On reliability assessment of weighted k-out-of-n systems with multiple types of components

ABSTRACT This paper studies the reliability function of a weighted $k$k-out-of-$n$n system with multiple types of components. A weighted system with $M$M different types of components is considered and two real-life scenarios for the operation of the system are defined. In the first one, it is assumed that the system operates if the accumulated weight of all active components is more than a predetermined threshold $k$k, otherwise the system fails. In the second scenario, the system is assumed to operate if the total weight of the operating components of type $i$i is more than or equal to the corresponding threshold ${k_i}$ki, for at least $m$m types of components, $m \le M$m≤M. Our approaches in assessing the reliability function of the system lifetime rely on the notion of survival signature corresponding to multi-type weighted $k$k-out-of-$n$n system. The formulas for the proposed survival signature are presented and the corresponding computational algorithms are also given. To examine the theoretical results, the reliability function and other aging characteristics of a wind farm (system) consisting of three plants located in three different regions are assessed numerically and graphically. Finally, an allocation problem to determine the best choice for the distribution of each type in the weighted $k$k-out-of-$n$n systems is studied.


Motivation and literature review
In recent years, weighted k-out-of-n systems have captured the attention of many researchers in reliability engineering because of a wide range of their applications in different fields.Many systems in the real world can be considered as weighted systems.For example, a heating, ventilation and air conditioning (HVAC) system is a weighted system in which one can consider the conditioning, cooling and heating units as weighted components according to their different capacities (see Samaniego & Shaked, 2008).The oil transmission systems in which pipes may have various transmission capacities and wind turbines with various generating capacities in power generation systems are also known examples of weighted k-out-of-n systems (see Eryilmaz & Kan, 2020).A system with n components is said to have a k-out-of-n structure, if it operates as long as at least k components out of the n components operate.In a weighted k-out-of-n system, the components have different loads or capacities (weight), say w i , i ¼ 1; 2; . . .; n, and each component contributes differently to the system performance according to its weight.A weighted k-out-of-n system then fails whenever the accumulated Wang et al. (2013), He et al. (2020), Han et al. (2021), Torrado (2021), Hu et al. (2021) and references therein.

Contributions of this paper
In this study, a weighted k-out-of-n system consisting of M, 2 � M � n, different types of components is considered such that there exist n i components of type i, i ¼ 1; 2; . . .; M, P M j¼1 n j ¼ n.We assume that the weight for the jth component of type i is w ðiÞ j , j ¼ 1; 2; . . .; n i .Two different scenarios for operation of the system described previously are considered: (a) In the first model, at a fixed time instant t, we assume the system operates if the accumulated weight of the functioning components is greater than or equal to a predetermined threshold k, where (b) In the second scenario, the system operates if the total weight of the functioning components of type i is more than or equal to the corresponding threshold k i , for at least m values of i 2 f1; 2; . . .; Mg, where Here, m 2 f1; 2; . . .; Mg is a prefixed value.
We assume that the components' lifetimes in each group are independent and identically distributed (i.i.d.) according to an absolutely continuous cumulative distribution function F i ðtÞ, i ¼ 1; 2; . . .; M. It is also assumed that all components are operating independently.For each of the parts (a) and (b), we obtain the formulas for the system survival signature based on which we compute the reliability function of weighted k-out-of-n systems with multiple types of components.As an application, the reliability function and other aging characteristics of a wind farm (system) consisting of three plants located in three different regions are assessed numerically and graphically.Some discussions on the optimal allocation of distributions to different types of components are also given.
In order to assess the reliability of the weighted systems, we consider two realistic scenarios that are different from the existing works in the literature that we reviewed in the previous subsection.In our approach, it is assumed that the components are categorized into M distinct groups, wherein each group the components may have a different lifetime distribution function from other groups.In addition, in the second model considered in the paper, we assume that not only the total capacity of the operating components in the system contributes to the system performance, but also the total capacity of operating components in each group directly affects the system performance.To this end, we propose the concept of survival signature associated with the weighted k-out-of-n systems consisting of several groups of components.
As the weighted k-out-of-n systems consisting of multiple-type components arise naturally in many real-life applications (see Section 3 for a real example), the achievements of the current study are practical for the managers and the system designers to evaluate the performance of such systems in order to preserve them in optimal conditions.Our proposed approaches to assess the reliability of weighted k-out-of-n systems with several types of components provide an opportunity for system designers to evaluate the system reliability under realistic scenarios that have not already been considered in the literature.
Hence, from the decision-making point of view, these results may allow the system managers to get more insights into the system performance for optimal maintenance and avoid unprofitable options.

Organization of the paper
The structure of the article is as follows: In Section 2, the definition of survival signature associated with coherent systems consisting of heterogeneous components and the corresponding reliability function are given.In Section 3, we precisely describe the weighted k-out-of-n system for each model (a) and (b), and then provide the exact formula for computing the corresponding survival signatures.Section 4 is devoted to applications and some illustrative examples in the real world.In particular, we examine our results for a wind farm (system) consisting of several wind turbines in three plants located in three different regions.In Section 5, using some empirical cases, a discussion is made on improving the system reliability in the sense of allocating components.The paper is finalized in Appendix Section by presenting some algorithms for computing the survival signatures of weighted k-out-of-n systems with different types of components, and an example for computing survival signature for a system with four types of components using the developed formulas.

System description
The concept of signature, introduced by Samaniego (1985), is a helpful tool that has been widely used for the reliability evaluation of n-component coherent systems.The signature associated with an n-component coherent system is a vector s ¼ ðs 1 ; s 2 ; . . .; s n Þ in which the ith element s i denotes the probability that the failure of the ith ordered component leads to the breakdown of the whole system.An exact formula for the reliability function of weighted k-out-of-n systems using signature vector s is obtained by Franko and Tütüncü (2016).The cited authors suggested an algorithm to compute the signature vector for such systems and derived various reliability indices via the system signature.
Although the system signature is useful for assessing the reliability properties of the coherent systems, it is only applicable for the systems with the same type of components.This limitation makes the signature inapplicable for most practical systems that may arise in real situations, like ones with multiple types of components having different life distributions.For the systems with M types of components, 1 � M � n, the concept of survival signature, introduced by Coolen and Coolen-Maturi (2012), is found to be very useful.Consider a coherent system with M different types of components.The components in group i are supposed to be i.i.d. with a common cumulative distribution function F i ðtÞ, and survival function � F i ðtÞ, i ¼ 1; 2; . . .; M. Let C i ðtÞ denote the number of operating components of group i at a time instant t.According to the i.i.d.assumption of components in each group, we have On the other hand, obtaining the reliability function of the weighted system at some time instant t requires knowing the previous probability for all groups of components at the same time t.Accordingly, it is necessary to obtain this probability for all types simultaneously.In other words, we should find P T M i¼1 C i ðtÞ ¼ l i f g À � which denotes the probability that the number of operating components in the first, second, . .., and Mth type are l 1 , l 2 , . .., and l M , respectively.From the independence assumption between different groups of components, we have Let Φ l 1 ; l 2 ; . . .; l M ð Þ represent the probability that the system operates with exactly l 1 components from the first type, exactly l 2 components from the second type, . .., and exactly l M components from the Mth type, where 0 � l i � n i , i ¼ 1; 2; . . .; M. The function Φ l 1 ; l 2 ; . . .; l M ð Þ is called the survival signature of the system.Notice that due to the i.i.d.assumption for the failure times of the n i components of type i, i ¼ 1; 2; . . .; M, all the state vectors corresponding to these components are equally likely to occur.Using the total probability low and using (1), the reliability of a weighted system can be written as Based on the concept of survival signature, several studies for systems with multiple types of components were posted.For recent references on this topic, see, for example, Feng et al. ( 2016), Eryilmaz et al. (2018), Hashemi et al. (2020), and references therein.In the next section, we present the reliability function of a weighted system consisting of heterogeneous components using the corresponding survival signature.
Before giving the models, we review the following assumptions: Assumption 1.A weighted system with n components is considered.
Assumption 2. The components of the system are selected from M different types, 1 � M � n.In the system, there exist n i components from type i where 1 � n i � n.
Assumption 3. The components from type i are supposed to be i.i.d. with a continuous distribution function F i t ð Þ and the reliability function � F i ðtÞ, i ¼ 1; 2; . . .; M.
Assumption 4. Complete independence is also assumed between the components in different types.
Assumption 5.The jth component of the type i has its own weight w ðiÞ j , j ¼ 1; 2; . . .; n i , i ¼ 1; 2; . . .; M.Then, the vector w ¼ ðw the corresponding component operates and ε ðiÞ j ðtÞ ¼ 0, otherwise.In the sequel, we will consider two scenarios for operation (or equivalently for the failure) of the system described previously.These two models are among the most commonly used models that have practical applications in engineering systems as we shall see in Section 4.

Model I
For the system described here, suppose that it operates if the total weight of the operating components at time t is greater than or equal to a predetermined threshold k.In other words, if ϕ 1 ðεðtÞÞ denotes the system state (the structure function) at t, then we have where Ið�Þ denotes the indicator function.In what follows, we obtain the survival signature for a weighted system with the above structure function.We restrict our attention to the class of weighted systems for which Φ l 1 ; l 2 ; . . .; l M ð Þ is non-decreasing in l i , i ¼ 1; 2; . . .; M, Φð0; 0; . . .; 0Þ ¼ 0, and Φ n 1 ; n 2 ; . . .
Under the assumptions, it is clear that for type i of components, there exists n i l i � � cases for the state vector corresponding to components of type i with exactly l i operating ones.In other words, there is � � number of ways to select l i components from the group i, and set the respective state vector v ðiÞ l i , i ¼ 1; 2; . . .; M, where v ðiÞ l i for components of type i.Furthermore, let V l 1 ;l 2 ;...;l M represent the set of all state vectors of the form for which exactly l 1 components from the first type, l 2 components from the second type,. .., and l M components from the Mth type operate.In other words, the set V l 1 ;l 2 ;...;l M contains all vectors of the size n that satisfies � � number of different vectors as different possible cases.Due to the i.i.d.assumption for the lifetimes of components of the same type, all these state vectors are equally likely, and hence if Φ I ðl 1 ; l 2 ; . . .; l M Þ denotes the survival signature of the system in the first model, we have The right-hand side of (3) denotes the probability that the system operates with an exactly specified number of components of each type.Note that the numerator P the number of permutations with exactly l 1 operating components from the first type, l 2 operating components from the second type, . .., and l M operating components from the Mth type for which the system is in the up state.In general, the computation of survival signature may be difficult and time-consuming.For this purpose, we have provided an algorithm for computing the survival signature of a weighted system, which is given in Appendix A.
Remark 2.1.The vectors in the set V l 1 ;l 2 ;...;l M for which the system is in the up state (i.e. the sum of the weights is greater than or equal to k) are, in fact, the so-called path vectors in which there are exactly l 1 operating components from the first type, l 2 operating components from the second type, . .., and l M operating components from the Mth type (see, Barlow and Proschan (1981) for the definition of path vectors).We also conclude that the path vectors of the system and hence the survival signature are strongly dependent on the components' weights and how the system failure is defined.Consequently, the survival signature will change according to a different threshold for the system or different models for the system failure.

Model II
Consider the weighted k-out-of-n system described at the beginning of this section.A different mechanism for the operation of such a system may be defined as follows: The system operates if the total weight of operating components of type i is greater than or equal to the pre-specific threshold k i , for at least m values of i 2 1; 2; . . .; M f g, 1 � m � M. We shall call such a structure as 'the generalized weighted ðk 1 ; k 2 ; . . .; k M Þ-out-of-n system'.In fact, in this model, components of type i constitute a weighted k i -out-of-n i subsystem, and the whole system acts as an m-out-of-M structure with M partial subsystems.In other words, the whole system fails at the time of the (M À m þ 1)th failure among its subsystems.Let εðtÞ represent the status vector of the system at time t.If ϕ m ðεðtÞÞ denotes the system state at time t, then In this model, for operating the system, it is not necessary that all the M types operate, but rather the system can operate with at least m types of components, m ¼ 1; 2; . . .; M. In other words, the system may be considered as an m-out-of-M structure consisting of weighted k i -out-of-n i subsystems.
Such systems arise in various situations of real life.As an example, consider a factory system that depends on a number, say, M of energy sources (electricity, oil, gas, etc.).Each source contains several lines with varying capabilities that provide the system with a specified kind of energy (the electricity source lines could be wind turbines with different capacities).Each source can be considered as a weighted subsystem which operates if it provides the device with at least a specified amount of energy, say, k i .The system can be supplied with M warning lights, one for each energy source.When each source fails (i.e. the delivered energy from this source becomes less than its threshold), the corresponding light turns off.The whole system is supposed to operate when at least M lights are active.Another example may be the wind power system consists of M plants that are located in different areas, and each plant contains n i turbines.Plant i is supposed to fail whenever the total delivered energy from all turbines in this plant becomes less than a predetermined threshold k i of the electricity, i.e. each plant forms a weighted k i -out-of-n i subsystem.Furthermore, when ðn À m þ 1Þ plants fail, it means that the wind speed generally comes down in most areas, and the wind power is no longer feasible in the economic sense to generate electricity.Hence, it is recommended to switch to another alternative source such as solar systems.
As we have pointed out earlier, the ith subsystem has the partial survival signature where Φ i ð Þ j ð Þ, j ¼ 1; 2; . . .; n i , indicates the probability that the subsystem i operates with exactly j components.Since for operating the whole system, it is required that at least m out of M subsystems operate, the system may operate with no functioning components in some types.On the other hand, because each subsystem surely fails when all of its components fail, we should take Φ i ð Þ 0 ð Þ ¼ 0 for i ¼ 1; 2; . . .; M. As in Model I, the survival signature of the system can be calculated as The independence of M groups of components makes it possible to provide the survival signature of the system differently and more simply as follows.Let us consider all the components in each group as a subsystem.Then it is clear that, in the model described above, the failure of the components in one subsystem will only contribute to the failure of that subsystem according to their weights, and does not affect the total weight of operating components in other subsystems.In contrast, each component will contribute to the system failure according to its weight.As a consequence, each group can be considered as a weighted subsystem and will have its own partial survival signature.For type i, let the vector be the partial survival signature, where the element Φ i ð Þ j ð Þ, i ¼ 1; 2; . . .; M, j ¼ 1; 2; . . .; n i , denotes the probability that the ith subsystem (group) operates with exactly j components.If represents the signature vector corresponding to this weighted subsystem (see, e.g.Franko & Tütüncü, 2016), then it can be , .We aim to obtain the system survival signature using the partial survival signatures of its types.Based on the failure model of the system described previouly, we have where l i ¼ 0; 1; . . .; n i , i ¼ 1; 2; . . .; M. In this case, as the ith subsystem operates with exactly l i components with probability Φ ðiÞ ðl i Þ, and by the independence between the subsystems, the probability that r of the subsystems operate with exactly l i components from subsystem i, will be equal to the product of the corresponding Φ i ð Þ l i ð Þ's.Thus, the summation on all these cases is taken as follows: Hereafter, for short, we denote ψ r ðl 1 ; l 2 ; . . .; l M Þ by ψ r .Furthermore, according to the inclusionexclusion principle (see Feller, 1968), to obtain the survival signature Φ m l 1 ; l 2 ; . . .; l M ð Þ given in relation ( 5), it is sufficient to know the values of ψ r 's, r ¼ 1; 2; . . .; M, as well as the overlaps for the operation of m; m þ 1; . . .; M subsystems occurring simultaneously.Then, to obtain the survival signature we can write (for a sketch of the proof, see Appendix E) where the values of Λ i 's for i ¼ m þ 1; . . .; M, indicate all the overlaps in the operation of m þ 1, m þ 2 . .., M subsystems simultaneously, and are given by and CðM; r; sÞ is given by For more details on deriving this result, see Appendix C.
In the special case in which m ¼ M, the system can be considered as a series structure consisting of M subsystems each of which having components of the same type, and the ith subsystem is a weighted k i -out-of-n i system, i ¼ 1; 2; . . .; M. In such a situation, we have In fact, by computing the survival signature for each group separately, it is easy to obtain the survival signature of the whole system just by using (9).
Remark 2.2.It is worth mentioning here that Equation ( 6) can be utilized to obtain the reliability function of a traditional k-out-of-n system having different component reliabilities.To be precise, if p i denotes the reliability of component i in a k-out-of-n system, then the system reliability hðp 1 ; p 2 ; . . .; p n Þ can be computed by taking M ¼ n and m ¼ k in Equation ( 6) as where and Λ kþ1 , Λ kþ2 ,. .., Λ n can be obtained using (7).

Applications and illustrative examples
Wind plants are among the most important sources of renewable energy.Generating electricity by wind farms has become one of the main sources of electricity in many countries.A wind plant consists of some wind turbines.In general, the ith wind plant generates a power P i equals to the sum of the real power produced by its constituent wind turbines as where n i denotes the number of wind turbines in the plant i, P j is the power released by the jth wind turbine and PL i represents the ith wind plant's collector system losses at the current operating state, see Louie and Sloughter (2014).Typically, a wind power generation system consists of one or more wind plants, and each plant may also contain more than one turbine.Each turbine has its own capacity.The wind plant is an example of a system with weighted components.Several factors influence the progress of generating energy in a wind generating system.Wind speed may be the most critical factor in the released energy of a wind turbine.Several studies have shown that the two-parameter Weibull distribution can be suitable for modeling the wind speed as a random variable, see, for example, Justus et al. (1978), Tuzuner and Yu (2008), and Louie and Sloughter (2014).The parameters α and β of the Weibull distribution can be estimated using a sample of wind speeds in each region (Tuzuner & Yu, 2008).For many wind plants located in different places (with considerable divergence), naturally, the wind speed and, as a result, the distribution which models the wind speed varies from one plant to the other.In other words, each plant has its own distribution parameters, which justifies the variation of turbine lifetimes in the system and the distribution assigned to each type.Louie and Sloughter (2014) indicated that the overall efficiency of the energy conversion process of a turbine depends on its design and operating state.Therefore, the capacity (weight) of a turbine probably differs from one turbine to another.This system is an appropriate example of weighted systems with multiple types of components.The plants in different regions may be considered as separate groups, each with its own distribution function according to the wind speed in that region.Turbines with various capacities of the same type may be taken as components with different weights.Such a system is typically connected to an electrical substation that would supply specific places with electricity.The failure of such a system is defined by the system administration team.For example, the system may be built to provide at least a predetermined value of megawatts, k, say, or the work requirements put more restrictions on the operational process.For example, a minimum threshold value of megawatts for every plant i may be determined, k i , say, such that in every plant, the total generated power must be more than or at least equal to the prespecified threshold k i .The following is a numerical example to illustrate the problem.
2 ¼ 4 and w ð2Þ 3 ¼ 8, and the third plant has two turbines with capacities w ð3Þ 1 ¼ 9, and w 3 ð Þ 2 ¼ 10, in megawatts.A Weibull distribution, Wðα; βÞ, with the probability density function is considered as a distribution model for the wind speed.According to the distance between the regions, the wind speed and hence the distribution that models the wind speed differs.It is assumed that the components of type i have a Weibull distribution with parameters ðα i ; β i Þ, i ¼ 1; 2; 3. Based on samples of wind speed in each type, one can estimate the distribution parameters using various methods of estimating, such as the maximum likelihood method (Tuzuner & Yu, 2008), see also Justus et al. (1978) in which estimating the parameters α i and β i was done using the mean and the standard deviation of the samples.The failure of such a system occurs whenever the accumulated outcome energy of the farm (i.e. the energy produced from the three plants together) becomes less than k ¼ 30 megawatts.Assume that the components of the first type are i.i.d. with Weibull distribution F 1 ,Wð1:5; 3Þ, the second group has i.i.d.components with distribution F 2 ,Wð2; 5Þ and the third group has i.i.d components with distribution F 3 ,Wð3; 6Þ.
At a time instant t, the structure function of the system is To obtain the system reliability function, we need to compute the survival signature Φ I l 1 ; l 2 ; l 3 ð Þ for all values l 1 ¼ 0; 1; 2; 3; 4; 5, l 2 ¼ 0; 1; 2; 3 and l 3 ¼ 0; 1; 2. Thus, the survival signature should be calculated for all possible values of l i ; i.e. for ðn The values of the survival signature of the system are present in Table 1.
Example 3.2.Consider the weighted system in Example 3.1.Let the system fail according to Model II with m ¼ 2; i.e. the system failure occurs when the total weight of the functional components in at least two groups becomes less than the corresponding thresholds.In these circumstances, the wind plant of type i should generate at least a predetermined amount k i of   electricity.Otherwise, the plant fails and generating energy will stop.Using the traditional counting algorithm, the survival signature of the system according to Model II is computed in Table 2 for all possible values of l i ¼ 0; 1; . . .; n i , i ¼ 1; 2; . . .; M. In this model, as mentioned in the previous section, each group of turbines can be treated as a distinct weighted subsystem.Accordingly, to obtain the survival signature of the system, it is required to compute the survival signature of each type, separately.If one takes the thresholds k 1 ¼ 5, k 2 ¼ 6 and k 3 ¼ 9 megawatts for the first, second and third types, respectively, then the corresponding survival signatures are as follows: Consequently, the elements of the system survival signature can be computed using Equation (6).For example, let us compute the element Φ 2 ð2; 1; 2Þ which indicates the probability that the system operates with exactly two components from the first type, one component from the second one, and two components from the third one.Using (6) with M ¼ 3 and m ¼ 2, we have To compute the required probability, one only need the elements Φ ð1Þ ð2Þ ¼ 9 10 , Φ ð2Þ ð1Þ ¼ 1 3 and Φ ð3Þ ð2Þ ¼ 1.The values of ψ 2 , ψ 3 and Λ 3 can then be obtained as and Thus, we get which equals to the highlighted element in Table 2 for the values l 1 ¼ 2, l 2 ¼ 1 and l 3 ¼ 2. Note that according to this failure model, when two (or more) of the subsystems (groups) fail, the system itself fails.Consequently, when any one of the groups has no operating components; i.e. l i ¼ 0 for at most one group, then the system may operate with the remaining two types.For example, it is observed that Φ 2 ð2; 1; 0Þ ¼ 3 10 , which shows that the system operates with exactly two components from the first type, one component from the second one and no operating component from the third one with probability 3 10 .Based on the distributions considered in Example 3.1, the system reliability functions and the corresponding hazard rates are plotted in Figure 2(a, b), respectively.It can be observed that, in contrast to Model I in Example 3.1, Case 3 leads to the highest system reliability for Model II.

Optimal allocation of components
As mentioned above, the Weibull distribution with two parameters is a suitable distribution for fitting the data corresponding to the wind speed.Tuzuner and Yu (2008) gave two methods for estimating the parameters α and β of the Weibull distribution.According to Tuzuner and Yu's (2008) results, depending on a sample of wind speed measurements in a specific region, one can estimate the parameters of the Weibull distribution.Consequently, for each region, there will be a particular Weibull distribution with some particular parameters α and β.Thus, it is possible to know the distribution characteristics of the area in which we want to install the air turbines in advance.Here, the fundamental problem lies in determining the best way to allocate the generating plant, to obtain the maximum reliability of the power generation system.
Regarding Example 3.1, if we have prior information about the random variable that models the wind speed in each of the three regions, then the goal is to distribute the three power plants to the three regions so that we get an integrated generation system with optimum reliability.The following example illustrates this point.
Example 3.1 (continued).Assume that the parameters of the Weibull distribution in the three regions are estimated as Wð1:2; 5Þ, Wð2:1; 7:5Þ and Wð2:8; 12Þ for the first, second and the third regions, respectively.The question is how to assign the three plants to the three regions to obtain the best reliability for the whole system.All possible cases for such an allocation problem are given in Table 3.The system reliability function in each case is plotted in Figure 3 with the threshold k ¼ 30.As figure shows, the ideal case that results in optimal system reliability is the case in which F 1 ðtÞ,Wð2:1; 7:5Þ, F 2 ðtÞ,Wð1:2; 5Þ and F 3 ðtÞ,Wð2:8; 12Þ.In fact, one recommends installing the first plant (which contains five turbines) in the second region, the second plant in the first region and the third plant in the third region to obtain the most reliable system in this case.However, the situation in which F 1 ðtÞ,Wð2:1; 7:5Þ, F 2 ðtÞ,Wð2:8; 12Þ and F 3 ðtÞ,Wð1:2; 5Þ is the worst case, and any of the other ones gives more desirable results in the system reliability point of view.
Another interesting problem that may arise in applied situations is the case that a system has two (or more) groups of components where each group has the same total weights; i.e.P n 1 j¼1 w ð1Þ j ¼ P n 2 j¼1 w ð2Þ j .Suppose that the environmental conditions and location of each  component determines the corresponding lifetime distribution.Then, the question is: Which group of components should be assigned to each of the two locations?We will address such an issue in the following example.
Example 4.1.Consider a system consists of two groups of components.The first group has n 1 ¼ 7 components with weights w 6 ¼ 5 and w ð1Þ 7 ¼ 5, and the second one has n 2 ¼ 3 components with weights w ð2Þ 1 ¼ 7, w ð2Þ 2 ¼ 8 and w ð2Þ 3 ¼ 9. Note that the sum of all weights in each group is the same.Suppose that the system fails according to Model I with k ¼ 27; that is, it fails when the total weight of all operating components becomes less than the threshold k ¼ 27.Further, assume that each group of components should be installed in one of the two locations L 1 and L 2 .Let the lifetime of each installed component in locations L 1 , and L 2 be distributed as Wð2:4; 8Þ and Wð1:3; 2:5Þ, respectively.The question that arises here is that what is the optimal allocation of two groups to the locations in order to get a more reliable system?To answer this question, let F 1 , and F 2 be the distributions of components in the first and the second groups, respectively, depending on the location of the installation.The reliability functions in the two apparent cases are plotted in Figure 4(a).It can be concluded that, to have more reliability in Model I, it is better to install the components of the first group in L 2 , and to assign the components of the second group to L 1 .Now, suppose that the system fails according to Model II with m ¼ 2, k 1 ¼ k 2 ¼ 22; i.e. it fails when the total weight of the active components in each of the two groups becomes less than the threshold k 1 ¼ k 2 ¼ 22.In this case, to get more reliability, the best way is to allocate the components of the first and second groups to the locations L 1 and L 2 , respectively, see Figure 4(b).

Conclusions
In the present paper, we obtained the survival signature for a weighted k-out-of-n system with multiple types of components.It was assumed that the components could be categorized into M groups, each consisting of components whose lifetimes are independent and identically distributed random variables.Two different models of failure for such a system are defined.In the first model, the system fails when the total weight of the operating components in the system becomes less than a determined threshold k.In the second model, the system failure occurs whenever the total weight of operating components in at least a specific number m of groups becomes less than the corresponding threshold for that group.In each model, the survival signature of the system is obtained as a function of the weights of components.In addition, for the second model, we developed a new method to get the survival signature of the system using the partial survival signatures of different groups of components.The method can also be applied for obtaining the reliability function of the traditional k-out-of-n systems with different reliabilities for the components.The system reliability functions for two models are then evaluated using the computed survival signature.To illustrate the theoretical results for each model, the reliability function of a wind farm (system) consisting of three plants located in three different regions was assessed numerically and graphically.Finally, some practical allocation problems for wind plant systems based on the obtained reliability function with some examples is investigated.Set of all state vectors of the system with exactly l i operating components of type i

Example 3. 1 .
Consider a wind farm (system) consists of three plants located in three different regions with considerable divergence.The first plant has five turbines with capacities w ð1Þ 1 ¼ 1, w ð1Þ 2 ¼ 2, w ð1Þ 3 ¼ 4, w ð1Þ 4 ¼ 5, and w ð1Þ 5 ¼ 6, in megawatts.The second plant contains three turbines with capacities w

Figure 1 .
Figure 1.The reliability functions; and (b) the hazard rates of the system in I for different Weibull distributions.

Figure 2 .
Figure 2. (a) The reliability functions; and (b) the hazard rates for the system in Model II for different Weibull distributions.

Figure 4 .
Figure 4. (a) the reliability functions for Model I; (b) the reliability functions for Model II.
Þ may be regarded as the weights vector for the system's components.

Table 3 .
All possible cases for three groups of components.
Weight of the jth component of type i wWeights vector of the system k System threshold (Minimum required weight/capacity for the operating system)Structure function of the system at time t in the first model of failure Φ I ðl 1 ; l 2 ; . . .; l M Þ Survival signature of the system in the first model of failure ϕ 2 ðεðtÞÞ System structure function at time t in the second model of failure Φ II ðl 1 ; l 2 ; . . .; l M Þ Survival signature of the system in the second model of failure Φ ðiÞ Survival signature of the type i Φ ðiÞ j Probability that the ith type operates with exactly j components v ðiÞ li State vector of the components of type i with exactly l i operating components