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Dynamic demand satisfaction probability of consecutive sliding window systems with warm standby components
Reliability Engineering and System Safety 189 (2019) 397–405
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Reliability Engineering and System Safety journal homepage: www.elsevier.com/locate/ress
Dynamic demand satisfaction probability of consecutive sliding window systems with warm standby components Gregory Levitina,b, Liudong Xingc, Hanoch Ben-Haimb, Hong-Zong Huanga,
T
⁎
a
Center for System Reliability and Safety, University of Electronic Science and Technology of China, Chengdu, Sichuan 611731, PR China The Israel Electric Corporation, P. O. Box 10, Haifa 31000, Israel c University of Massachusetts, Dartmouth, MA 02747, USA b
A R T I C LE I N FO
A B S T R A C T
Keywords: Consecutive sliding window system Dynamic performance distribution Warm standby Demand satisfaction probability Optimization
Motivated by practical applications such as heating systems, radar, and sensor monitoring, this paper models and analyzes a linear consecutive multi-state sliding window system with warm standby components (CSWS-WS). The system contains n linearly ordered components, each being a warm standby configuration of multiple elements with heterogeneous time-to-failure distributions and nominal performances. Thus, depending on the currently operating (online) element, each component may exhibit multiple states, corresponding to different failure behaviors and performance rates. The system function depends on the accumulated performance (sum of performance rates) of r consecutive components, referred to as a r-sized window. The system is considered being failed if the accumulated performance in each of at least m consecutive overlapping r-sized windows is lower than a random demand. To evaluate the reliability (demand satisfaction probability) of a CSWS-WS, a probabilistic model is first presented to determine the dynamic performance distribution of each warm standby component; a universal generating function-based method is then suggested for obtaining the dynamic demand satisfaction probability (DSP). Based on the DSP evaluation, the optimal element distribution and sequencing problem is formulated and solved for the CSWS-WS system. As demonstrated through examples, solutions to the considered optimization problems can facilitate a proper choice of element distribution and activation sequencing, maxmizing the minimum instananeous DSP or expected DSP over a certain mission time.
1. Introduction The sliding window system (SWS) in the context of system reliability engineering was first introduced in early 2000s [1,2]. The SWS is originated from the linear consecutive k-out-of-r-from-n: F system, which contains n ordered components and fails if there exists a group of r consecutive components where at least k of them have failed. Though being mentioned in various works related to testing for non-random clustering, service systems, radar detection, quality control and inspections [3–7], the linear consecutive k-out-of-r-from-n: F system was not formally introduced until Griffith's work in [8]. Following its introduction, exact and approximate reliability evaluation methods were developed for different types of consecutive k-out-of-r-from-n: F systems. Examples of the exact methods include the universal generating function-based method [9] and the binary decision diagrams-based method [10]. Simulations [11], Boole–Bonferroni and Hunter–Worsley bounds [12,13] are examples of methods providing approximate system
reliability results. The SWS generalized the consecutive k-out-of-r-from-n: F system model by considering multi-state components with more than two different states, ranging from complete failure to perfect functioning [1,2]. Each state is characterized by a different performance rate. The SWS is considered being failed when the sum of any r consecutive components’ performance rates is lower than a desired system demand. The reliability evaluation of SWS has been considered in [1]; optimization problems such as the optimal allocation of multi-state components [2] and the optimal load distribution [14] have been solved using the genetic algorithms. In [15], the SWS model was further generalized to the case of m consecutive overlapping windows, each window containg r consecutive components. This generalized SWS model was named a consecutive sliding window system (CSWS). The performance rate of a r-sized window (i.e., a group with r consecutive components) is the sum of performance rates of the components belonging to the group. The CSWS
⁎ Corresponding author at: Center for System Reliability and Safety, University of Electronic Science and Technology of China, Chengdu, Sichuan 611731, PR China. E-mail addresses: [email protected] (G. Levitin), [email protected] (L. Xing), [email protected] (H.-Z. Huang).
https://doi.org/10.1016/j.ress.2019.05.002 Received 9 January 2019; Received in revised form 19 March 2019; Accepted 6 May 2019 Available online 14 May 2019 0951-8320/ © 2019 Elsevier Ltd. All rights reserved.
Reliability Engineering and System Safety 189 (2019) 397–405
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Acronyms
performance for any r-sized group) lth realization of random demand D(t) probability of the lth realization of random demand, i.e., hl(t) = Pr{D(t) = dl} Gj(t) performance DSCTP of the jth component fh(t), Fh(t) pdf, cdf of time-to-failure distribution of element h gh nominal performance of element h ζh time-to-failure deceleration factor of element h pj,k(t) probability that performance level of component j at time t is gs(j,k), i.e., pj,k(t) = Pr(Gj(t) = gs(j,k)) Tk random time when the last element from sequence s(j,1), …, s(j,k) fails in the operation mode qk(t) pdf of Tk ef,l(t) conditional probability that m consecutive r-sized groups starting from component f fail at time t given that the demand level is dl El(t) conditional system failure probability at time t given that the demand is dl y˜f , k vector of performances of the fth r-sized group while being in state k p˜f , k (t ) probability of the kth state of the fth r-sized group σ(y) summation of items of vector y u-function denoting pmf of random performance of comuj(z, t) ponent j at time t Uf(z,t) vector-u-function denoting state distribution for the fth rsized group at time t ⊗ composition operator over a vector-u-function and a udl hl(t)
cdf cumulative distribution function CSWS linear consecutive multi-state sliding window system CSWS-WS CSWS with warm standby components DSCTP discrete-state continuous-time process DSP demand satisfaction probability pdf probability density function pmf probability mass function SWS linear multi-state sliding window system u-function universal generating function representing the pmf of a scalar random variable vector-u-function universal generating function representing the distribution of a random vector Nomenclature ξ n Nj m r Ω τ s(j,k) a(t) A(τ) D(t)
total number of elements in the CSWS-WS total number of components in the CSWS-WS number of elements in component j maximum allowable number of failed consecutive groups in the CSWS-WS number of consecutive components in the group or window considered permutation representing distribution and sequencing of elements mission time index of element activated after k − 1 failures in component j instantaneous DSP of CSWS-WS expected DSP during the CSWS-WS mission DSCTP of demand (minimum allowable cumulative
←
ϕ(y,g) cf,k 1(A)
function shift operator over vector y and value g integer counter of failed consecutive groups Boolean function: return 1 if A is True; 0 if A is False
the genetic algorithm. All of the afore-mentioned works assume static performance distributions of CSWS components. In this paper we extend the model by assuming that each component is a fault-tolerant warm standby configuration consisting of non-identical binary elements (as demonstrated
fails if the performance rate for each of at least m consecutive overlapping r-sized windows is lower than the pre-specified demand D. An extended universal generating function-based method was suggest in [15] for the CSWS reliability evaluation. The optimal allocation of multi-state components in a CSWS was later considered in [16] using
Fig. 1. Example of a CSWS-WS with n = 6 and r = 3. 398
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The cumulative performance of any group of r consecutive components (r-sized window) needs to meet a desired random demand D(t), which is also modeled as a DSCTP with d = {d0,…, dL}, h(t) = {h0(t), …, hL(t)}, and hl(t) = Pr{D(t) = dl} for l = 0, 1, …L. Consider the rsized window beginning with location f (i.e., group of r consecutive components located in positions from f to f + r − 1). This group fails if its cumulative performance is lower than the desired demand D(t), that is,
in Fig. 1), i.e., a CSWS with warm standby components (CSWS-WS). We evaluate a demand satisfaction probability (DSP) for the CSWS-WS considering that the components’ dynamic performances and the demand constitute discrete-state continuous-time processes (DSCTPs). As demonstrated in [2,16], the distribution of elements among components (locations) can greatly affect the system reliability. As demonstrated in [17], different elements activation sequences may lead to different reliabilities of a warm standby system with heterogeneous elements. Thus, for the CSWS-WS considered in this work, having a given set of heterogeneous elements, it is possible to achieve considerable improvements in the system DSP through choosing proper distribution and sequencing of those elements. Therefore, based on the evaluation of the DSP, another contribution of this work is to solve the optimal element distribution and sequencing problem for the general CSWS-WS model. The paper presents a new methodology of CSWS-WS optimization based on dynamic system performance metrics. This methodology combines a convolution-based method of determining stochastic performance processes for warm standby components and the algorithm for determining the DSP of CSWS adopted from [15]. The rest of the paper is organized as follows. Section 2 describes the CSWS-WS model illustrated by a practical example of a heating system, and gives the formulation of the DSP metrics and the optimization problems. Section 3 describes a technique used for determining standby component performance distribution. Section 4 discusses an algorithm for evaluating the instantaneous DSP of CSWS-WS with given element distribution and sequencing. Illustrative examples of instantaneous DSP evaluation are presented in Section 5. Illustrative examples of solutions to the formulated optimization problems are presented in Section 6. Section 7 concludes the work and points out a few directions of future research.
f +r−1
∑
Wf (t ) =
Gj (t ) < D (t ). (1)
j=f
Consider the set of m overlapping consecutive r-sized windows beginning from position s. There are m + r − 1 components in the set. The first window spans components located at positions s, s + 1, …, s + r − 1. The last, i.e., the mth window spans components located at positions s + m − 1, s + m, …, s + m + r − 2. Thus, the failure condition of the set (each one of m consecutive r-sized windows fails) is indicated by (2). s+m−1
∏
s+m−1
1(Wf (t ) < D (t )) =
f =s
∏ f =s
1
⎛ ⎜ ⎝
f +r−1
∑ j=f
⎞ Gj (t ) < D (t ) = 1. ⎟ ⎠
(2)
For the considered CSWS-WS with n linearly ordered components, the sets of m overlapping r-sized windows can start from locations 1, …, n − r − m + 2. The entire system is reliable if (2) is not true for any set of m consecutive windows. Therefore, the instantaneous system DSP a (t) is defined as
a (t ) = Pr
2. System model and problem formulation
⎧ n−r−m+2 ⎡ s+m−1 ⎛ f +r−1 ⎫ ⎞⎤ ∑ ⎢ ∏ 1 ∑ Gj (t ) < D (t ) ⎟ ⎥ = 0⎬ ⎨ s=1 ⎢ f =s ⎜ j=f ⎝ ⎠⎥ ⎣ ⎦ ⎩ ⎭
(3)
The expected system DSP during mission time τ can thus be obtained as
2.1. System description
A (τ ) =
The CSWS-WS considered consists of n linearly ordered locations. Each location j is deployed with an ordered set of elements Sj = {s(j,1), …, s(j,Nj)} composing a 1-out-of-Nj warm standby component. Specifically, each warm standby component contains one element that is on-line and operating, and (Nj − 1) redundant elements that serve as standby spares (staying in a warm standby mode) [17–19]. In the case of the on-line element malfunctioning, based on a pre-determined sequence a standby element (if available) is activated to replace the faulty element and take over its mission task. A standby element may have different readiness levels, ranging from completely-ready to not-readyat-all. A completely-ready element, also known as a hot standby element works in parallel with the online element(s) and is fully exposed to the operational stresses. A not-ready-at-all element, also known as a cold standby element is unpowered and is fully shielded from the operational stresses [20,21]. In general the readiness level of a warm standby element is in-between the two extremes to tradeoff the high standby cost (consumption of energy and materials) of the hot standby and the long restoration delay of the cold standby [18,20]. Besides the diversity in the readiness level, different standby elements may have different performance rates and time-to-failure distirbutions causing the dynamic performance distribution of each CSWS-WS component. In particular, each element s(j,k) is characterized by its nominal performance gs(j,k) and probability density function (pdf) of its time-to-failure fs(j,k)(t). Depending on the element currently operating in the moment, performance of a component can change dynamically. Mathematically, the performance Gj(t) of each component j can be considered as a DSCTP, whose probability mass function (pmf) at time instance t is described using two vectors {gs(j,0),gs(j,1),…, gs(j,Nj)} and {pj,0(t),pj,1(t),…, N pj,Nj(t)} with pj,k(t) = Pr(Gj(t) = gs(j,k)). Note that ∑k =j 0 pj, k = 1and gs(j,0) = 0 corresponds to the case where all the elements belonging to the jth component have failed.
1 τ
τ
∫ a (t ) dt. 0
(4)
A practical example of the CSWS-WS model is a heating system, which is used to provide desired temperature along a line with moving parts. There are n locations for deploying heaters. At each location, a number of heaters compose a warm standby configuration. The temperature at each point of the line depends on the cumulative effect of heaters at r closest locations. Due to different purchase and exploitation history, the heaters at the same location may have different time-tofailure distributions and nominal power. Depending on the power of the heater that is currently operating, the heating effect in each location can vary discretely. Due to random changes in the ambient temperature, the desired temperature or demand D(t) can also vary. Though the ambient temperature changes continuously, its variation can be divided into several zones, corresponding to different discrete values of the temperature demand. To provide the desired temperature (not less than certain specified temperature value at each point of the line), the heaters in any r adjacent locations should occupy states where their accumulated radiation intensity exceeds the desired minimum D(t). If the heating intensity becomes insufficient (dropping below the desired level) along a significant portion of the line (spanning m consecutive overlapping heater windows), the parts have adequate time to cool down and deteriorate leading to the system failure. Such a situation takes place when in each one of at least m consecutive overlapping windows of r heaters, the heating intensity is insufficient. The CSWSWS systems can also be found in applications such as radar systems [15] and sensor mointoring systems [22]. 2.2. Formulation of optimization problems When a heterogeneous set of ξ functionally equivalent elements 399
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t
exists, the distribution of the elements among n CSWS-WS components and elements activation sequence within each component can affect the DSP of the considered CSWS-WS. The distribution and secuencing of ξ elements in n components can be determined by permutation Ω of a string consisting of ξ + n − 1 different integer numbers ranging from 1 to ξ + n − 1. In such a string, numbers from 1 to ξ represent the elements and numbers greater than n serve as separators among different components. All the numbers located between two separators represent elements belonging to the same component and activated according to their order in the string. Such string representation of the elements’ distribution is chosen following [2,16] as it is convenient for implementation in the genetic algorithm [23] operating with numerical strings (see Section 6). Table 1 presents examples of solutions represented by different integer strings for a CSWS-WS system with n = 3 and ξ = 7. Solutions with the first or last number greater than ξ or with two adjacent numbers that are both greater than ξ correspond to empty components (designated by “-” sign). The last two rows of Table 1 present examples of such strings. The optimal element allocation and sequencing problem is to find the permutation Ω that maximizes the minimum instantaneous DSP achieved during the mission time τ
Ω = arg max ⎧min a (t , Ω)⎫ ⎬ ⎨ Ω ⎭ ⎩ 0
qk (t ) = qk − 1 (t ) Fs (j, k ) (t ) +
0
Ω
(7)
where 0 ≤ ςs(j, k) ≤ 1 is a deceleration factor of element s(j,k), used to reflect lower stresses experienced by the element in the standby mode than in the operation mode based on the cumulative exposure model [17]. According to (7), qk(t) for k = 2,…, Ni can be obtained in an iterative manner. The probability that element s(j,1) is in operation at time t, i.e. component j operates with performance level gs(j,1) provided by this element at time t (Pr(Gj(t) = gs(j,1))) can be determined by (8), representing the probability that element s(j,1) does not fail before time t:
pj,1 (t ) = 1 − Fs (j,1) (t ).
(8)
The probability that element s(j,k) is in operation at time t, i.e. component j operates with performance level gs(j,k) at time t (Pr (Gj(t) = gs(j,k))) can be determined as the probability that Tk−1 = t-θ for any 0 ≤ θ ≤t and element s(j,k) that has waited in the standby mode for time t-θ does not fail before spending at least time θ in the operation mode: t
pj, k (t ) =
∫ qk−1 (t − θ)[1 − Fs (j,k) (ςs (j,k) (t − θ) + θ)] dθ.
(9)
0
(5)
The probability that all the elements of component j fail at time t, i.e., the performance of component j is zero, can be determined as
or the expected system DSP during the mission time τ
Ω = arg max{A (τ , Ω)}.
∫ qk−1 (t − θ) fs (j,k) (ςs (j,k) (t − θ) + θ) dθ
Nj
(6)
pj,0 (t ) = 1 −
∑ pj,k (t ).
(10)
k=1
2.3. Assumptions 4. Evaluation algorithm of CSWS-WS instantaneous DSP
The proposed methodology has the following assumptions:
The universal generating function (u-function) technique, firstly introduced in [24], has proved to be effective for reliability analysis of various multi-state systems [25]. In this section, we extend this technique for evaluating the instantaneous DSP of the considered CSWS-WS.
1 Element replacement time is negligible compared to the mission time. 2 Elements cannot be redistributed among components during the mission. 3 Element activation sequence within each component cannot be changed during the mission. 4 No common cause failures can occur during the mission. 5 Mechanisms of element failure detection and elements replacement are perfectly reliable.
4.1. u-function for an individual component In general, the polynomial in (11) defines the u-function of a DSCTP X(t) that can assume K possible values, and pk(t) = Pr(X(t) = xk). K
3. Determining component performance distribution
u (z , t ) =
∑ pk (t ) z xk .
(11)
k=1
Consider the jth component with Nj elements configured in a 1-outof-Nj warm standby structure for the considered CSWS-WS. Define Tk as a random variable representing the time when the last element from sequence s(j,1),…, s(j,k) fails during the operation mode. The pdf of this variable is qk(t). For k = 1, q1(t) = fs(j,1)(t) since only one element s(j,1) belongs to the sequence. Next we discuss how to obtain qk(t) for k = 2, …,Nj with qk−1(t) and fs(j,k)(t). There exist two scenarios where the failure of the last element from the sequence s(j,1),…, s(j,k) takes place at time t:
Particularly, the u-function in (12) defines the performance distribution of component j in the considered CSWS-WS; it represents all possible states of component j by relating the probability of each state pj,k(t) = Pr(Gj(t) = gs(j,k)) to the performance realization gs(j,k). Hj
uj (z , t ) =
∑ pj,k (t ) z gs (j,k).
(12)
k=1
Table 1 Examples of strings corresponding to different element distribution and sequencing solutions for n = 3 and ξ = 7.
1) Tk = Tk−1 = t: the failure of the last element from sequence s(j,1), …, s(j,k − 1) occurs at time t, whereas element s(j,k) fails in the standby mode before t, which can occur with probability Fs(j,k)(t). 2) 2). Tk = t, Tk−1 = t − θ: the failure of the last element from sequence s(j,1),…, s(j,k − 1) occurs at time t − θ (0≤ θ ≤t), and element s(j,k) fails at time t (spending time t − θ in the standby mode and then funcitoning for time θ in the operation mode). Thus, given qk−1(t) and fs(j,k)(t), the pdf of Tk can be obtained as 400
Ω
Element activation sequences
1,5,2,9,3,8,4,7,6 2,4,8,3,1,9,6,7,5 2,3,8,9,5,4,6,7,1 8,4,2,1,9,5,6,7,3
Component 1 1,5,2 2,4 2,3 –
Component 2 3 3,1 – 4,2,1
Component 3 4,7,6 6,7,5 5,4,6,7,1 5,6,7,3
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Fig. 2. Instantaneous DSP of CSWS-WS with ξ = 10, n = 5, for r = 2 and different m.
4.2. vector-u-function for a r-sized group/window
Table 2 Best obtained element allocation solutions for CSWS-WS with ξ = 10, n = 5, r = 2. m
min a(t)
A(τ)
Element allocation
1 2 3 4
0.5454 0.8852 0.9870 0.9989
0.8620 0.9757 0.9980 0.9999
(1,3)(2,4)(5,7,9)(6,8,10)(-) (-)(1,2,3,5)(4,6,8)(7,9,10)(-) (-)(1,2,3,5)(4,6,8,10)(7,9)(-) (2,4,6)(1,3,8)(-)(5,7,9,4)(-)
To represent the performance distribution of the fth r-sized group (i.e., the group of r consecutive s-independent components starting from the fth component), the vector-u-function is derived in this subsection, which represents distribution of combination of random performance values of the components belonging to the group: Gf = {Gf(t), …,Gf+r−1(t)}. Each combination of states of individual components constituting the group defines a state of the group. Thus, the number of possible combinations of states of components constituting the group defines Kf, the total number of different states of the fth group. Because each r component j can assume Nj + 1 states, Kf = ∏ j = 1 (Nf + j − 1 + 1) . The performance values of components of the fth group in any state k are denoted by the realization yf,k of the random vector Gf. The occupation probability of any state of the group is evaluated as the product of probabilities of corresponding states of individual components constituting the group. Thus, applying the operator over u-functions of components belonging to the fth group (i.e., r consecutive components starting from
Table 3 Parameters of elements for Example 2. i ηi βi ζi gi
1,2 100 1.5 0.2 4
3,4 80 1.0 0.4 6
5,6 120 1.3 0.8 7
7,8 100 1.1 0.6 5
9,10 70 1.2 0.3 3
11,12 180 1.0 0.7 5
13,14 120 1.7 0.2 4
15,16 150 1.1 0.8 6
Fig. 3. Instantaneous DSP of CSWS-WS with ξ = 16, n = 10, for r = 3, different m and constant demands. 401
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Table 4 Best obtained solutions of problem (5) for CSWS-WS with ξ = 16, n = 10, r = 3. D(t)
10
15
m
min a(t)
A(τ)
Element distribution
1 2 3 4 1 2 3 4
0.406 0.835 0.900 0.991 0.014 0.108 0.385 0.870
0.796 0.964 0.981 0.999 0.302 0.531 0.781 0.973
(-)(13,14)(5,3,4)(-)(7,1,2)(15,16)(11,9)(6,8)(12,10)(-) (-)(11,1)(12,9,10)(5,6,7,2)(-)(15,16,4,3)(-)(8,13,14)(-)(-) (-)(-)(11,12)(7,8,3)(15,16,4)(5,13)(6,14)(9,10,1,2)(-)(-) (-)(-)(-)(11,7,1,9)(5,15,3,13)(6,16,4,14)(12,8,2)(-)(10)(-) (11)(5)(12,1)(7,13)(6)(2,14)(15,3)(16,4)(8,9,10)(-) (-)(7,13)(11)(15,3)(12,4)(9,10,1)(5,6)(16,2)(8,14)(-) (-)(-)(15,11)(1,13,9)(16,3,4)(5,6)(7,14,10)(12,8,2)(-)(-) (-)(-)(-)(7,1,13,9)(5,6,15,3)(16,11,4,2)(12,8,14,10)(-)(-)(-)
Table 5 Comparison of the best obtained solutions of problems (5) and (6) for CSWS-WS with ξ = 16, n = 10, r = 3 and D(t) = 15. m
Problem
min a(t)
A(τ)
Element distribution
2
max(A(τ)) max(min a(t)) max(A(τ)) max(min a(t))
0.107 0.108 0.373 0.385
0.582 0.531 0.783 0.781
(-)(7,1)(11,12)(15,3)(9,13)(5,4)(10,14)(16,6)(8,2)(-) (-)(7,13)(11)(15,3)(12,4)(9,10,1)(5,6)(16,2)(8,14)(-) (-)(-)(15,3,4)(11,7,8)(13,1)(16,14,9)(12,2,10)(5,6)(-)(-) (-)(-)(15,11)(1,13,9)(16,3,4)(5,6)(7,14,10)(12,8,2)(-)(-)
3
Table 6 Random demand DSCTP. l dl hl(t)
1 15 0.3sin(0.047t) + 0.3
2 12 0.1sin(0.047t) + 0.1
3 10 0.12sin(0.047t + π) + 0.3
4 8 1 − h1(t) − h2(t) − h3(t)
components is in state k, and vector y˜f , k = (yf,k(1),…, yf,k(r)) contains values of performance rates of components belonging to the fth group in state k. Thus, the cumulative performance of the group in this state is r obtained as σ( y˜f , k ) = ∑ j = 1 yf , k (j ) . 4.3. vector-u-functions for all the r-sized groups For the considered CSWS-WS with n components, there exist exactly n − r + 1 r-sized groups and each component belongs to no more than r such groups. The following procedure is applied to obtain the vector-ufunctions for all the r-sized groups. 1 Define the vector-u-function (15), where vector y˜0 contains r zeros.
U1 − r (z , t ) = z y˜0 .
(15)
2 Define the operator ⊗ (16) over vector-u-function for the fth group ←
Uf(z,t) and u-function of individual component j u j(z,t).
Uf (z , t )⊗ uj (z , t )
Fig. 4. Probabilities of the random demand realizations.
←
Kf
=
the fth component) defined in (13), one obtains the vector-u-function corresponding to the fth group:
Nj
Kf
Nj
∑ p˜f ,k (t ) z y˜ f ,k ⊗ ∑ pj,h (t ) z gj,h= ∑ ∑ p˜f ,k (t ) pj,h (t ) z ϕ (˜yf ,k,gj,h). ← k=1
h=0
k=1 h=0
(16)
Uf (z , t ) = ⊗(uf (z , t ), uf + 1 (z , t ), ...,uf + r − 1 (z , t )) Nf
Nf + 1
∑ ∑
=
k1= 0 k2= 0
Nf + r − 1
∑
...
pf , k1 (t ) × pf + 1, k2 (t ). ..
.
The operator ϕ in (16) over arbitrary vector y˜ and value g assigns y (s − 1) = y(s) for 1 < s ≤ r and y(r) = g. Thus, the operator ϕ removes performance value of the first (leftmost) component of the fth group, and adds performance value of the next component (not considered yet) to the group. The order of components belonging to the group is preserved during the operation. Hence, applying operator ⊗ over the
kr = 0
×pf + r − 1, kr (t ) z { gs (f , k1), gs (f + 1, k2),..., gs (f + r − 1, kr ) }
(13)
Simplifying the representation in (13), one obtains the vector-ufunction in (14) that defines all Kf possible states of the fth group of r consecutive components.
←
vector-u-function representing the fth group's performance distribution, one obtains the vector-u-function representing the performance distribution of the f + 1th group.
Kf
Uf (z , t ) =
∑ p˜f ,k (t ) z y˜f ,k , k=1
(14)
1 Obtain vector-u-functions for all possible r-sized groups U1(z,t), …,
where p˜f , k (t ) is the probability that the fth group of r consecutive 402
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Fig. 5. Instantaneous DSP of CSWS ξ = 16, n = 10, for r = 3 and different m and stochastic demand. Table 7 Comparison of the best obtained solutions of problems (5) and (6) for CSWS-WS with ξ = 16, n = 10, r = 3 and stochastic demand. m
Problem
min a(t)
A(τ)
Element distribution
1
max(A(τ)) max(min a(t)) both both max(A(τ)) max(min a(t))
0.342 0.357 0.632 0.879 0.9899 0.9900
0.598 0.552 0.784 0.932 0.9959 0.9958
(13)(7,14)(5,3)(1,2)(15)(16,4)(11,9)(6,8)(12,10)(-) (7)(1,13)(5,3)(11,9)(15,10)(6,2)(8,14)(16,4)(12)(-) (-)(11,1)(15,9)(5,12)(7,2)(8,13)(6,3)(16,4)(10,14)(-) (-)(-)(15,13)(16,2,10)(6,11,9)(7,8,1)(5,3,4)(12,14)(-)(-) (-)(-)(-)(11,7,1,9)(5,15,3,13)(6,16,4,14)(12,8,2,10)(-)(-)(-) (-)(-)(-)(11,7,13,9)(5,15,3,1)(6,16,4,2)(12,8,14,10)(-)(-)(-)
2 3 4
Mf
Un−r+1(z,t) by applying the operator ⊗ sequentially for j = 1, …, n ← as
Uj + 1 − r (z , t ) = Uj − r (z , t )⊗ uj (z , t ). ←
U^f , l (z , t )⊗ uj (z , t ) = ←
Nj
∑ p˜f ,k (t ) z cf ,k,y˜f ,k ⊗ ∑ pj,h (t ) z gj,h ← h=0
k=1 Mf
(17)
=
Nj
∑ ∑ p˜f ,k (t ) pj,h (t ) z
⎛ ⎞ ρ ⎜cf , k , σ ⎜⎛ϕ (y˜f , k , gj, h) ⎟⎞ ⎟, ϕ (y˜f , k , gj, h) ⎝ ⎠⎠ ⎝
=
k=1 h=0 Mf + 1
∑ U1(z,t), the vector-u-function for the first r-sized group is obtained after applying operator ⊗ r times. In Uf(z,t) (f > 0), the value y(s) of
f ,l
4.4. Counting m failed consecutive groups
δ (U^f , l (z , t )) =
(18)
(21)
4.5. Evaluating conditional system failure probability and instantaneous DSP
where Mf denotes the number of distinguishable combinations of cf,k and y˜f , k . The counter has an initial value of 0. The procedure in (15) and (16) is revised as
U^1 − r , l (z , t ) = z 0, y˜0
Mf
∑ p˜f ,k (t )1(cf ,k = m) k=1
Mf k=1
f,k
rence probabilities of such terms, and obtains the overall probability that the system failure is caused by components represented by the vector-u-function Uf(z, t) at time t given that the demand level is dl.
For detecting m failed r-sized groups for a particular demand level dl, an integer counter cf,k is incorporated into the vector-u-function (14) as:
∑ p˜f ,k (t ) z cf ,k,y˜ f ,k ,
(20)
cg + 1 if x < dl where ρ (cg , x ) = ⎧ ⎨ ⎩ 0 if x ≥ dl . The counter has value of m only when m consecutive r-sized groups fail. The system failure is indicated by the terms p˜f , k (t ) z cf , k, y˜f , k of U^ (t , z ) with c = m. The operator δ defined in (21) collects occur-
←
vector y˜f , k is the performance rate of component f − 1 + s in state k of the fth group.
U^f , l (z , t ) =
p˜f + 1, k (t ) z cf + 1, k, y˜f + 1, k
k=1
Let ef,l(t) denote the conditional probability that m consecutive rsized groups starting from component f (containing components from f to f + m + r − 2) fail at time t given that the demand level is dl. If these m consecutive groups starting from component f are failed, the entire system always fails regardless the states of components not belonging to those m groups. Thus, the conditional failure probability of the entire
(19)
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each component. The cumulative performance of any r = 2 adjacent components is 8, which is insufficient to meet the demand D(t)=10. Thus, the instantaneous DSP of the CSWS-WS in the mission beginning is 0. The demand can be satisfied only after failure of the first elements of the components and activation of the standby element with performance 8. As the probability of failure of the first element and activation of the standby one increases in time, the instantaneous DSP of the CSWS-WS also increases. It becomes decreasing when the probability that both elements belonging to the same component fail increases considerably. Though the element distribution II is impractical as it guarantees zero DSP at the mission beginning, it illustrates the possibility of the non-monotonic behavior of the instantaneous DSP even when the demand is constant.
system at time t given that the demand is dl can be evaluated as the summation of occurrence probabilities of the mutually exclusive events n−r−m+1
El (t ) = e1,l (t ) + e2,l (t )(1 − e1,l (t )) + en − r − m + 2, l (t )
∏
(1 − ej, l (t ))
j=1
(22) where e1,l(t)=δ (U^m, l (z , t )) . To evaluate e2,l(t)(1 − e1,l(t)) in (22), all the terms with Cm, k = m are removed from U^ (z , t ) (excluding cases where the system failure is m, l
caused by failures of the first m r-sized groups), the operator ⊗ is ap←
plied over the truncated vector-u-function U^m, l (z , t ) and u-function (z , t ) , and then the operator δ is applied over u (z,t) to obtain U^ m + 1, l
m+r
U^m + 1, l (z , t ) . i−1 Recursively, to evaluate ei, l (t ) ∏ j = 1 (1 − ej, l (t )) in (22), all the terms (z , t ) , then with c are removed from U^ =m
6. Optimization examples Based on the evaluation of DSP, the optimal element distribution problems formulated in (5) and (6) can be solved. Both of them are complicated combinatorial optimization problems with (n + ξ − 1)! possible solutions. Considering reasonable time limitations, the brute force search technique is not realistic even for moderate numbers of components and elements. The genetic algorithms (GA) based on the evolutionary search principle were proven to be an effective optimization tool in reliability engineering [26–28]. Based on the solution representation illustrated in Table 1, the steady state GA, named GENITOR is applied in this work to solve the proposed optimization problems. Refer to [16,23] for a description of the GENITOR algorithm.
m + i − 1, l
m + i − 1, k
U^m + i, l (z , t ) = U^m + i − 1, l (z , t )⊗ um + r + i − 1 (z , t ) is obtained, and finally ←
δ (U^m + i, l (z , t )) is applied. Having the DSCTP D(t), one obtains the instantaneous DSP of CSWSWS as L
a (t ) = 1 −
∑ hl (t ) El (t ).
(23)
l=0
The following pseudo-code summarizes the algorithm obtaining the instantaneous DSP of CSWS-WS.
6.1. Example 1 1. Initialization: a(t) = 1; 2. For l = 0,…,L: 2.1. Set El(t) = 0; U^1 − r , l (t , z ) = z 0, y˜0 ( y˜0 contains r zeros). 2.2. For j = 1,…,n: uj (z , t ) , and simplify the 2.1.1. Obtain U^j − r + 1, l (z , t ) = U^j − r , l (z , t )⊗ ←
For the same example CSWS-WS evaluated in Section 5, Table 2 presents the best obtained element distributions (the indices of elements are presented in parentheses for each component according to their activation sequence) and corresponding values of the minimum and average DSP during the mission. The element distributions obtained as solutions of problems (5) (maximizing the minimum instantaneous DSP during the mission) and (6) (maximizing the expected system DSP during the mission) coincide. The rightmost subfigure in Fig. 2 presents the instantaneous DSP of the example system for the best obtained element distributions for each value of m (no more than four elements can be located in the same position). It can be seen that much greater values of a(t) are achieved compared with those obtained using element distributions I and II in Fig. 2. Observe that in the optimal element distribution solutions some components remain empty (which is designated by “-” sign). Such phenomenon is common in sliding window systems because in each point the performance of the system is determined by the cumulative effect of several components and keeping some components empty allows enhancement of other components. The detailed discussion of this effect can be found in [29].
obtained vector-u-function by combining like terms (summing terms with the same exponents). 2.2.2. If j ≥ m + r − 1, then add the value δ (U^j − r + 1, l (z , t )) to El(t), (t , z ) . = m from U^ and remove all of the terms with c j−r+1,k
j − r + 1, l
2.3. Subtract hl(t)El(t) from a(t). As can be seen from the pseudo-code above the computational complexity of the algorithm is O(nL), i.e., it is a linear function of the number of demand levels L and the number of components n. 5. Illustrative example of instantaneous DSP evaluation Consider a CSWS-WS with ξ = 10 elements distributed among n = 5 components (positions). The time-to-failure of all the elements obeys the Weibull distribution with cdf Fi(t) = 1 − exp(−(t/100)1.5). The deceleration factor of all the elements is ζi = 0.2. The nominal performances are gi = 4 for element i = 1,3,5,7,9 and gi = 8 for element i = 2,4,6,8,10. The mission time is τ = 100, and the demand is constant D(t) = 10. Fig. 2 presents instantaneous DSP of the example CSWS-WS during the mission for r = 2, different values of m and two different element distributions I and II. In both element distributions, all components contain identical set of two elements with different nominal performances. In element distribution I the elements with greater nominal performance are activated first (in the beginning of the mission), whereas in element distribution II the elements with lower nominal performance are activated first. The instantaneous DSP of the example CSWS-WS decreases in time for element distribution I, however it behaves non-monotonically for element distribution II. Indeed, at the beginning of the mission distribution II presumes activation of elements with performance gi=4 in
6.2. Example 2 Consider a CSWS-WS with ξ = 16 elements distributed among n = 10 components. The time-to-failure of all the elements obeys the Weibull distribution Fi(t) = 1 − exp(−(t/ηi)βi) with parameters presented in Table 3. The nominal performances and the deceleration factors of all the elements are also presented in Table 3. Fig. 3 presents instantaneous DSP of the best obtained solutions of problem (5) for this example CSWS-WS with r = 3, different values of m and two values of the constant demand. As in the previous example, the instantaneous DSP increases in m because more consecutive failed groups of components are allowed. The best obtained element distributions and the DSP metrics are presented in Table 4. The comparison of solutions to problems (5) and (6) for this example CSWS-WS with r = 3 and D(t) = 15 is given in Table 5. 404
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Consider now random demand described by DSCTP, presented in Table 6 and Fig. 4. Fig. 5 presents instantaneous DSP of the CSWS-WS for different m and this stochastic demand for the best obtained solutions of problems (5) and (6) (For m = 2 and m = 3 the best obtained solutions for (5) and (6) coincide). The corresponding element distributions are presented in Table 7. It can be seen that for all the obtained element distributions the instantaneous system DSP behaves non-monotonically. Whereas for a constant demand the DSP monotonically decreases (see Fig. 3), for variable demands it begins increasing from certain time because the probabilities of the high demand levels (l = 1 and l = 2) decrease and, therefore, the probability that the random demand is satisfied increases. Then, because of the further system deterioration (failures of elements) the DSP becomes decreasing even for low demands.
network reliability. J Appl Probab 1985;22:386–93. [4] Saperstain B. The generalized birthday problem. J Am Stat Assoc 1972;67:425–8. [5] Saperstain B. On the occurrence of n successes within N Bernoulli trails. Technometrics 1973;15:809–18. [6] Naus J. Probabilities for a generalized birthday problem. J Am Stat Assoc 1974;69:810–5. [7] Nelson J. Minimal-order models for false-alarm calculations on sliding windows. IEEE Trans Aerosp Electron Syst 1978;AES-14:351–63. [8] Griffith W. On consecutive k-out-of-n failure systems and their generalizations. In: Basu AP, editor. Reliability and quality control. Elsevier (North-Holland); 1986. p. 157–65. [9] Levitin G. Consecutive k-out-of-r-from-n system with multiple failure criteria. IEEE Trans Reliab 2004;53:394. 388. [10] Dutuit Y, Rauzy A. New insights into the assessment of k-out-of-n and related systems. Reliab Engg Syst Saf 2001;72:303–14. [11] Psillakis Z. A simulation algorithm for computing failure probability of a consecutive-k-out-of-r-from-n:f system. IEEE Trans Reliab 1995;44:523–31. [12] Habib A, Szántai T. New bounds on the reliability of the consecutive k-out-of-rfrom-n: F system. Reliab Eng Syst Saf 2000;68:97–104. [13] Radwan T, Habib A, Alseedy R, Elsherbeny A. Bounds for increasing multi-state consecutive k-out-of-r-from-n: F system with equal components probabilities. Appl Math Model 2011;35(5):2366–73. [14] Xiao H, Peng R, Wang W, Zhao F. Optimal element loading for linear sliding window systems. Proc Inst Mech Eng Part O 2016;230(1):75–84. [15] Levitin G, Ben Haim H. Consecutive sliding window systems. Reliab Eng Syst Saf 2011;96:1367–74. [16] Xiang Y, Levitin G, Dai Y. Optimal allocation of multi-state components in consecutive sliding window systems. IEEE Trans Reliab 2013;62(1):267–75. [17] Levitin G, Xing L, Dai Y. Optimal sequencing of warm standby elements. Comput Ind Eng 2013;65(August (4)):570–6. [18] Levitin G, Xing L, Dai Y. "Mission Cost and Reliability of 1-out-of-N Warm Standby Systems with Imperfect Switching Mechanisms. IEEE Trans Syst Man Cybern 2014;44(September (9)):1262–71. [19] Zhai Q, Peng R, Xing L, Yang J. Reliability of demand-based warm standby systems subject to fault level coverage. Appl Stochastic Models in Bus Ind 2015;31(May/ June (3)):380–93 Special Issue on Mathematical Methods in Reliability. [20] Levitin G, Xing L, Dai Y. Optimal backup distribution in 1-out-of-N cold standby systems. IEEE Trans Syst Man Cybern 2015;45(April (4)):636–46. [21] Xing L, Tannous O, Dugan JB. Reliability analysis of non-repairable cold-standby systems using sequential binary decision diagrams. IEEE Trans Syst Man Cybern Part A 2012;42(May (3)):715–26. [22] Ma S, Si G, Yue W, Ding Z. An online monitoring measure consistency computing algorithm by sliding window in multi-sensor system. Proc. of IEEE international conference on mechatronics and automation, Harbin. 2016. p. 2185–90. [23] Whitley D. The GENITOR algorithm and selective pressure: why rank-based allocation of reproductive trials is best. In: Schaffer D, editor. Proc. 3th international conference on genetic algorithms. Morgan Kaufmann; 1989. p. 116–21. [24] Ushakov I. Universal generating function. Sov J Comput Syst Sci 1986;24(5):118–29. [25] Levitin G. Universal generating function in reliability analysis and optimization. London: Springer-Verlag; 2005. [26] Gen M, Kim J. GA-based reliability design: state-of-the-art survey. Comput Ind Eng 1999;37:151–5. [27] Levitin G. Genetic algorithms in reliability engineering. Guest editorial. Reliab Eng Syst Saf 2006;91(9):975–6. [28] G. Levitin. Computational intelligence in reliability engineering. Evolutionary techniques in reliability analysis and optimization. Springer; 2007. [29] Levitin G. Uneven allocation of elements in linear multi-state sliding window system. Eur J Oper Res 2004;163:418–33. [30] Levitin G, Xing L, Dai Y. Non-homogeneous 1-out-of-N warm standby systems with random replacement times. IEEE Trans Reliab 2015;64(June (2)):819–28. [31] Levitin G. Common supply failures in linear multi-state sliding window systems. Reliab Eng Syst Saf 2003;82:55–62. [32] Sakurahara T, Schumock G, Reihani S, Kee E, Mohaghegh Z. Simulation-informed probabilistic methodology for common cause failure analysis. Reliab Eng Syst Saf 2019;185:84–99.
7. Conclusion and future work The existing works on modeling CSWS systems have assumed static performance distributions of their constituent components. This paper advances the state-of-the-art by considering a CSWS with components subject to dynamic performance distributions due to the warm-standby configuration of heterogeneous functional elements. The considered system called CSWS-WS fails if the accumulated performance in each of at least m consecutive r-sized windows cannot meet a desired random demand. A vector-u-function technique is suggested to evaluate the dynamic demand satisfaction probability of the considered CSWS-WS. The GA is further applied to solve the optimal element distribution and sequencing problem, maxmizing the minimum instananeous DSP or expected DSP over a certain mission time. As demonstrated through examples, the proposed methodology can facilitate effective choice of element allocation among different components and activation sequencing for each component, leading to significantly better DSP. The proposed methodology has assumed negligible element replacement time. Based on [30] we plan to extend the method to consider random replacement time in the case of the online element malfunctioning. We are also interested in modeling imperfect element fault detection and replacement mechanisms based on [18]. The present research can also be extended in the future to consider effects of common-cause failures [31,32] that can impact multiple components and elements simultaneously. Acknowledgment This work was supported in part by the National Natural Science Foundation of China (Grant no. 51875089). References [1] Levitin G. Linear multi-state sliding window systems. IEEE Trans Reliab 2003;52:63–269. [2] Levitin G. Optimal allocation of components in a linear multi-state sliding window system. Reliab Eng Syst Saf 2002;76:245–54. [3] Tong Y. A rearrangement inequality for the longest run with an application in
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Reliability Engineering and System Safety journal homepage: www.elsevier.com/locate/ress
Dynamic demand satisfaction probability of consecutive sliding window systems with warm standby components Gregory Levitina,b, Liudong Xingc, Hanoch Ben-Haimb, Hong-Zong Huanga,
T
⁎
a
Center for System Reliability and Safety, University of Electronic Science and Technology of China, Chengdu, Sichuan 611731, PR China The Israel Electric Corporation, P. O. Box 10, Haifa 31000, Israel c University of Massachusetts, Dartmouth, MA 02747, USA b
A R T I C LE I N FO
A B S T R A C T
Keywords: Consecutive sliding window system Dynamic performance distribution Warm standby Demand satisfaction probability Optimization
Motivated by practical applications such as heating systems, radar, and sensor monitoring, this paper models and analyzes a linear consecutive multi-state sliding window system with warm standby components (CSWS-WS). The system contains n linearly ordered components, each being a warm standby configuration of multiple elements with heterogeneous time-to-failure distributions and nominal performances. Thus, depending on the currently operating (online) element, each component may exhibit multiple states, corresponding to different failure behaviors and performance rates. The system function depends on the accumulated performance (sum of performance rates) of r consecutive components, referred to as a r-sized window. The system is considered being failed if the accumulated performance in each of at least m consecutive overlapping r-sized windows is lower than a random demand. To evaluate the reliability (demand satisfaction probability) of a CSWS-WS, a probabilistic model is first presented to determine the dynamic performance distribution of each warm standby component; a universal generating function-based method is then suggested for obtaining the dynamic demand satisfaction probability (DSP). Based on the DSP evaluation, the optimal element distribution and sequencing problem is formulated and solved for the CSWS-WS system. As demonstrated through examples, solutions to the considered optimization problems can facilitate a proper choice of element distribution and activation sequencing, maxmizing the minimum instananeous DSP or expected DSP over a certain mission time.
1. Introduction The sliding window system (SWS) in the context of system reliability engineering was first introduced in early 2000s [1,2]. The SWS is originated from the linear consecutive k-out-of-r-from-n: F system, which contains n ordered components and fails if there exists a group of r consecutive components where at least k of them have failed. Though being mentioned in various works related to testing for non-random clustering, service systems, radar detection, quality control and inspections [3–7], the linear consecutive k-out-of-r-from-n: F system was not formally introduced until Griffith's work in [8]. Following its introduction, exact and approximate reliability evaluation methods were developed for different types of consecutive k-out-of-r-from-n: F systems. Examples of the exact methods include the universal generating function-based method [9] and the binary decision diagrams-based method [10]. Simulations [11], Boole–Bonferroni and Hunter–Worsley bounds [12,13] are examples of methods providing approximate system
reliability results. The SWS generalized the consecutive k-out-of-r-from-n: F system model by considering multi-state components with more than two different states, ranging from complete failure to perfect functioning [1,2]. Each state is characterized by a different performance rate. The SWS is considered being failed when the sum of any r consecutive components’ performance rates is lower than a desired system demand. The reliability evaluation of SWS has been considered in [1]; optimization problems such as the optimal allocation of multi-state components [2] and the optimal load distribution [14] have been solved using the genetic algorithms. In [15], the SWS model was further generalized to the case of m consecutive overlapping windows, each window containg r consecutive components. This generalized SWS model was named a consecutive sliding window system (CSWS). The performance rate of a r-sized window (i.e., a group with r consecutive components) is the sum of performance rates of the components belonging to the group. The CSWS
⁎ Corresponding author at: Center for System Reliability and Safety, University of Electronic Science and Technology of China, Chengdu, Sichuan 611731, PR China. E-mail addresses: [email protected] (G. Levitin), [email protected] (L. Xing), [email protected] (H.-Z. Huang).
https://doi.org/10.1016/j.ress.2019.05.002 Received 9 January 2019; Received in revised form 19 March 2019; Accepted 6 May 2019 Available online 14 May 2019 0951-8320/ © 2019 Elsevier Ltd. All rights reserved.
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Acronyms
performance for any r-sized group) lth realization of random demand D(t) probability of the lth realization of random demand, i.e., hl(t) = Pr{D(t) = dl} Gj(t) performance DSCTP of the jth component fh(t), Fh(t) pdf, cdf of time-to-failure distribution of element h gh nominal performance of element h ζh time-to-failure deceleration factor of element h pj,k(t) probability that performance level of component j at time t is gs(j,k), i.e., pj,k(t) = Pr(Gj(t) = gs(j,k)) Tk random time when the last element from sequence s(j,1), …, s(j,k) fails in the operation mode qk(t) pdf of Tk ef,l(t) conditional probability that m consecutive r-sized groups starting from component f fail at time t given that the demand level is dl El(t) conditional system failure probability at time t given that the demand is dl y˜f , k vector of performances of the fth r-sized group while being in state k p˜f , k (t ) probability of the kth state of the fth r-sized group σ(y) summation of items of vector y u-function denoting pmf of random performance of comuj(z, t) ponent j at time t Uf(z,t) vector-u-function denoting state distribution for the fth rsized group at time t ⊗ composition operator over a vector-u-function and a udl hl(t)
cdf cumulative distribution function CSWS linear consecutive multi-state sliding window system CSWS-WS CSWS with warm standby components DSCTP discrete-state continuous-time process DSP demand satisfaction probability pdf probability density function pmf probability mass function SWS linear multi-state sliding window system u-function universal generating function representing the pmf of a scalar random variable vector-u-function universal generating function representing the distribution of a random vector Nomenclature ξ n Nj m r Ω τ s(j,k) a(t) A(τ) D(t)
total number of elements in the CSWS-WS total number of components in the CSWS-WS number of elements in component j maximum allowable number of failed consecutive groups in the CSWS-WS number of consecutive components in the group or window considered permutation representing distribution and sequencing of elements mission time index of element activated after k − 1 failures in component j instantaneous DSP of CSWS-WS expected DSP during the CSWS-WS mission DSCTP of demand (minimum allowable cumulative
←
ϕ(y,g) cf,k 1(A)
function shift operator over vector y and value g integer counter of failed consecutive groups Boolean function: return 1 if A is True; 0 if A is False
the genetic algorithm. All of the afore-mentioned works assume static performance distributions of CSWS components. In this paper we extend the model by assuming that each component is a fault-tolerant warm standby configuration consisting of non-identical binary elements (as demonstrated
fails if the performance rate for each of at least m consecutive overlapping r-sized windows is lower than the pre-specified demand D. An extended universal generating function-based method was suggest in [15] for the CSWS reliability evaluation. The optimal allocation of multi-state components in a CSWS was later considered in [16] using
Fig. 1. Example of a CSWS-WS with n = 6 and r = 3. 398
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The cumulative performance of any group of r consecutive components (r-sized window) needs to meet a desired random demand D(t), which is also modeled as a DSCTP with d = {d0,…, dL}, h(t) = {h0(t), …, hL(t)}, and hl(t) = Pr{D(t) = dl} for l = 0, 1, …L. Consider the rsized window beginning with location f (i.e., group of r consecutive components located in positions from f to f + r − 1). This group fails if its cumulative performance is lower than the desired demand D(t), that is,
in Fig. 1), i.e., a CSWS with warm standby components (CSWS-WS). We evaluate a demand satisfaction probability (DSP) for the CSWS-WS considering that the components’ dynamic performances and the demand constitute discrete-state continuous-time processes (DSCTPs). As demonstrated in [2,16], the distribution of elements among components (locations) can greatly affect the system reliability. As demonstrated in [17], different elements activation sequences may lead to different reliabilities of a warm standby system with heterogeneous elements. Thus, for the CSWS-WS considered in this work, having a given set of heterogeneous elements, it is possible to achieve considerable improvements in the system DSP through choosing proper distribution and sequencing of those elements. Therefore, based on the evaluation of the DSP, another contribution of this work is to solve the optimal element distribution and sequencing problem for the general CSWS-WS model. The paper presents a new methodology of CSWS-WS optimization based on dynamic system performance metrics. This methodology combines a convolution-based method of determining stochastic performance processes for warm standby components and the algorithm for determining the DSP of CSWS adopted from [15]. The rest of the paper is organized as follows. Section 2 describes the CSWS-WS model illustrated by a practical example of a heating system, and gives the formulation of the DSP metrics and the optimization problems. Section 3 describes a technique used for determining standby component performance distribution. Section 4 discusses an algorithm for evaluating the instantaneous DSP of CSWS-WS with given element distribution and sequencing. Illustrative examples of instantaneous DSP evaluation are presented in Section 5. Illustrative examples of solutions to the formulated optimization problems are presented in Section 6. Section 7 concludes the work and points out a few directions of future research.
f +r−1
∑
Wf (t ) =
Gj (t ) < D (t ). (1)
j=f
Consider the set of m overlapping consecutive r-sized windows beginning from position s. There are m + r − 1 components in the set. The first window spans components located at positions s, s + 1, …, s + r − 1. The last, i.e., the mth window spans components located at positions s + m − 1, s + m, …, s + m + r − 2. Thus, the failure condition of the set (each one of m consecutive r-sized windows fails) is indicated by (2). s+m−1
∏
s+m−1
1(Wf (t ) < D (t )) =
f =s
∏ f =s
1
⎛ ⎜ ⎝
f +r−1
∑ j=f
⎞ Gj (t ) < D (t ) = 1. ⎟ ⎠
(2)
For the considered CSWS-WS with n linearly ordered components, the sets of m overlapping r-sized windows can start from locations 1, …, n − r − m + 2. The entire system is reliable if (2) is not true for any set of m consecutive windows. Therefore, the instantaneous system DSP a (t) is defined as
a (t ) = Pr
2. System model and problem formulation
⎧ n−r−m+2 ⎡ s+m−1 ⎛ f +r−1 ⎫ ⎞⎤ ∑ ⎢ ∏ 1 ∑ Gj (t ) < D (t ) ⎟ ⎥ = 0⎬ ⎨ s=1 ⎢ f =s ⎜ j=f ⎝ ⎠⎥ ⎣ ⎦ ⎩ ⎭
(3)
The expected system DSP during mission time τ can thus be obtained as
2.1. System description
A (τ ) =
The CSWS-WS considered consists of n linearly ordered locations. Each location j is deployed with an ordered set of elements Sj = {s(j,1), …, s(j,Nj)} composing a 1-out-of-Nj warm standby component. Specifically, each warm standby component contains one element that is on-line and operating, and (Nj − 1) redundant elements that serve as standby spares (staying in a warm standby mode) [17–19]. In the case of the on-line element malfunctioning, based on a pre-determined sequence a standby element (if available) is activated to replace the faulty element and take over its mission task. A standby element may have different readiness levels, ranging from completely-ready to not-readyat-all. A completely-ready element, also known as a hot standby element works in parallel with the online element(s) and is fully exposed to the operational stresses. A not-ready-at-all element, also known as a cold standby element is unpowered and is fully shielded from the operational stresses [20,21]. In general the readiness level of a warm standby element is in-between the two extremes to tradeoff the high standby cost (consumption of energy and materials) of the hot standby and the long restoration delay of the cold standby [18,20]. Besides the diversity in the readiness level, different standby elements may have different performance rates and time-to-failure distirbutions causing the dynamic performance distribution of each CSWS-WS component. In particular, each element s(j,k) is characterized by its nominal performance gs(j,k) and probability density function (pdf) of its time-to-failure fs(j,k)(t). Depending on the element currently operating in the moment, performance of a component can change dynamically. Mathematically, the performance Gj(t) of each component j can be considered as a DSCTP, whose probability mass function (pmf) at time instance t is described using two vectors {gs(j,0),gs(j,1),…, gs(j,Nj)} and {pj,0(t),pj,1(t),…, N pj,Nj(t)} with pj,k(t) = Pr(Gj(t) = gs(j,k)). Note that ∑k =j 0 pj, k = 1and gs(j,0) = 0 corresponds to the case where all the elements belonging to the jth component have failed.
1 τ
τ
∫ a (t ) dt. 0
(4)
A practical example of the CSWS-WS model is a heating system, which is used to provide desired temperature along a line with moving parts. There are n locations for deploying heaters. At each location, a number of heaters compose a warm standby configuration. The temperature at each point of the line depends on the cumulative effect of heaters at r closest locations. Due to different purchase and exploitation history, the heaters at the same location may have different time-tofailure distributions and nominal power. Depending on the power of the heater that is currently operating, the heating effect in each location can vary discretely. Due to random changes in the ambient temperature, the desired temperature or demand D(t) can also vary. Though the ambient temperature changes continuously, its variation can be divided into several zones, corresponding to different discrete values of the temperature demand. To provide the desired temperature (not less than certain specified temperature value at each point of the line), the heaters in any r adjacent locations should occupy states where their accumulated radiation intensity exceeds the desired minimum D(t). If the heating intensity becomes insufficient (dropping below the desired level) along a significant portion of the line (spanning m consecutive overlapping heater windows), the parts have adequate time to cool down and deteriorate leading to the system failure. Such a situation takes place when in each one of at least m consecutive overlapping windows of r heaters, the heating intensity is insufficient. The CSWSWS systems can also be found in applications such as radar systems [15] and sensor mointoring systems [22]. 2.2. Formulation of optimization problems When a heterogeneous set of ξ functionally equivalent elements 399
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t
exists, the distribution of the elements among n CSWS-WS components and elements activation sequence within each component can affect the DSP of the considered CSWS-WS. The distribution and secuencing of ξ elements in n components can be determined by permutation Ω of a string consisting of ξ + n − 1 different integer numbers ranging from 1 to ξ + n − 1. In such a string, numbers from 1 to ξ represent the elements and numbers greater than n serve as separators among different components. All the numbers located between two separators represent elements belonging to the same component and activated according to their order in the string. Such string representation of the elements’ distribution is chosen following [2,16] as it is convenient for implementation in the genetic algorithm [23] operating with numerical strings (see Section 6). Table 1 presents examples of solutions represented by different integer strings for a CSWS-WS system with n = 3 and ξ = 7. Solutions with the first or last number greater than ξ or with two adjacent numbers that are both greater than ξ correspond to empty components (designated by “-” sign). The last two rows of Table 1 present examples of such strings. The optimal element allocation and sequencing problem is to find the permutation Ω that maximizes the minimum instantaneous DSP achieved during the mission time τ
Ω = arg max ⎧min a (t , Ω)⎫ ⎬ ⎨ Ω ⎭ ⎩ 0
qk (t ) = qk − 1 (t ) Fs (j, k ) (t ) +
0
Ω
(7)
where 0 ≤ ςs(j, k) ≤ 1 is a deceleration factor of element s(j,k), used to reflect lower stresses experienced by the element in the standby mode than in the operation mode based on the cumulative exposure model [17]. According to (7), qk(t) for k = 2,…, Ni can be obtained in an iterative manner. The probability that element s(j,1) is in operation at time t, i.e. component j operates with performance level gs(j,1) provided by this element at time t (Pr(Gj(t) = gs(j,1))) can be determined by (8), representing the probability that element s(j,1) does not fail before time t:
pj,1 (t ) = 1 − Fs (j,1) (t ).
(8)
The probability that element s(j,k) is in operation at time t, i.e. component j operates with performance level gs(j,k) at time t (Pr (Gj(t) = gs(j,k))) can be determined as the probability that Tk−1 = t-θ for any 0 ≤ θ ≤t and element s(j,k) that has waited in the standby mode for time t-θ does not fail before spending at least time θ in the operation mode: t
pj, k (t ) =
∫ qk−1 (t − θ)[1 − Fs (j,k) (ςs (j,k) (t − θ) + θ)] dθ.
(9)
0
(5)
The probability that all the elements of component j fail at time t, i.e., the performance of component j is zero, can be determined as
or the expected system DSP during the mission time τ
Ω = arg max{A (τ , Ω)}.
∫ qk−1 (t − θ) fs (j,k) (ςs (j,k) (t − θ) + θ) dθ
Nj
(6)
pj,0 (t ) = 1 −
∑ pj,k (t ).
(10)
k=1
2.3. Assumptions 4. Evaluation algorithm of CSWS-WS instantaneous DSP
The proposed methodology has the following assumptions:
The universal generating function (u-function) technique, firstly introduced in [24], has proved to be effective for reliability analysis of various multi-state systems [25]. In this section, we extend this technique for evaluating the instantaneous DSP of the considered CSWS-WS.
1 Element replacement time is negligible compared to the mission time. 2 Elements cannot be redistributed among components during the mission. 3 Element activation sequence within each component cannot be changed during the mission. 4 No common cause failures can occur during the mission. 5 Mechanisms of element failure detection and elements replacement are perfectly reliable.
4.1. u-function for an individual component In general, the polynomial in (11) defines the u-function of a DSCTP X(t) that can assume K possible values, and pk(t) = Pr(X(t) = xk). K
3. Determining component performance distribution
u (z , t ) =
∑ pk (t ) z xk .
(11)
k=1
Consider the jth component with Nj elements configured in a 1-outof-Nj warm standby structure for the considered CSWS-WS. Define Tk as a random variable representing the time when the last element from sequence s(j,1),…, s(j,k) fails during the operation mode. The pdf of this variable is qk(t). For k = 1, q1(t) = fs(j,1)(t) since only one element s(j,1) belongs to the sequence. Next we discuss how to obtain qk(t) for k = 2, …,Nj with qk−1(t) and fs(j,k)(t). There exist two scenarios where the failure of the last element from the sequence s(j,1),…, s(j,k) takes place at time t:
Particularly, the u-function in (12) defines the performance distribution of component j in the considered CSWS-WS; it represents all possible states of component j by relating the probability of each state pj,k(t) = Pr(Gj(t) = gs(j,k)) to the performance realization gs(j,k). Hj
uj (z , t ) =
∑ pj,k (t ) z gs (j,k).
(12)
k=1
Table 1 Examples of strings corresponding to different element distribution and sequencing solutions for n = 3 and ξ = 7.
1) Tk = Tk−1 = t: the failure of the last element from sequence s(j,1), …, s(j,k − 1) occurs at time t, whereas element s(j,k) fails in the standby mode before t, which can occur with probability Fs(j,k)(t). 2) 2). Tk = t, Tk−1 = t − θ: the failure of the last element from sequence s(j,1),…, s(j,k − 1) occurs at time t − θ (0≤ θ ≤t), and element s(j,k) fails at time t (spending time t − θ in the standby mode and then funcitoning for time θ in the operation mode). Thus, given qk−1(t) and fs(j,k)(t), the pdf of Tk can be obtained as 400
Ω
Element activation sequences
1,5,2,9,3,8,4,7,6 2,4,8,3,1,9,6,7,5 2,3,8,9,5,4,6,7,1 8,4,2,1,9,5,6,7,3
Component 1 1,5,2 2,4 2,3 –
Component 2 3 3,1 – 4,2,1
Component 3 4,7,6 6,7,5 5,4,6,7,1 5,6,7,3
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Fig. 2. Instantaneous DSP of CSWS-WS with ξ = 10, n = 5, for r = 2 and different m.
4.2. vector-u-function for a r-sized group/window
Table 2 Best obtained element allocation solutions for CSWS-WS with ξ = 10, n = 5, r = 2. m
min a(t)
A(τ)
Element allocation
1 2 3 4
0.5454 0.8852 0.9870 0.9989
0.8620 0.9757 0.9980 0.9999
(1,3)(2,4)(5,7,9)(6,8,10)(-) (-)(1,2,3,5)(4,6,8)(7,9,10)(-) (-)(1,2,3,5)(4,6,8,10)(7,9)(-) (2,4,6)(1,3,8)(-)(5,7,9,4)(-)
To represent the performance distribution of the fth r-sized group (i.e., the group of r consecutive s-independent components starting from the fth component), the vector-u-function is derived in this subsection, which represents distribution of combination of random performance values of the components belonging to the group: Gf = {Gf(t), …,Gf+r−1(t)}. Each combination of states of individual components constituting the group defines a state of the group. Thus, the number of possible combinations of states of components constituting the group defines Kf, the total number of different states of the fth group. Because each r component j can assume Nj + 1 states, Kf = ∏ j = 1 (Nf + j − 1 + 1) . The performance values of components of the fth group in any state k are denoted by the realization yf,k of the random vector Gf. The occupation probability of any state of the group is evaluated as the product of probabilities of corresponding states of individual components constituting the group. Thus, applying the operator over u-functions of components belonging to the fth group (i.e., r consecutive components starting from
Table 3 Parameters of elements for Example 2. i ηi βi ζi gi
1,2 100 1.5 0.2 4
3,4 80 1.0 0.4 6
5,6 120 1.3 0.8 7
7,8 100 1.1 0.6 5
9,10 70 1.2 0.3 3
11,12 180 1.0 0.7 5
13,14 120 1.7 0.2 4
15,16 150 1.1 0.8 6
Fig. 3. Instantaneous DSP of CSWS-WS with ξ = 16, n = 10, for r = 3, different m and constant demands. 401
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Table 4 Best obtained solutions of problem (5) for CSWS-WS with ξ = 16, n = 10, r = 3. D(t)
10
15
m
min a(t)
A(τ)
Element distribution
1 2 3 4 1 2 3 4
0.406 0.835 0.900 0.991 0.014 0.108 0.385 0.870
0.796 0.964 0.981 0.999 0.302 0.531 0.781 0.973
(-)(13,14)(5,3,4)(-)(7,1,2)(15,16)(11,9)(6,8)(12,10)(-) (-)(11,1)(12,9,10)(5,6,7,2)(-)(15,16,4,3)(-)(8,13,14)(-)(-) (-)(-)(11,12)(7,8,3)(15,16,4)(5,13)(6,14)(9,10,1,2)(-)(-) (-)(-)(-)(11,7,1,9)(5,15,3,13)(6,16,4,14)(12,8,2)(-)(10)(-) (11)(5)(12,1)(7,13)(6)(2,14)(15,3)(16,4)(8,9,10)(-) (-)(7,13)(11)(15,3)(12,4)(9,10,1)(5,6)(16,2)(8,14)(-) (-)(-)(15,11)(1,13,9)(16,3,4)(5,6)(7,14,10)(12,8,2)(-)(-) (-)(-)(-)(7,1,13,9)(5,6,15,3)(16,11,4,2)(12,8,14,10)(-)(-)(-)
Table 5 Comparison of the best obtained solutions of problems (5) and (6) for CSWS-WS with ξ = 16, n = 10, r = 3 and D(t) = 15. m
Problem
min a(t)
A(τ)
Element distribution
2
max(A(τ)) max(min a(t)) max(A(τ)) max(min a(t))
0.107 0.108 0.373 0.385
0.582 0.531 0.783 0.781
(-)(7,1)(11,12)(15,3)(9,13)(5,4)(10,14)(16,6)(8,2)(-) (-)(7,13)(11)(15,3)(12,4)(9,10,1)(5,6)(16,2)(8,14)(-) (-)(-)(15,3,4)(11,7,8)(13,1)(16,14,9)(12,2,10)(5,6)(-)(-) (-)(-)(15,11)(1,13,9)(16,3,4)(5,6)(7,14,10)(12,8,2)(-)(-)
3
Table 6 Random demand DSCTP. l dl hl(t)
1 15 0.3sin(0.047t) + 0.3
2 12 0.1sin(0.047t) + 0.1
3 10 0.12sin(0.047t + π) + 0.3
4 8 1 − h1(t) − h2(t) − h3(t)
components is in state k, and vector y˜f , k = (yf,k(1),…, yf,k(r)) contains values of performance rates of components belonging to the fth group in state k. Thus, the cumulative performance of the group in this state is r obtained as σ( y˜f , k ) = ∑ j = 1 yf , k (j ) . 4.3. vector-u-functions for all the r-sized groups For the considered CSWS-WS with n components, there exist exactly n − r + 1 r-sized groups and each component belongs to no more than r such groups. The following procedure is applied to obtain the vector-ufunctions for all the r-sized groups. 1 Define the vector-u-function (15), where vector y˜0 contains r zeros.
U1 − r (z , t ) = z y˜0 .
(15)
2 Define the operator ⊗ (16) over vector-u-function for the fth group ←
Uf(z,t) and u-function of individual component j u j(z,t).
Uf (z , t )⊗ uj (z , t )
Fig. 4. Probabilities of the random demand realizations.
←
Kf
=
the fth component) defined in (13), one obtains the vector-u-function corresponding to the fth group:
Nj
Kf
Nj
∑ p˜f ,k (t ) z y˜ f ,k ⊗ ∑ pj,h (t ) z gj,h= ∑ ∑ p˜f ,k (t ) pj,h (t ) z ϕ (˜yf ,k,gj,h). ← k=1
h=0
k=1 h=0
(16)
Uf (z , t ) = ⊗(uf (z , t ), uf + 1 (z , t ), ...,uf + r − 1 (z , t )) Nf
Nf + 1
∑ ∑
=
k1= 0 k2= 0
Nf + r − 1
∑
...
pf , k1 (t ) × pf + 1, k2 (t ). ..
.
The operator ϕ in (16) over arbitrary vector y˜ and value g assigns y (s − 1) = y(s) for 1 < s ≤ r and y(r) = g. Thus, the operator ϕ removes performance value of the first (leftmost) component of the fth group, and adds performance value of the next component (not considered yet) to the group. The order of components belonging to the group is preserved during the operation. Hence, applying operator ⊗ over the
kr = 0
×pf + r − 1, kr (t ) z { gs (f , k1), gs (f + 1, k2),..., gs (f + r − 1, kr ) }
(13)
Simplifying the representation in (13), one obtains the vector-ufunction in (14) that defines all Kf possible states of the fth group of r consecutive components.
←
vector-u-function representing the fth group's performance distribution, one obtains the vector-u-function representing the performance distribution of the f + 1th group.
Kf
Uf (z , t ) =
∑ p˜f ,k (t ) z y˜f ,k , k=1
(14)
1 Obtain vector-u-functions for all possible r-sized groups U1(z,t), …,
where p˜f , k (t ) is the probability that the fth group of r consecutive 402
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Fig. 5. Instantaneous DSP of CSWS ξ = 16, n = 10, for r = 3 and different m and stochastic demand. Table 7 Comparison of the best obtained solutions of problems (5) and (6) for CSWS-WS with ξ = 16, n = 10, r = 3 and stochastic demand. m
Problem
min a(t)
A(τ)
Element distribution
1
max(A(τ)) max(min a(t)) both both max(A(τ)) max(min a(t))
0.342 0.357 0.632 0.879 0.9899 0.9900
0.598 0.552 0.784 0.932 0.9959 0.9958
(13)(7,14)(5,3)(1,2)(15)(16,4)(11,9)(6,8)(12,10)(-) (7)(1,13)(5,3)(11,9)(15,10)(6,2)(8,14)(16,4)(12)(-) (-)(11,1)(15,9)(5,12)(7,2)(8,13)(6,3)(16,4)(10,14)(-) (-)(-)(15,13)(16,2,10)(6,11,9)(7,8,1)(5,3,4)(12,14)(-)(-) (-)(-)(-)(11,7,1,9)(5,15,3,13)(6,16,4,14)(12,8,2,10)(-)(-)(-) (-)(-)(-)(11,7,13,9)(5,15,3,1)(6,16,4,2)(12,8,14,10)(-)(-)(-)
2 3 4
Mf
Un−r+1(z,t) by applying the operator ⊗ sequentially for j = 1, …, n ← as
Uj + 1 − r (z , t ) = Uj − r (z , t )⊗ uj (z , t ). ←
U^f , l (z , t )⊗ uj (z , t ) = ←
Nj
∑ p˜f ,k (t ) z cf ,k,y˜f ,k ⊗ ∑ pj,h (t ) z gj,h ← h=0
k=1 Mf
(17)
=
Nj
∑ ∑ p˜f ,k (t ) pj,h (t ) z
⎛ ⎞ ρ ⎜cf , k , σ ⎜⎛ϕ (y˜f , k , gj, h) ⎟⎞ ⎟, ϕ (y˜f , k , gj, h) ⎝ ⎠⎠ ⎝
=
k=1 h=0 Mf + 1
∑ U1(z,t), the vector-u-function for the first r-sized group is obtained after applying operator ⊗ r times. In Uf(z,t) (f > 0), the value y(s) of
f ,l
4.4. Counting m failed consecutive groups
δ (U^f , l (z , t )) =
(18)
(21)
4.5. Evaluating conditional system failure probability and instantaneous DSP
where Mf denotes the number of distinguishable combinations of cf,k and y˜f , k . The counter has an initial value of 0. The procedure in (15) and (16) is revised as
U^1 − r , l (z , t ) = z 0, y˜0
Mf
∑ p˜f ,k (t )1(cf ,k = m) k=1
Mf k=1
f,k
rence probabilities of such terms, and obtains the overall probability that the system failure is caused by components represented by the vector-u-function Uf(z, t) at time t given that the demand level is dl.
For detecting m failed r-sized groups for a particular demand level dl, an integer counter cf,k is incorporated into the vector-u-function (14) as:
∑ p˜f ,k (t ) z cf ,k,y˜ f ,k ,
(20)
cg + 1 if x < dl where ρ (cg , x ) = ⎧ ⎨ ⎩ 0 if x ≥ dl . The counter has value of m only when m consecutive r-sized groups fail. The system failure is indicated by the terms p˜f , k (t ) z cf , k, y˜f , k of U^ (t , z ) with c = m. The operator δ defined in (21) collects occur-
←
vector y˜f , k is the performance rate of component f − 1 + s in state k of the fth group.
U^f , l (z , t ) =
p˜f + 1, k (t ) z cf + 1, k, y˜f + 1, k
k=1
Let ef,l(t) denote the conditional probability that m consecutive rsized groups starting from component f (containing components from f to f + m + r − 2) fail at time t given that the demand level is dl. If these m consecutive groups starting from component f are failed, the entire system always fails regardless the states of components not belonging to those m groups. Thus, the conditional failure probability of the entire
(19)
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each component. The cumulative performance of any r = 2 adjacent components is 8, which is insufficient to meet the demand D(t)=10. Thus, the instantaneous DSP of the CSWS-WS in the mission beginning is 0. The demand can be satisfied only after failure of the first elements of the components and activation of the standby element with performance 8. As the probability of failure of the first element and activation of the standby one increases in time, the instantaneous DSP of the CSWS-WS also increases. It becomes decreasing when the probability that both elements belonging to the same component fail increases considerably. Though the element distribution II is impractical as it guarantees zero DSP at the mission beginning, it illustrates the possibility of the non-monotonic behavior of the instantaneous DSP even when the demand is constant.
system at time t given that the demand is dl can be evaluated as the summation of occurrence probabilities of the mutually exclusive events n−r−m+1
El (t ) = e1,l (t ) + e2,l (t )(1 − e1,l (t )) + en − r − m + 2, l (t )
∏
(1 − ej, l (t ))
j=1
(22) where e1,l(t)=δ (U^m, l (z , t )) . To evaluate e2,l(t)(1 − e1,l(t)) in (22), all the terms with Cm, k = m are removed from U^ (z , t ) (excluding cases where the system failure is m, l
caused by failures of the first m r-sized groups), the operator ⊗ is ap←
plied over the truncated vector-u-function U^m, l (z , t ) and u-function (z , t ) , and then the operator δ is applied over u (z,t) to obtain U^ m + 1, l
m+r
U^m + 1, l (z , t ) . i−1 Recursively, to evaluate ei, l (t ) ∏ j = 1 (1 − ej, l (t )) in (22), all the terms (z , t ) , then with c are removed from U^ =m
6. Optimization examples Based on the evaluation of DSP, the optimal element distribution problems formulated in (5) and (6) can be solved. Both of them are complicated combinatorial optimization problems with (n + ξ − 1)! possible solutions. Considering reasonable time limitations, the brute force search technique is not realistic even for moderate numbers of components and elements. The genetic algorithms (GA) based on the evolutionary search principle were proven to be an effective optimization tool in reliability engineering [26–28]. Based on the solution representation illustrated in Table 1, the steady state GA, named GENITOR is applied in this work to solve the proposed optimization problems. Refer to [16,23] for a description of the GENITOR algorithm.
m + i − 1, l
m + i − 1, k
U^m + i, l (z , t ) = U^m + i − 1, l (z , t )⊗ um + r + i − 1 (z , t ) is obtained, and finally ←
δ (U^m + i, l (z , t )) is applied. Having the DSCTP D(t), one obtains the instantaneous DSP of CSWSWS as L
a (t ) = 1 −
∑ hl (t ) El (t ).
(23)
l=0
The following pseudo-code summarizes the algorithm obtaining the instantaneous DSP of CSWS-WS.
6.1. Example 1 1. Initialization: a(t) = 1; 2. For l = 0,…,L: 2.1. Set El(t) = 0; U^1 − r , l (t , z ) = z 0, y˜0 ( y˜0 contains r zeros). 2.2. For j = 1,…,n: uj (z , t ) , and simplify the 2.1.1. Obtain U^j − r + 1, l (z , t ) = U^j − r , l (z , t )⊗ ←
For the same example CSWS-WS evaluated in Section 5, Table 2 presents the best obtained element distributions (the indices of elements are presented in parentheses for each component according to their activation sequence) and corresponding values of the minimum and average DSP during the mission. The element distributions obtained as solutions of problems (5) (maximizing the minimum instantaneous DSP during the mission) and (6) (maximizing the expected system DSP during the mission) coincide. The rightmost subfigure in Fig. 2 presents the instantaneous DSP of the example system for the best obtained element distributions for each value of m (no more than four elements can be located in the same position). It can be seen that much greater values of a(t) are achieved compared with those obtained using element distributions I and II in Fig. 2. Observe that in the optimal element distribution solutions some components remain empty (which is designated by “-” sign). Such phenomenon is common in sliding window systems because in each point the performance of the system is determined by the cumulative effect of several components and keeping some components empty allows enhancement of other components. The detailed discussion of this effect can be found in [29].
obtained vector-u-function by combining like terms (summing terms with the same exponents). 2.2.2. If j ≥ m + r − 1, then add the value δ (U^j − r + 1, l (z , t )) to El(t), (t , z ) . = m from U^ and remove all of the terms with c j−r+1,k
j − r + 1, l
2.3. Subtract hl(t)El(t) from a(t). As can be seen from the pseudo-code above the computational complexity of the algorithm is O(nL), i.e., it is a linear function of the number of demand levels L and the number of components n. 5. Illustrative example of instantaneous DSP evaluation Consider a CSWS-WS with ξ = 10 elements distributed among n = 5 components (positions). The time-to-failure of all the elements obeys the Weibull distribution with cdf Fi(t) = 1 − exp(−(t/100)1.5). The deceleration factor of all the elements is ζi = 0.2. The nominal performances are gi = 4 for element i = 1,3,5,7,9 and gi = 8 for element i = 2,4,6,8,10. The mission time is τ = 100, and the demand is constant D(t) = 10. Fig. 2 presents instantaneous DSP of the example CSWS-WS during the mission for r = 2, different values of m and two different element distributions I and II. In both element distributions, all components contain identical set of two elements with different nominal performances. In element distribution I the elements with greater nominal performance are activated first (in the beginning of the mission), whereas in element distribution II the elements with lower nominal performance are activated first. The instantaneous DSP of the example CSWS-WS decreases in time for element distribution I, however it behaves non-monotonically for element distribution II. Indeed, at the beginning of the mission distribution II presumes activation of elements with performance gi=4 in
6.2. Example 2 Consider a CSWS-WS with ξ = 16 elements distributed among n = 10 components. The time-to-failure of all the elements obeys the Weibull distribution Fi(t) = 1 − exp(−(t/ηi)βi) with parameters presented in Table 3. The nominal performances and the deceleration factors of all the elements are also presented in Table 3. Fig. 3 presents instantaneous DSP of the best obtained solutions of problem (5) for this example CSWS-WS with r = 3, different values of m and two values of the constant demand. As in the previous example, the instantaneous DSP increases in m because more consecutive failed groups of components are allowed. The best obtained element distributions and the DSP metrics are presented in Table 4. The comparison of solutions to problems (5) and (6) for this example CSWS-WS with r = 3 and D(t) = 15 is given in Table 5. 404
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Consider now random demand described by DSCTP, presented in Table 6 and Fig. 4. Fig. 5 presents instantaneous DSP of the CSWS-WS for different m and this stochastic demand for the best obtained solutions of problems (5) and (6) (For m = 2 and m = 3 the best obtained solutions for (5) and (6) coincide). The corresponding element distributions are presented in Table 7. It can be seen that for all the obtained element distributions the instantaneous system DSP behaves non-monotonically. Whereas for a constant demand the DSP monotonically decreases (see Fig. 3), for variable demands it begins increasing from certain time because the probabilities of the high demand levels (l = 1 and l = 2) decrease and, therefore, the probability that the random demand is satisfied increases. Then, because of the further system deterioration (failures of elements) the DSP becomes decreasing even for low demands.
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7. Conclusion and future work The existing works on modeling CSWS systems have assumed static performance distributions of their constituent components. This paper advances the state-of-the-art by considering a CSWS with components subject to dynamic performance distributions due to the warm-standby configuration of heterogeneous functional elements. The considered system called CSWS-WS fails if the accumulated performance in each of at least m consecutive r-sized windows cannot meet a desired random demand. A vector-u-function technique is suggested to evaluate the dynamic demand satisfaction probability of the considered CSWS-WS. The GA is further applied to solve the optimal element distribution and sequencing problem, maxmizing the minimum instananeous DSP or expected DSP over a certain mission time. As demonstrated through examples, the proposed methodology can facilitate effective choice of element allocation among different components and activation sequencing for each component, leading to significantly better DSP. The proposed methodology has assumed negligible element replacement time. Based on [30] we plan to extend the method to consider random replacement time in the case of the online element malfunctioning. We are also interested in modeling imperfect element fault detection and replacement mechanisms based on [18]. The present research can also be extended in the future to consider effects of common-cause failures [31,32] that can impact multiple components and elements simultaneously. Acknowledgment This work was supported in part by the National Natural Science Foundation of China (Grant no. 51875089). References [1] Levitin G. Linear multi-state sliding window systems. IEEE Trans Reliab 2003;52:63–269. [2] Levitin G. Optimal allocation of components in a linear multi-state sliding window system. Reliab Eng Syst Saf 2002;76:245–54. [3] Tong Y. A rearrangement inequality for the longest run with an application in
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