Reliability analysis of the triple modular redundancy system under step-partially accelerated life tests using Lomax distribution

Triple modular redundancy (TMR) is a robust technique utilized in safety-critical applications to enhance fault-tolerance and reliability. This article focuses on estimating the distribution parameters of a TMR system under step-stress partially accelerated life tests, where each component included in the system follows a Lomax distribution. The study aims to analyze the system’s reliability and mean residual lifetime based on the estimated parameters. Various estimation techniques, including maximum likelihood, percentile, least squares, and maximum product of spacings, are explored. Additionally, the optimal stress change time is determined using two criteria. An illustrative example supported by two actual data sets is presented to showcase the methodology’s application. By conducting Monte Carlo simulations, the assessment of the estimation methods’ effectiveness reveals that the maximum likelihood method outperforms the other three methods in terms of both accuracy and performance, as indicated by the numerical outcomes. This research contributes to the understanding and practical implementation of TMR systems in safety-critical industries, potentially saving lives and preventing catastrophic events.

Designing the triple modular redundancy system.Within engineering systems, redundancy pertains to the replication of vital components within a system, aimed at enhancing reliability and reducing the likelihood of hazards.Systems that incorporate threefold replication of key elements are categorized as TMR systems.
A fault represents a tangible imperfection that has the potential to result in malfunctions or errors within a system.Many critical areas like electric power distribution networks, patient life support setups, transportation, nuclear reactor oversight, automobiles, aviation, and spacecraft, are required to function reliably and securely even when confronted with such imperfections, see, for example, [1][2][3][4] .Such systems demand a consistent and dependable level of service.The capability of a system to keep on performing its intended function in the presence of a fault is known as fault-tolerance 5 .Fault-tolerant systems are those that can endure more demanding environmental circumstances and external disruptions.Consequently, designers now possess a pair of choices when it comes to managing faults: either masking them or tolerating them.Errors and system failures are avoided by the technique of "fault masking".A common example of such a technique is majority voting systems.In either scenario, redundancy (the inclusion of additional parts, units, or equipment that are not necessary for the regular running of the system) is necessary.The earliest kind of this redundancy is called TMR.At least two out of the three system output levels must be voted on in this design.
TMR is a powerful technique used in safety-critical applications to achieve high levels of fault-tolerance and reliability.By triplicating critical components and comparing their outputs, it ensures consistent operation even in the presence of faults or failures.This redundancy approach is vital in industries where the consequences of system malfunctions can be severe, potentially saving lives and preventing catastrophic events.
The majority of modules (2 out of 3) are compatible with the output within such designs.This implies that the system can tolerate a maximum of one individual module failure, which can be mitigated (masked).

Accelerated life tests.
Ongoing enhancements in manufacturing design have led to a significant increase in the reliability of products and materials.However, acquiring data about the longevity of these enhanced reliability tools and materials through testing has become a more intricate, expensive, and time-intensive process.In situations like these, determining the hazard rates of products requires subjecting them to stress levels more severe than those encountered in the manufacturing phase.This approach, known as the "accelerated life test" (ALT), involves testing products at elevated stress levels.The hazard rates observed in ALT can then be used to approximate the expected durability of these products under typical usage conditions.
There are primarily three ALT techniques, see Ref. 9 .The first technique is referred to as the constant-stress ALT, which sustains a constant stress level throughout the entire lifespan of the tested products, see, for example 10 .The stress placed on a test product is continually increased over time in the second technique, which is known as the progressive-stress ALT, see, for example 11,12 .Several authors have explored the third technique of ALT, termed the step-stress ALT.In this method, the applied stress level is altered either after a specific number of failures have been observed or at certain predetermined time intervals.
The ALT necessitates a well-defined relation (such as the inverse power rule, Arrhenius model, log-linear model, etc.) between the stress applied to the products and their average lifespan.The parameters involved in these relations must be estimated.Often, establishing such a relation is intricate and cannot be assumed beforehand.Consequently, the extrapolation of ALT data for practical use conditions becomes unfeasible.This is where partially ALT (PALT) offers a practical solution.PALT involves determining the acceleration factor, which represents the ratio of the average lifespan under standard usage to that under accelerated conditions (as referenced in Ref. 9 ).Once this factor is determined, it becomes feasible to extend the application of accelerated test data to actual usage conditions.
Lomax distribution.The Lomax distribution (LD) 22 , often known as Pareto distribution of the second kind, was proposed by Lomax as a model for business failure data.It is commonly used in reliability engineering, where it is applied to model failure rates, lifetime distributions, and survival probabilities.It is also used in various fields for modeling data with heavy tails, such as insurance claims, income distribution, and extreme value analysis.
The following are, respectively, the probability density function (PDF), cumulative distribution function (CDF), reliability function (RF), and hazard rate function (HRF) of the LD, for t > 0, where β > 0 and γ > 0 are the scale and shape parameters of the distribution, respectively. (1) 1.The CDF of the LD possesses a closed-form expression, simplifying the process of estimation.
2. The LD is characterized by a heavy right tail, indicating that extreme values occur with higher probability compared to many other distributions.3. The parameter ( γ ) grants the LD more flexibility to control the tail heaviness.4. The parameter ( β ) determines the spread of the distribution.5.The LD's HRF exhibits a decreasing trend.This characteristic could be advantageous for entities experiencing early failures.However, it might also pose a limitation, as certain scenarios might involve a bathtub-shaped hazard rate.In such instances, a Weibull distribution could be more appropriate for modeling such a situation than LD.
K-out-of-N system.The reliability analysis of a system often concerns analyzing K-out-of-N systems.A K-out-of-N system is operational when K or more out of N units work (in other words it fails when fewer than K units are functional), whereas for series(parallel is the reliability of a given unit, then the K-out-of-N system's reliability is the probability that N − K or fewer units have failed by time t (or-at least K units are functional) The new in this article is to analyze and estimate the reliability of the TMR system under step-stress PALTs where each component's lifetime has a LD, through estimating the included parameters.The mean residual lifetime (MRL) as well as the HRF of the considered system are also estimated.Four different methods are considered in the estimation procedure such as the maximum likelihood (ML) estimate (MLE), percentile estimate (PE), least squares (LS) estimate (LSE), and maximum product of spacings (MPS) estimate (MPSE).Two optimal criteria are proposed to determine the ideal stress change time in the step-stress PALTs.
The article is structured into nine distinct sections, with the remaining sections outlined as follows: "Reliability of the triple modular redundancy system" presents the derivation of the reliability of the TMR system.In "Model description and mean residual lifetime", the model is elaborated upon, accompanied by the calculation of the MRL."Estimation methods" delves into the study of various estimation methods, including ML, percentile, LS, and MPS methods, for determining unknown parameters and parameter functions."Common optimality criteria" introduces two criteria for the optimal timing of stress changes.The exploration of an illustrative example based on real-world data is conducted in "Real data set".Subsequently, in "Simulation Study", the application of Monte Carlo simulations is detailed."Simulation results" presents selected simulation outcomes.Lastly, the article concludes with "Concluding remarks", which offers closing observations and a glimpse into potential future research endeavors.

Reliability of the triple modular redundancy system
Suppose that three units (components) A 1 , A 2 , and A 3 are structured in a TMR system which is set up to allow majority voting.For the system to function effectively, at least two of the three units must be in operation.Assume that T is a random variable (with a realization t) that represents the TMR system's lifetime.If the voter's reliability is one, the reliability of the TMR system, at time t, is as follows: If then It is clear that Eq. ( 6) coincides with Eq. ( 5) when K = 2 and N = 3 .Therefore, the TMR system's CDF and PDF are provided, respectively, by where f (t) = −dR (t) dt is the PDF of lifetime of a given unit.Using PDF (1) and RF (3) and for (β, γ > 0) , the CDF, PDF, RF, and HRF of the TMR system are given, respectively, by The PDF, CDF, RF, and HRF of the LD and TMR system are plotted in Fig. 2 with β = 0.5 and γ = 0.9.
In the subsequent section, both the model's description and its mean residual lifetime are provided.

Model description and mean residual lifetime
This section is devoted to applying the step-stress PALT to the TMR system whose lifetime T is considered a random variable (with realization t) subjecting to CDF (9).The PDF, SF, and HRF of T are provided, respectively, by (10), (11), and (12).Two techniques may be used to raise the stress in the step-stress PALTs, one of them is called tampered random variable, see Ref. 23 , and the other one is called tampered failure rate (TFR), see Ref. 13 .For the TMR system under the TFR technique, as soon as a change time point η is reached, the stress raises and remains constant.This would be done by multiplying the primary HRF h TMR (t) by an undefined factor α(> 1) (which may rely on η ).Let h ⋆ (t) denote the total HRF in PALT.Then, the TFR model is provided by For the total lifetime T in Model ( 13), the RF is provided by The function R 2 (t) defined in RF ( 14) is deduced as follows: www.nature.com/scientificreports/By substituting RF (10) in (15), RF (15) becomes Based on ( 10), ( 14) and ( 16), RF (14) of T under PALT becomes The corresponding CDF, PDF, and HRF are given, respectively, by

Mean residual lifetime.
For the purposes of studying survival analysis, the MRL is crucial.Assuming the system has lived up until time t 0 , the MRL is defined as the expected residual lifetime as follows:

Estimation methods
In this section, the parameters α, β, and γ of the TMR system under step-stress PALT with CDF (18) and PDF (19) are estimated using four estimation methods which are the ML, percentile, least squares, and MPS.RF (17), HRF (20), and MRL function ( 21) are also estimated using the invariance property (IP).
It is common knowledge that the method of ML estimation is enjoyed with the IP.This property does not apply well to the estimation of a parametric function when the approach of parameter estimation is switched from ML to any other conventional method.The IP of the MPS estimation was studied by Anatolyev and Kosenok 24 , who came to the conclusion that it shares the same property as ML estimation.However, different authors have attempted to adopt this IP for other best estimators of the parameters to estimate a function of the parameters in various contexts, see, for example, Srinivasa Rao and Kantam 25 .
) is an ordered observed random sample taken from a population with PDF (19), where n 1 (n 2 = n − n 1 ) is a random number denoting the number of failures under normal(accelerated) stress condition.Consequently, the likelihood function can be expressed as follows: In general, it is simpler to determine the MLEs of the underlying parameters by maximizing the natural logarithm of the likelihood function than the likelihood function itself.Thus, using PDF (19), if L = ln L(α, β, γ ; t) , then Based on (23), by solving the likelihood equations with respect to α, β , and γ after equating them to zero, the MLEs ( α, β, γ ) of (α, β, γ ) can be obtained.This procedure can be done as follows: From ( 24), the following equation can be used to calculate α as a function of β and γ, By substituting α(β, γ ) in ( 25) and ( 26), the MLEs of β and γ can be produced by solving the likelihood equations ∂L ∂β = 0 and ∂L ∂γ = 0 , with regard to β and γ by utilizing any numerical iteration method.
The MLEs of RF, HRF, and MRL function at time t 0 , denoted by R(t 0 ) , ĥ(t 0 ) and m(t 0 ) , are obtained by sub- stituting ( α, β, γ ) in Eqs. ( 17), (20), and (21), respectively.Remark 4.1 It can be noticed that: 1.The MLEs of α , β , and γ don't exist if there are no failures observable under normal stress circumstances ( n 1 = 0 and n 2 = n).2. The MLE of α does not exist if there are no failures observable under accelerated stress circumstances ( n 1 = n and n 2 = 0).3. The MLEs of α , β , and γ exist if there are at least two failures observable during the test one of them is under normal stress circumstances ( n 1 > 0 and n 2 > 0).
The initial two aspects outlined in Remark 4.1 could potentially serve as limitations when applying the model under certain specific data.
In light of the widespread asymptotic normality theory for MLEs, each of the following standard quantities , and ( γ − γ )/ Var( γ ) can subject to a normal distribution with mean Vol.:(0123456789) www.nature.com/scientificreports/0 and variance 1, where Var( α) , Var( β) , and Var( γ ) stand for the variances, of the MLEs, which correspond to the main diagonal of the inverse of observed Fisher information matrix (FIM) given by where where the caret ˆ means that the derivative is calculated at the MLEs of the parameters.Matrix ( 28) is useful in determining confidence intervals (CIs) for the unknown parameters.It is simple to determine the elements of Matrix (29).
A negative value for the positive parameter often appears in the lower bound of NACI.Hence, Meeker and Escobar 26 recommended adopting a log transformation CI (LTCI) for this parameter.Therefore, for i = 1, 2, 3 , ln μi − ln µ i Var(ln μi ) ∼ N(0, 1) , according to the normal approximation of log-transformed MLE, where Var(ln μi ) = Var( μi ) μi 2 .Thus, a LTCI for µ i , with confidence coefficient 100(1 − ρ)% , could be given as follows: The second estimation method, percentile estimation, is discussed in the following subsection.
Percentile estimation.Kao 27 proposed a percentile approach to compute estimates for unknown parameters.This method proves suitable for estimating an unknown parameter when collecting data using a closedform CDF.It involves adjusting a linear relationship between the percentile values obtained from the sample and the corresponding theoretical points derived from the CDF.
Suppose that, for i = 1, . . ., n , t (i) is a realization of the ith order statistic T (i) in a random sample of size n taken from a population with PDF (19).If i is an estimate of F(t (i) ) , then minimization of the next equation with regard to α, β and γ , will yield the PEs α , β , and γ.

Common optimality criteria
In this section, the focus shifts to the matter of ascertaining the most suitable time point η for elevating the stress level.3][34][35][36] .

A-optimality criterion
The primary goal of this criterion is to maximize the FIM's trace, which provides a measure of total information based on marginal knowledge about the parameters.It is preferred to use this criterion when the estimates are at most moderately correlated.

D-optimality criterion
The primary goal of this criterion is to maximize the FIM's determinant which offers a comprehensive assessment of variability by taking into consideration the correlation between the estimates.When the estimates are strongly correlated, this criterion is preferred.Consequently, maximization(minimization) of Maximize{tr(�)}.

Maximization of the tr(�) [ | |
] can be achieved by solving the equation ∂tr(�) ∂η = 0 ∂|�| ∂η = 0 , with regard to η .The solution cannot be represented in a closed-form.An iterative method must be used to reach a numerical solution.

Real data set
Two real sets of data are taken into consideration in this section to clarify the inference techniques carried out in "Estimation methods" and the optimality criteria stated in "Common optimality criteria".
• The first real set of data was proposed in 37 .The reliability properties of solar lighting equipment were evalu- ated using a step-stress test of two levels.As soon as the time point η = 5 is reached, the temperature is raised from 293 to 353 K when some devices are placed through a life test at their regular working temperature (in hundred hours).The data set is listed in Table 1 where one can see that 15 devices failed under the first stress level, while 16 devices failed under the second stress level.• The second real set of data was proposed in Ref. 38 .A prototype small/micro unmanned aerial vehicle's (SUAV) reliability characteristics for civil applications, such as disaster management, fire monitoring, and forest aerial surveys, were evaluated using a step-stress test of two levels.The SUAV is an aircraft that is operated remotely by a ground operator while being autonomously controlled by an automatic system.Dual independent propulsion systems were included with the SUAV.In the event that both systems malfunction, the SUAV loses its flying height prior to finishing its mission.The SUAV is not affected by the failure of either propulsion system, despite the failure time being captured digitally.The stress factor whose level was altered during the test, in this instance, is the wind speed.η = 15 is the stress change time.Table 1, in the last rows, lists the data set.
Now, it is reasonable to inquire whether CDF ( 18) matches the two real sets of data that are shown in Table 1.A test statistic, related to Kolmogorov-Smirnov (KS), and its associated p-value are employed.It is evident from the results shown in Table 2 that every p-value exceeds 0.05 indicating that the model with CDF (18) matches the two real sets of data presented well.Drawing the empirical CDF alongside CDF (18) for each real set of data further illustrates this point, see Figure 3.
Based on the data displayed in Tables 1, 3 displays the MLEs, MPSEs, PEs, LSEs, and WLSEs of the parameters α , β , and γ .Also, it displays the estimates of RF and HRF as well as the MRL.Furthermore, the NACIs and LTCIs of the model parameters in addition to the optimal value of increasing stress level ( η ) under two optimality criteria, η * A and η * D , are calculated and presented in Table 4.

Simulation study
One of the valuable techniques for evaluating and contrasting the efficiency of estimation methods is the utilization of Monte Carlo simulation.Within this section, we engage in numerical exploration through Monte Carlo simulation to appraise the effectiveness and proficiency of the estimation approaches and optimality criteria discussed in the previous sections.The ensuing sequence of actions is outlined below: 1. Provide values for the parameters (α, β, γ ) , the sample size n, and η (stress change time).
2. From the uniform distribution with support (0, 1), generate a sample of n observations,  18) together with empirical CDF for the data given in Table 1.
First data set α β γ Ŝ(5.5) ĥ(5.5) m(5.www.nature.com/scientificreports/ 3.For i = 1, . . ., n , generate two observations t 1,i and t 2,i from CDFs F 1 (t) and F 2 (t) , respectively, given by (18).Therefore, the observations (t 1 , t 2 , . . ., t n ) under PALT from CDF (18) can be obtained as follows: where, 4. Find the value of n 1 , the number of observations that occur under normal stress conditions, as follows: 5. Calculate the MLEs, MPSEs, PEs, LSEs, NACIs, and LTCIs of (α, β, γ ). 6. Calculate the MLEs, MPSEs, PEs, and LSEs of the RF and HRF as well as the MRL (at time t = 5.0).7. Calculate the values of η * D and η * A (the optimal values at which the stress can be raised).8. Replicate the above steps M (= 5000) times.9.The average of estimates with their mean squared error (MSE), and relative absolute bias (RAB) of ̺ over M samples are calculated according to the following: where ̺ is an estimate of ̺. 10.Use Step 9 to calculate the average of estimates of (α, β, γ ) in addition to the mean of the estimates of RF, HRF, and MRL function with their MSEs and RABs.11.The average of the RABs (ARAB1 and ARAB2) and average of the MSEs (AMSE1 and AMSE2) are also calculated.12.The NACIs and LTCIs, with confidence coefficient 95%, of the parameters (α, β, γ ) are calculated.The average interval lengths (AILs) in addition to the coverage probabilities (COVPs) of the NACIs and LTCIs are also calculated.Further, the average of the AILs (AAIL) is calculated.

Simulation results
The findings of the computations shown in Tables 5, 6, 7, 8 and 9 can be summarised as follows: 1.The results assure that the MLEs of the parameters, RF, HRF, and MRL function (at time t = 5.0 ) are better than the PEs, MPSEs, and LSEs through fewer values of the ARAB1s, ARAB2s, AMSE1s, and AMSE2s.2. Through the ARAB1s, ARAB2s, AMSE1s, and AMSE2s, the PEs of the parameters, RF, HRF, and MRL function (at time t = 5.0 ) are superior to the MPSEs and LSEs.3. Due to lower AAIL values, NACIs perform better than LTCIs.Table 6.MPSEs of (α, β, γ ) , RF, HRF, as well as the MRL with their MSEs, RABs, ARAB1s, ARAB2s, AMSE1s, and AMSE2s based on 5, 000 simulations.Except in rare situations, which may be consequent to fluctuations in the data, the above findings are accurate.

Concluding remarks
Under step-stress PALTs, we have estimated the distribution parameters of the TMR system, with each component's lifetime following a LD.Additionally, we have calculated the RF, HRF, and MRL of the TMR system.Various estimation methods, including ML, percentile, LS, and MPS, have been taken into account.The determination of the optimal stress change time has been achieved using two distinct optimality criteria.An illustrative instance involving two actual datasets has been examined.Additionally, a Monte Carlo simulation has been utilized to analyze the effectiveness of the estimation methodologies.The assessment of these estimation techniques has been executed, uncovering that the ML method outperforms the remaining three methods in accuracy and performance as indicated by the numerical results.
Future research.Neutrosophic statistics is an extension of classical statistics that deals with uncertain, indeterminate, and inconsistent data.It is based on the concept of neutrosophy.It offers a new framework for modeling and analyzing data coming from complex process or uncertain environment.It has been applied in various fields, such as decision-making, risk assessment, medical diagnosis, and pattern recognition, where uncertainties play a significant role.Therefore, the current study can be extended using neutrosophic statistics as future research, see, for example, 39,40 .

Figure 2 .
Figure 2. The PDF, CDF, RF, and HRF of the LD and TMR system.

Figure 3 .
Figure 3. CDF (18) together with empirical CDF for the data given in Table1.

Table 1 .
The two real sets of data.

Table 4 .
NACIs, and LTCIs of (α, β, γ ) and optimal stress change values, η * D and η * A based on the real data given in Table1.

Table 9 .
AILs and COVPs (in %) of 95% CIs of α, β and γ in additional to the optimal value of η based on 5000 simulations.