α } ^{ U }  X _{4α } ^{ L }  X _{4α } ^{ U }  MTBF _{ α } ^{ L }  MTBF _{ α } ^{ U } 

0.00  0.003  0.005  0.006  0.008  0.066  0.074  120.053  166.667 
0.10  0.0031  0.0049  0.0061  0.0079  0.0664  0.0736  121.843  163.654 
0.20  0.0032  0.0048  0.0062  0.0078  0.0668  0.0732  123.679  160.732 
0.30  0.0033  0.0047  0.0063  0.0077  0.0672  0.0728  125.561  157.897 
0.40  0.0034  0.0046  0.0064  0.0076  0.0676  0.0724  127.491  155.144 
0.50  0.0035  0.0045  0.0065  0.0075  0.068  0.072  129.471  152.47 
0.60  0.0036  0.0044  0.0066  0.0074  0.0684  0.0716  131.502  149.873 
0.70  0.0037  0.0043  0.0067  0.0073  0.0688  0.0712  133.588  147.349 
0.80  0.0038  0.0042  0.0068  0.0072  0.0692  0.0708  135.729  144.895 
0.90  0.0039  0.0041  0.0069  0.0071  0.0696  0.0704  137.928  142.508 
1.00  0.004  0.004  0.007  0.007  0.07  0.07  140.187  140.187 
Failure rate estimation is one of the most accurate (and most expensive) reliability analysis activities. An inaccurate estimation of failure rate leads to an inaccurate or incorrect evaluation of reliability, availability, MTTF, and MTBF of systems. In most cases, accelerated tests are utilized for accurate estimation of failure rates. It should be noted, however, that creating genuine conditions in these tests is considerably costly and sometimes not feasible. Therefore, it is more appropriate to consider failure rate values as intervals. Upper bound, lower bound, and central values for failure rates and repair rate can be expressed through triangular fuzzy numbers. As a result, MTTF and MTBF adopt upper bound, lower bound, and central values as well. More specifically, independent failure rate and dependent failure rates each can be expressed as triangular fuzzy numbers. The functioning condition of redundant systems causes them to undergo failure at different times. Therefore, it is possible to determine an interval for failure rate using previous failure data. In centrifugal water pumping redundant systems, fuzzy failure rates can also be collected by taking previous failure data into consideration. Repair rate can also be considered as triangular fuzzy numbers. In this case, the maintenance operator can determine an interval for the maintenance operation. Using the model presented in this paper, the MTBF of a redundant system with two types of dependent failures as well as fuzzy failure rates and fuzzy repair rate can be evaluated.
Discussion and conclusion
Most dependent failures reduce MTTF and MTBF in redundant systems. Therefore, the frequency of dependent failure events in redundant systems must be minimized and appropriate corrective actions must be taken. However, redundant systems undergo both independent failures and dependent failures. In this paper, using Markov chain and transposed matrix, the MTBF function for a 2outof3 redundant system is developed by taking into consideration two types of dependent failures. First, the alphafactor model and capacity flow model are briefly outlined. Since the failure rates are usually preferred to be expressed as linguistic terms, the failure and maintenance rates are expressed as triangular fuzzy numbers, and, the upper bound, lower bound, and central values for the fuzzy MTBF of a 2outof3 redundant system are determined by applying Zadeh’s extension principle, concepts of αcuts, and NLP. To validate the model, the results were compared with those obtained by the model developed by Huang et al. (2008). The comparison revealed that considering dependent failures in the MTBF function leads to the reduction of the MTBF of redundant systems. Obviously, by investigating and integrating dependent failures in MTBF evaluation models for redundant systems, more applicable and realistic MTBF models can be developed.
System failure in dynamic environment is another variety of dependent failures (XiaoFei and Min 2014). To evaluate the MTBF of redundant systems, it is assumed that the components operate in a static environment. Therefore, the models developed under such assumptions may not be appropriate for redundant systems operating in dynamic conditions. It is recommended that future studies develop a model to evaluate the MTBF of redundant system, incorporating CCF, cascade failure in dynamic environments. The dynamic model may also incorporate uncertainty of the realworld problems.
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