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Timedependent reliability of (weighted) koutofn systems with identical component deterioration
Journal of Infrastructure Preservation and Resilience volume 2, Article number: 3 (2021)
Abstract
The performance of civil infrastructure systems is vital in supporting a community’s functionalities. Reliability assessment of these systems is a powerful approach to evaluate whether the system performance is desirably safe under the impacts of resistance degradation and nonstationary loads. A koutofn system is a widelyused logic model for a system with n components, which survives (works) if at least k components work. Its special cases include a series or a parallel system. Furthermore, a weighted koutofn system has components with positive integer weights and the system survives if the total weight of working components reaches the predefined threshold k. This paper proposes a method for estimating the timedependent reliability of both ordinary and weighted koutofn systems, taking into account the effects of resistance deterioration, resistance correlation and load nonstationarity, for which a mathematical solution is derived. The applicability of the proposed method is illustrated through reliability evaluation of a representative koutofn system.
Introduction
Civil infrastructure systems are expected to function with an acceptable level of serviceability and safety during their service lives. The aggressive environmental or operating conditions in service, however, could threaten the system performance significantly. Taking into account the uncertainties associated with these safetythreatening factors, which are often difficult or even impossible to predict exactly, structural reliability assessment is a powerful tool to evaluate a system’s capability of fulfilling the safety requirements during a reference period of interest [1–4].
An important ingredient in reliability assessment is to model the degradation of structural resistance (e.g., strength, stiffness, and others) at both the component and system levels [5–8]. On the other hand, the external load process could be nonstationary on the temporal scale in terms of occurrence frequency and/or magnitude [9–11]. For example, for structures in cycloneprone areas, the future cyclone winds could be affected by the impact of climate change [9, 10]. As a result, it is important to incorporate the factors of resistance deterioration and load nonstationarity in structural reliability assessment. Mori and Ellingwood [12] developed a method for estimating structural timedependent reliability, where the load process was modeled by a homogeneous Poisson process. This work was later improved by Li et al [13] so that the nonstationarity in the load process can also be considered in structural reliability assessment.
Most of infrastructure systems consist of multiple structures or components [14]. One of the key features of a system is the interaction between different components [15–17]. Furthermore, correlation may also arise between the performances of different components due to common design provisions and construction practices [18,19]. Wang et al [20] developed an approach for estimating the timedependent reliability of an aging series system considering resistance correlation of different components. Wang and Zhang [21] investigated the seismic resilience of a power grid system, where the impact of correlation and deterioration of component resistances was considered.
The concept of a koutofn system has been extensively employed in engineering practice, which refers to such a system with totally n components that the system survives (works) if at least k components normally work. With this definition, the system is written as a koutof n:G system. Intuitively, the system reduces to a series or parallel system when k=n or k=1 respectively. Note that there exists an alternative definition for koutofn system, with which the system fails if at least k components fail. Both definitions serve as a complement for each other and should be used with careful instruction. In this paper, the koutof n:G system will be discussed, and will be simply referred to as a koutofn system unless otherwise stated.
A generalized case of a koutofn system is known as a weighted koutofn system [22], where each component has a positive integer weight. Correspondingly, the system works when the total weight of working components reaches the threshold k. For example, consider two substations from an electricity distribution network, which are connected directly by different transmission lines (components) [23]. The voltage of each transmission line could differ, resulting in different weights accordingly. When a certain level of voltage is needed for electricity transmission between the two substations, giving the required threshold k, the system (consisting of these transmission lines) can be modeled by a weighted koutofn system. In the presence of natural hazards such as earthquakes, the resistance of each component is reflected by the seismic fragility curve, which could degrade due to the impact of environmental conditions [21]. One can also refer to [24] for other examples of a weighted koutofn system. When the weight for each component equals unity, the weighted koutofn system reduces to an ordinary one.
The reliability assessment of a koutofn system has been widely discussed in previous studies [25–27]. Wang et al [28] proposed a method for timedependent reliability of a koutofn system with common cause failure through systemlevel loadstrength interference analysis. Zhang et al [29] developed a reliability model for a loadsharing koutofn system subjected to discrete loads, where the loadsharing effect was reflected by modeling the load distribution variation and strength damage after component failures directly. However, these studies did not consider the impacts of component resistance (in terms of correlation and deterioration) and load nonstationarity on the timedependent reliability of koutofn systems.
In an attempt to compute the reliability of a weighted koutofn system, recursive formulas were developed in [30,31]. Eryilmaz and Tutuncu [32] investigated the reliability of a weighted koutofn system by modeling the component interdependency as a Markov type. Eryilmaz [33] studied the reliability of a koutofn system with random weights and proposed a recursive formula to compute the system state probabilities. FaghihRoohi et al [34] presented a dynamic model for availability assessment of multistate weighted koutofn system and optimized the component availability and capacity through a genetical algorithm. Coit et al [35] proposed a reliability model for dynamic koutofn system considering component partnership, where the system reliability at time t is measured by the instantaneous system performance. Franko et al [36] studied the impact of cold standby component on the reliability of weighted koutofn systems with two types of components. Zhang [37] performed reliability analysis of koutofn systems with heterogeneous components and random weights. Hamdan et al [38] developed an optimal preventive maintenance model for weighted koutofn systems on the basis of cost analysis. Yet the timedependent reliability assessment of a weighted koutofn system has to be addressed by considering the timevariation of both component resistances and external loads.
This paper presents a method for the timedependent reliability of an aging (weighted) koutofn system, taking into account the impacts of component resistance deterioration, resistance correlation and nonstationary load effect. It is shown that the proposed reliability method is a generalized form of that for a single component, series system or parallel system. The implementation of the proposed method is also discussed. The applicability of the developed reliability method is demonstrated through an illustrative example.
Timedependent reliability assessment
Reliability of a single component
Consider the reliability of a component over time interval [0,T]. Significant load events (e.g., earthquake events) occur randomly in time with random intensities. Using a Poisson process to model the occurrence of the load events, let λ(t) be the timevariant occurrence rate (t denotes time), and F_{S}(s,t) the cumulative distribution function (CDF) of the load effect at time t conditional on occurrence. Mathematically, if the load effect at time t is S(t), then Pr(S(t)≤s)=F_{S}(s,t), where Pr() denotes the probability of the event in the brackets.
The resistance at time t, R(t), is modeled as follows,
where g(t) is the deterioration function, and R_{0} is the initial resistance. The deterioration function may take different shapes (e.g., linear, squareroot and parabolic), depending on the dominant deterioration mechanism [2,39].
With this, the timedependent reliability within [0,T],L(T), is estimated by [13]
where f_{R}(r) is the probability density function (PDF) of the initial resistance.
Reliability of a series or parallel system
For a series system with n components (structures), let R_{j} be the initial resistance of component j (j=1,2,…n), and g_{j}(t) the deterioration function of component j=1,2,…n, which is assumed to be independent of the load process. Suppose that the load process is modeled by a Poisson process with a timevariant occurrence rate of λ(t) and a timevariant CDF of load effect F_{S}(s,t). An occurring load event with magnitude s induces a structural action c_{j}·s in component j (e.g., moment, shear, etc) for j=1,2,…n. With this, the timedependent series system reliability, L_{ss}(T), is given by [20]
where f_{R}(r) is the joint PDF of the initial resistances R={R_{1},R_{2},…R_{n}}, and r={r_{1},r_{2},…r_{n}}.
For a parallel system consisting of n components with the same configuration as that of the series system as discussed earlier, if each c_{j} is independent of the number of working components, and the deterioration function of each component is identical, denoted by g(t), the timedependent parallel system reliability, L_{ps}(T), is given by [40]
In Eqs. (3) and (4), if the component resistances are statistically independent, then \(f_{\mathbf {R}}(\mathbf {r}) = \prod _{i=1}^{n} f_{R_{i}}(r_{i})\), where \(f_{R_{i}}(r_{i})\) is the PDF of R_{i}. However, taking into account the correlation between different component resistances, one would need to reasonably model the probabilistic behaviour of f_{R}(r). With this regard, the copula function is a promising tool to describe correlated random variables [40,41].
Reliability of a koutofn system
For a koutofn system with n components (structures), the system survives if at least k components work, as mentioned before. Its timedependent reliability will be discussed in this section. Assume that the resistance deterioration function for each component is identical, denoted by g(t), and is independent of the load process. It is also assumed that the value of c_{j}(j=1,2,…n) is independent of the number of working components.
We introduce \(\mathcal {M}(\mathbf {a},k)\), which is a function of vector a (with n elements) and integer k≤n. The function \(\mathcal {M}\) returns the kth largest element of a, which will be used in the following derivation. It is straightforward to observe that \(\mathcal {M}(\mathbf {a},1)=\max (\mathbf {a})\) and \(\mathcal {M}(\mathbf {a},n)=\min (\mathbf {a})\).
The hazard function, denoted by h(t) at time t, represents the probability of structural failure at time t provided structural survival up to time t. It can be related to the timedependent reliability L(t) according to [3,40]
or equivalently,
Consider the hazard function h_{kn}(t) for the koutofn system at time t, conditional on R={R_{1},R_{2},…R_{n}}=r={r_{1},r_{2},…r_{n}}. Let \(\mathbf {a}(t) = \left [\frac {r_{1} g(t)}{c_{1}},\frac {r_{2} g(t)}{c_{2}},\ldots \frac {r_{n} g(t)}{c_{n}}\right ]\). By the definition of hazard function, for Δt→0,h_{kn}(t)Δt equals the probability that the system fails within time interval (t,t+Δt] provided the system’s survival within [0,t]. The system failure is caused by the occurrence of a significant load event with probability λ(t)Δt, with which
where S(t) is the load effect at time t conditional on load occurrence as before. It is noticed that in Eq. (7), h_{kn}(t) represents the instantaneous failure probability of the system. The condition of system survival before time t refers to the case that at least k components work at any time τ∈[0,t] (not necessarily all the n components working at time τ).
Remark 1
Letting \(\mathcal {N}_{f}(\tau)\)denote the number of working components at time τ, the probability of structural survival up to time t, L(t), is determined by
If there are n_{t} load events occurring within [0,t] at times \(\phantom {\dot {i}\!}t_{1},t_{2},\ldots t_{n_{t}}\) respectively, it follows that
Eq. 9 is explained by the fact that, with an identical deterioration function for each component, the condition of \(\bigcap _{i=1}^{n_{t}} S(t_{i})\leq \mathcal {M}(\mathbf {a}(t_{i}),k)\) guarantees that the k components with the greatest resistances at the initial time survive at any time τ_{1}∈[0,t]. Furthermore, at time τ_{2}∈(t,t+dt], where dt→0, if one load event occurs (with probability λ(t)dt) and the magnitude satisfies \(S(\tau _{2})> \mathcal {M}(\mathbf {a}(\tau _{2}),k)\), then the system fails at time τ_{2}.
According to Eq. (7), the timedependent reliability for a reference period of [0,T], conditional on R=r, is estimated by
which is further rewritten as follows taking into account the uncertainties associated with the component initial resistances,
where f_{R}(r) is the joint PDF of R={R_{1},R_{2},…R_{n}}, and r={r_{1},r_{2},…r_{n}} as before. Furthermore, Eq. (11) can be simplified as follows,
where \(f_{\mathcal {M}(\mathbf {a}(0),k)}(r)\) is the PDF of \(\mathcal {M}(\mathbf {a}(0),k)\). Eq. 12 implies that the (n+1)fold integral in Eq. (11) can be converted into a twofold integral if \(f_{\mathcal {M}(\mathbf {a}(0),k)}(r)\) is known. In Eq. (12), if treating \(\mathcal {M}(\mathbf {a}(0),k)\) as a generalized initial resistance, then Eq. (12) is consistent with Eq. (2) by considering an equivalent component having a resistance of \(\mathcal {M}(\mathbf {a}(0),k)\) for the system.
Eq. 12 (or (11)) is the proposed method for timedependent reliability of an aging koutofn system, where the nonstationarity of loads can be reflected by the timevariation of λ(t) and F_{S}(·,t).
Remark 2
Note that Eq. (11) is consistent with Eqs. (3) and (4), since a series system is equivalent to an noutofn system and a parallel system is simply a 1outofn system. In fact, in Eq. (11), letting k=n gives \(\mathcal {M}(\mathbf {a}(t),k)= \min _{j=1}^{n}\frac {r_{j} g(t)}{c_{j}}\), and similarly, \(\mathcal {M}(\mathbf {a}(t),1)= \max _{j=1}^{n}\frac {r_{j} g(t)}{c_{j}}\).
Remark 3
For an ndimensional vector a and an integer k≤n, if \(\overline {\mathcal {M}}(\mathbf {a},k)\) returns the kth smallest element of a, then \(\mathcal {M}(\mathbf {a},k) = \overline {\mathcal {M}}(\mathbf {a},n+1k)\) and \(\overline {\mathcal {M}}(\mathbf {a},k) = \mathcal {M}(\mathbf {a},n+1k)\). The implementation of \(\mathcal {M}\) and \(\overline {\mathcal {M}}\) can be realized through some commercial software such as Matlab (https://www.mathworks.com). For example, the builtin function mink(a,k) in Matlab returns a vector containing the k smallest elements of a. Illustratively, if a=[1 2 3 4 5], then \(\mathcal {M}(\mathbf {a},2)=\overline {\mathcal {M}}(\mathbf {a},4)=\texttt {max(mink(a,4))}=4\).
Remark 4
Note that Eq. (12) has been based on a koutof n:G system. If considering a koutof n:F system (that is, the system fails if at least k components fail), then the hazard function for the system is
which is consistent with that in Eq. (7).
Remark 5
It is noticed that in Eq. (12), the deterioration process of each component has been assumed to be deterministic. This is applicable for many engineering cases where the variation of the deterioration process is small [42]. However, for a deterioration process with large variation, Eq. (12) needs to be modified slightly to incorporate the effect of uncertainty associated with component deterioration. With this regard, it was shown in [13] that a fullycorrelated process is reasonable to describe the component resistance deterioration. That being the case, letting f_{G(T)}(g) be the PDF of g(T), the twofold integral in Eq. (12) would become threefold by replacing the item \(f_{\mathcal {M}(\mathbf {a}(0),k)}(r)\mathrm {d}r\) with \(f_{\mathcal {M}(\mathbf {a}(0),k)}(r)f_{G(T)}(g) \mathrm {d}g\mathrm {d}r\), if taking into account the uncertainty associated with the component deterioration process.
Reliability of a weighted koutofn system
In this section, the timedependent reliability of a weighted koutofn system will be discussed, which is by nature a generalized form of Eq. (12). Recall that the system works if the sum of weights associated with the working components is no less than the threshold k.
As before, let \(\mathbf {a}(t) = \left [\frac {r_{1} g(t)}{c_{1}},\frac {r_{2} g(t)}{c_{2}},\ldots \frac {r_{n} g(t)}{c_{n}}\right ]\), and a_{s}(t) the sorted a(t) in a descending order. Let b be an ndimensional vector representing the weights of the elements in a_{s}(0). We can determine such a positive integer k_{s} that
Clearly, k_{s}=k if b_{i}=1 for ∀i=1,2,…n.
Note that \(\mathcal {M}(\mathbf {a}(t),k_{s})=\mathcal {M}(\mathbf {a}_{s}(t),k_{s})\). With this, the hazard function h_{wkn}(t) at time t, conditional on R=r, is determined by
Similar to Eq. (10), the timedependent reliability for a service period of [0,T], conditioned on R=r, is estimated by
which, using the law of total probability, is rewritten as follows considering the uncertainty of R,
By referring to Eq. (12), Eq. (17) can be further reduced to
where \(f_{\mathcal {M}(\mathbf {a}(0),k_{s})}(r)\) is the PDF of \(\mathcal {M}(\mathbf {a}(0),k_{s})\).
Eq. 18 (or (17)) presents the proposed timedependent reliability method for a weighted koutofn system in the presence of resistance deterioration and correlation. It has a similar form to Eq. (12) except the item k_{s} involved, due to the impact of the weights for each component.
Implementation of timedependent reliability assessment
For the (weighted) koutofn system reliability in Eqs. (11) and (17), if the expressions of \(f_{\mathcal {M}(\mathbf {a}(0),k)}(r)\) and \(f_{\mathcal {M}(\mathbf {a}(0),k_{s})}(r)\) are unaccessible, one would need to solve the multifold integrals numerically. This is usually timeconsuming to conduct, and one could alternatively assess the reliability via Monte Carlo simulation, which is especially powerful in dealing with multidimensional problems with mathematical simplicity and robustness [20,40,43,44]. Technically, one can first generate a sample for R, denoted by r, and then compute the cores of Eqs. (11) and (17) numerically with R=r, with which the multifold integration can be evaluated via simulation according to
where ∗=kn or wkn, and E() denotes the mean value of the random variable in the brackets.
Illustrative example
In this section, an illustrative koutofn system will be used to show the applicability of the proposed method in Eqs. (12) and (18).
Structural configuration
Consider a koutof5 system with totally five components, which have identical physical configuration and load conditions. Table 1 presents a summary of the probabilistic models of resistance and loads (dead load and live load). Two live load models, namely LL1 and LL2, are considered, which are associated with an occurrence rate of 1.0/year. The first is representative of a stationary load process, and the second is a nonstationary load process with an increasing mean load magnitude. Suppose that each load event will induce identical load effect to each component, with which c_{j}=1 for j=1,2,…5. The deterioration of resistance for each component is identical and deterministic, taking a form of g(t)=1−ηt^{α}, where η and α are two parameters reflecting the rate and shape of the deterioration process. The deterioration function evaluated at the end of 50 years equals 0.8. The initial resistances of the components are identically distributed and equally correlated pairwise with a correlation coefficient of ρ=0.5, unless noted otherwise. The Gaussian copula is used to describe the joint behaviour of different component resistances at the initial time.
System reliability assessment
The system reliability is discussed in this section. Figure 1 presents the timedependent failure probabilities P_{f,kn}(T)=1−L_{kn}(T) for reference periods up to 50 years associated with a single component and a 3outof5 system with k=3 and a linear deterioration function. The two live load models as summarized in Table 1 are considered. Note that a(0)=R since c_{j}=1 for each j. The probability of failure of a single component is calculated according to Eq. (2), while the system failure probability is estimated with 100,000 replications of simulation. From Fig. 1 it is seen that, for either a component or a system, a greater load intensity leads to a greater probability of failure as expected. Furthermore, for either a component or a system, the logarithm of failure probability increases approximately linearly with the duration of reference period after 20 years. In the presence of the same load condition, the system reliability is greater than that of a single component because of the redundancy of the 3outof5 system. In fact, as shown in Table 2, the mean value of \(\mathcal {M}(\mathbf {R},3)\) is close to that of component resistance, while the standard deviation of \(\mathcal {M}(\mathbf {R},3)\) is smaller, yielding a greater system reliability.
One can use the results in Fig. 1 to estimate the system’s performance (e.g., the service life) under the context of reliabilitybased assessment. For instance, for the 3outof5 system with a target reliability index of β=3.5 (corresponding to a failure probability of 2.33×10^{−4}), the service lives associated with live load models 1 and 2 (c.f. Table 1) are 24.5 and 19.5 years respectively. The difference between the predicted service lives is evident of the importance of considering the future varying trend of loads in an attempt to reasonably estimate the system performance.
The timedependent failure probabilities for 4outof5 and 5outof5 (series) systems are presented in Figs. 2 and 3, respectively. The deterioration shape is linear for all components and the two live load models in Table 1 (LL1 and LL2) are used. Similar to Fig. 1, a greater load intensity results in a lower reliability due to the enhanced load risks. The failure probability of a 4outof5 system is close to that of a single component, due to the closeness between the probabilistic behaviour of \(\mathcal {M}(\mathbf {R},4)\) and component resistance at the lower tail. As shown in Table 2, both the mean value and standard deviation of \(\mathcal {M}(\mathbf {R},4)\) are smaller than those of the component resistance, and these two factors have opposite effects on the system reliability. In Fig. 3, the probability of failure of a series system is greater than that of a single component, which is consistent with that reported in [20].
Comparing Figs. 1 through 3 it can be seen that when n is fixed, a greater k results in a smaller reliability. Correspondingly, with a target reliability index of 3.5, the predicted service life of the system is shortened with a greater value of k. This can be explained by observing the monotonicity of \(\mathcal {M}\). In fact, for two integers k_{1} and k_{2} satisfying 1≤k_{1}≤k_{2}≤n, it follows that \(\mathcal {M}(\mathbf {a},k_{1})\geq \mathcal {M}(\mathbf {a},k_{2})\) by the definition of \(\mathcal {M}\).
The dependence of system failure probability on the number of components, n, is presented in Fig. 4, where the load model LL2 in Table 1 is used. A greater number of components results in a smaller failure probability of the system, due to the enhanced system redundancy.
The timedependent failure probabilities for a weighted koutof5 system are shown in Fig. 5, with a linear deterioration model and LL2. It is assumed that the weights for the five components are 1,2,3,4 and 5 respectively. Note that in this case, k could be greater than n while k_{s} varies between 1 and n. It is observed from Fig. 5 that a smaller value of k gives a smaller failure probability due to the weaker requirement of system survival. Furthermore, with both k and n fixed, the system failure probability associated with a weighted system is smaller than that of an ordinary system because k_{s}≤k.
Roles of resistance correlation and deterioration
Figures 6 and 7 examine the effect of resistance deterioration function on the timedependent reliability of a 4outof5 system, where LL2 in Table 1 is used. The component resistance degrades by 20% over 50 years in Fig. 6 and 30% in Fig. 7. The different values of α indicate different dominant deterioration mechanisms. It can be seen that a severer resistance deterioration leads to a greater failure probability due to the increased probability of load effect exceeding the resistance. Furthermore, a squareroot deterioration shape (with α=0.5) results in the greatest failure probability, followed by linear and parabolic deterioration models. This is because, with the same g(50), the deterioration mainly occurs at the early stage with α=0.5, which gives the smallest resistance over the time period of 50 years.
The timedependent failure probabilities associated with a weighted 4outof5 system subjected to LL2 are presented in Figs. 8 and 9, where the weights for the five components are 1 through 5 respectively. It is observed that a severer resistance deterioration or a smaller value of α leads to a greater failure probability, which is consistent with the observations from Figs. 6 and 7. Furthermore, the failure probability of a weighted system is smaller than that of an ordinary one due to the fact that k_{s}≤k.
Figures 10 and 11 show the dependence of timedependent 4outof5 system failure probability on the resistance correlation ρ. The LL2 in Table 1 is used and the component resistance degrades linearly by 20% over 50 years. The system is an ordinary one in Fig. 10 and weighted in Fig. 11. A greater resistance correlation results in a greater failure probability due to the increased variation of \(\mathcal {M}(\mathbf {R},4)\). In fact, as ρ increases from 0.3 to 0.9, the standard deviation of \(\mathcal {M}(\mathbf {R},4)\) increases from 471.3 to 673.7 kN ·m. While the mean of \(\mathcal {M}(\mathbf {R},4)\) increases from 3286.0 to 3486.0 kN ·m simultaneously, the system reliability is less sensitive to this increase in mean. However, the relationship between the system reliability and the resistance correlation, as revealed in Figs. 10 and 11, does not necessarily hold for any k. For example, Fig. 12 shows the timedependent failure probability for a 5outof5 (series) system, where the structural configuration is the same as that in Fig. 10. It can be seen that a greater resistance correlation results in a smaller failure probability due to the fact that the increase of mean value of \(\mathcal {M}(\mathbf {R},5)\) dominates in the system reliability compared with the increase of standard deviation. Thus, one should carefully identify the resistance correlation when assessing the system reliability in practice.
Finally, it is noticed that for a koutofn system, the reliabilities with k=1 (parallel system) and k=n (series system) are the upper and lower bounds for the system reliability, respectively, if the resistance deterioration for each component is identical. This can be seen by recalling Eq. (12), where \(\mathcal {M}(\mathbf {a}(0),n)\leq \mathcal {M}(\mathbf {a}(0),k) \leq \mathcal {M}(\mathbf {a}(0),1)\). A more rigorous proof is as follows. Since
it follows that,
or equivalently,
According to Eq. (7), one has
with which
This relationship can be further extended to a weighted koutofn system, and the proof is similar (by replacing k with k_{s}, which varies between 1 and n at any time t).
Concluding remarks
In this paper, a new method is presented for the estimation of timedependent reliability of aging koutofn systems (for both ordinary and weighted ones), considering the nonstationarity in the external loads and the component resistance deterioration and correlation. An illustrative example is presented to demonstrate the applicability of the proposed method.
Analytical results show that, for a component or a koutofn system (either ordinary or weighted), the increase of mean load intensity has a significant impact on the failure probability. For a koutofn system, an increase of component number leads to a smaller failure probability. Furthermore, a severer resistance deterioration or a squareroot deterioration shape results in a greater failure probability due to the increased risks of load effect exceeding structural resistance. The reliabilities associated with a parallel system and a series system serve as the upper and lower bounds of the koutofn system reliability. However, the relationship between the system reliability and the resistance correlation is not necessarily monotonic, suggesting the importance of reasonably identifying component resistance correlation in the estimate of system safety level.
It is finally noticed that the koutofn system considered in this paper has been assumed to have identical component resistance deterioration. More research efforts are needed in the future to generalize the case to that with different deterioration processes of component resistance.
Availability of data and materials
All data generated or analysed during this study are included in this published article.
Abbreviations
 CDF:

cumulative distribution function
 PDF:

probability density function
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The author would like to acknowledge the thoughtful suggestions of two anonymous reviewers, which substantially improved the present paper.
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The research described in this paper was supported by the ViceChancellor’s Postdoctoral Research Fellowship from the University of Wollongong. This support is gratefully acknowledged.
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Wang, C. Timedependent reliability of (weighted) koutofn systems with identical component deterioration. J Infrastruct Preserv Resil 2, 3 (2021). https://doi.org/10.1186/s43065021000181
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DOI: https://doi.org/10.1186/s43065021000181