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Hindawi Publishing Corporation Journal of Quality and Reliability Engineering Volume 2015, Article ID 212303, 20 pages http://dx.doi.org/10.1155/2015/212303
Review Article Consecutive-Type Reliability Systems: An Overview and Some Applications
Ioannis S. Triantafyllou
Department of Statistics and Insurance Science, University of Piraeus, 80 Karaoli and Dimitriou Street, 18534 Piraeus, Greece
Correspondence should be addressed to Ioannis S. Triantafyllou; [email protected]
Received 28 July 2014; Revised 10 April 2015; Accepted 17 April 2015
Academic Editor: Xiaohu Li
Copyright © 2015 Ioannis S. Triantafyllou. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
The family of consecutive-type reliability systems is under investigation. More specifically, an up-to-date presentation of almost all generalizations of the well-known consecutive k-out-of-n: F system that have been proposed in the literature is displayed, while several recent and fundamental results for each member of the aforementioned family are stated.
1. Introduction depends only on its structure and not on the specific underly- ing distribution of 𝑋𝑖.Inotherwords,𝑠𝑖(𝑛) is the proportion 𝑘 𝑛 𝐹 Alinear(circular)consecutive -out-of- : system consists of permutations, among the 𝑛! equally likely permutations of 𝑛 of components which are linearly (circularly) arranged 𝑋1,𝑋2,...,𝑋𝑛, which result in a minimal cut set failing upon 𝑘 and the system fails if and only if at least consecutive the occurrence of 𝑋𝑖:𝑛. components fail. The most popular applications of these The signature of the system, which was first introduced systems pertain to computer network, telecommunication, by Samaniego [3], is closely related to many well-known pipeline network modeling, engineering, or integrated cir- reliability characteristics, a fact turning it to a very useful tool 𝑘 𝑛 𝐹 cuitry design. The consecutive -out-of- : system has for studying coherent systems and their ageing properties. For been subject of substantial research interest for many years example, the reliability polynomial of a structure can be easily and a lot of generalizations have been suggested in order expressed in terms of its signature. More precisely, Samaniego to accommodate more flexible operation principles. For a [3] proved that for any coherent system with independent 𝑘 𝑛 𝐹 detailed presentation of the consecutive -out-of- : systems and identical components which have absolutely continu- and some generalizations, the interested reader is referred to ous cumulative density functions, system’s reliability can be the excellent monograph of Kuo and Zuo [1]ortheworkof expressed as Chang et al. [2]. 𝑛 Let 𝑇 be the lifetime of a reliability system with 𝑛 com- 𝑅 (𝑡) =𝑃(𝑇>𝑡) = ∑ 𝑠𝑖 (𝑛) 𝑃(𝑋𝑖:𝑛 >𝑡). (2) ponents and 𝑋1,𝑋2,...,𝑋𝑛 its components’ lifetimes. If we 𝑖=1 assume that 𝑋1,𝑋2,...,𝑋𝑛 are exchangeable (and therefore Navarro and Rychlik [4] proved that the above represen- identically distributed but not necessarily independent), the 𝑋 ,𝑋 ,...,𝑋 signature of the system is defined as the probability vector tation also holds true when the lifetimes 1 2 𝑛 have an absolutely continuous exchangeable distribution (this (𝑠1(𝑛),2 𝑠 (𝑛),...,𝑠𝑛(𝑛)) with property had been mentioned earlier by Kochar et al. [5]). Moreover, Eryilmaz and Bayramoglu [6]usedthesystem 𝑠 (𝑛) =𝑃(𝑇=𝑋 ), 𝑖= , ,...,𝑛, 𝑖 𝑖:𝑛 1 2 (1) signature in order to evaluate the extreme residual lifetimes of the remaining components after the complete failure of the where 𝑋1:𝑛 ≤𝑋2:𝑛 ≤ ⋅⋅⋅ ≤ 𝑋𝑛:𝑛 are the order statistics of system. the sample 𝑋1,𝑋2,...,𝑋𝑛. It can be easily verified that, in Both reliability function and signature of a structure the exchangeable case, the signature of a reliability system can be evaluated based on either recursive formulas or 2 Journal of Quality and Reliability Engineering explicit expressions. In some cases, where neither of the In order to launch the above recurrence scheme, a set of aforementioned methods can be established for a specific initial of conditions would be necessary. Observing that for system, approximating and limiting results are available. In 𝑛≥𝑘+1 the following ensues the present review article, the family of consecutive-type 𝐿 𝐿 reliability systems is under investigation. More specifically, 𝑅1,𝑘,𝑛 (𝑝1,𝑝2,...,𝑝𝑛)=𝑅1,𝑘,𝑛−1 (𝑝1,𝑝2,...,𝑝𝑛) Section 2 offers an up-to-date presentation of almost all −𝑝 ( −𝑝 )⋅⋅⋅( −𝑝 ) generalizations of the well-known consecutive 𝑘-out-of-𝑛: 𝐹 𝑛−𝑘 1 𝑛−𝑘+1 1 𝑛 (4) system that have been proposed and studied in the literature. ⋅𝑅𝐿 (𝑝 ,𝑝 ,...,𝑝 ) For each structure, the corresponding fundamental work and 1,𝑘,𝑛−𝑘−1 1 2 𝑛 selected results are displayed either briefly or in detail. In while Section 3, applications of these reliability systems in several fields are described, while Section 4 presents a full-detailed 𝑛 𝑅𝐿 (𝑝 ,𝑝 ,...,𝑝 )= − ∏ ( −𝑝), diagram which connects the aforementioned structures by 𝑚,𝑘,𝑚𝑘 1 2 𝑛 1 1 𝑖 (5) 𝑖=1 giving the information under which conditions a system can be treated as a special case of another one. we have at hand the set of initial values needed to evaluate reliability of the system. 2. Family of Consecutive-Type Note that for the i.i.d case (e.g., 𝑝1 =𝑝2 =⋅⋅⋅=𝑝𝑛 =𝑝), Reliability Systems the recurrence of Theorem 1 reduces to the following form: 𝐿 In this section, we study in detail almost all generalizations of 𝑅𝑚,𝑘,𝑛 (𝑝) the well-known consecutive 𝑘-out-of-𝑛: 𝐹 systems that have 𝐿 been proposed in the literature till now. For each reliability =𝑅𝑚,𝑘,𝑛−1 (𝑝) structurethatisincludedinthefamilyofconsecutive- 𝑚 (6) type systems, the general operational structure is described, 𝑠𝑘 𝐿 𝐿 − ∑ 𝑝𝑞 [𝑅𝑚−𝑠+1,𝑘,𝑛−𝑠𝑘−1 (𝑝) −𝑚−𝑠,𝑘,𝑛−𝑠𝑘− 𝑅 1 (𝑝)] , while several important and some recent relevant results are 𝑠=1 displayed. 𝑛≥𝑘𝑚+1. 2.1. 𝑚-Consecutive-𝑘-out-of-𝑛: 𝐹 Systems. An 𝑚-consecutive- 𝑘 𝑛 𝐹 𝑛 𝐶 -out-of- : system consists of components and fails if and Theorem 2 (Alevizos et al. [9]). Let 𝑅𝑚,𝑘,𝑛(𝑝1,𝑝2,...,𝑝𝑛) only if there exist at least 𝑚 nonoverlapping runs of 𝑘 con- denote the reliability function of a circular 𝑚-consecutive-𝑘- secutive failed components. This system was first introduced out-of-𝑛: 𝐹 system, where 𝑝𝑖 (𝑞𝑖 = 1 −𝑝𝑖) is the reliability 𝐶 by Griffith [7] and since then it has attracted a considerable (unreliability) of its 𝑖th component. Then 𝑅𝑚,𝑘,𝑛(𝑝1,𝑝2,...,𝑝𝑛) research attraction. In the sequel, we present the main results satisfies the following recurrence: appearing in the literature, such as recurrence relations and 𝐶 𝐿 closed formulas for the evaluation of the reliability function 𝑅𝑚,𝑘,𝑛 (𝑝1,𝑝2,...,𝑝𝑛)=𝑝𝑛𝑅𝑚,𝑘,𝑛−1 (𝑝1,𝑝2,...,𝑝𝑛) and signature vector of 𝑚-consecutive-𝑘-out-of-𝑛: 𝐹 systems. 𝐶 It is worth mentioning that the aforementioned system −𝑝𝑛 −𝑞𝑛𝑅𝑚,𝑘,𝑛−1 (𝑝1,𝑝2,...,𝑝𝑛) generalizes the well-known consecutive-𝑘-out-of-𝑛: 𝐹 system (for 𝑚=1), while for 𝑘=1the𝑚-consecutive-𝑘-out-of-𝑛: 𝐹 𝑚 𝑠𝑘−1 − ∑ ∑ (𝑞 𝑞 ⋅⋅⋅𝑞𝑝 ) system reduces to an ordinary 𝑚-out-of-𝑛: 𝐹 system. 1 2 𝑖 𝑖+1 𝑠=1 𝑖=0 (7) 2.1.1. Recursive Schemes for the Reliability of 𝑚-Consecutive-𝑘- ⋅(𝑞𝑛𝑞𝑛−1 ⋅⋅⋅𝑞𝑛−𝑠𝑘+𝑖+1𝑝𝑛−𝑠𝑘+𝑖) out-of-𝑛: 𝐹 Systems. The following theorems provide recur- rence for the calculation of reliability of an 𝑚-consecutive-𝑘- 𝐿,𝑖+2 ×(𝑅𝑚−𝑠+1,𝑘,𝑛−𝑠𝑘−2 (𝑝1,𝑝2,...,𝑝𝑛) out-of-𝑛: 𝐹 system. 𝐿,𝑖+2 𝐿 −𝑅𝑚−𝑠,𝑘,𝑛−𝑠𝑘−2 (𝑝1,𝑝2,...,𝑝𝑛)) , Theorem 1 (Papastavridis [8]). Let 𝑅𝑚,𝑘,𝑛(𝑝1,𝑝2,...,𝑝𝑛) denote the reliability function of a linear 𝑚-consecutive-𝑘-out- 𝑛≥𝑘𝑚+ 𝑅𝐿,𝑖+2(𝑝 ,𝑝 ,...,𝑝 ) 𝑛 𝐹 𝑝 𝑖 where 2 and 𝑑,𝑘,𝑐 1 2 𝑛 denotes the of- : system, where 𝑖 is the reliability of its th component. 𝑑 𝑘 𝑐 𝐹 𝑅𝐿 (𝑝 ,𝑝 ,...,𝑝 ) reliability of a linear -consecutive- -out-of- : subsystem Then 𝑚,𝑘,𝑛 1 2 𝑛 satisfies the following recurrence with components 𝑖+2,𝑖+3,...,𝑛−𝑠𝑘+𝑖−1. relation: To launch the above recursive scheme, a set of initial of 𝐿 𝐿 𝑅𝑚,𝑘,𝑛 (𝑝1,𝑝2,...,𝑝𝑛)=𝑅𝑚,𝑘,𝑛 (𝑝1,𝑝2,...,𝑝𝑛) conditions would be necessary. Observing that the following 𝑚 𝑠𝑘 ensue − ∑ 𝑝 ∏𝑞 [𝑅𝐿 (𝑝 ,𝑝 ,...,𝑝 ) 𝐶 𝑛−𝑠𝑘 𝑛−𝑠𝑘+𝑖 𝑚−𝑠+1,𝑘,𝑛−𝑠𝑘−1 1 2 𝑛 (3) (i) 𝑅𝑚,𝑘,𝑛 = 1,for𝑛<𝑘𝑚, 𝑠=1 𝑖=1 𝐶 𝑛 (ii) 𝑅 = 1 −∏ 𝑞𝑖,for𝑛=𝑘𝑚, −𝑅𝐿 (𝑝 ,𝑝 ,...,𝑝 )] , 𝑚,𝑘,𝑛 𝑖=1 𝑚−𝑠,𝑘,𝑛−𝑠𝑘−1 1 2 𝑛 𝐶 𝑛 𝑛 (iii) 𝑅𝑚,𝑘,𝑛 = 1 − ∏𝑖=1𝑞𝑖 − ∑𝑖=1 𝑞1𝑞2 ⋅⋅⋅𝑞𝑖−1𝑝𝑖𝑞𝑖+1 ⋅⋅⋅𝑞𝑛,for 𝐿 where 𝑛≥𝑘𝑚+1 and 𝑅0,𝑘,0(𝑝1,𝑝2,...,𝑝𝑛)=0. 𝑛=𝑘𝑚+1, Journal of Quality and Reliability Engineering 3
𝐶 we have at hand the set of initial values needed to evaluate Theorem 5 (Makri and Philippou [10]). Let 𝑅𝑚,𝑘,𝑛(𝑝) denote reliability of the system. the reliability function of a circular 𝑚-consecutive-𝑘-out-of-𝑛: Note that for the i.i.d case (e.g., 𝑝1 =𝑝2 = ⋅⋅⋅ = 𝑝𝑛 =𝑝), 𝐹 system, where 𝑝(𝑞) is the common reliability (unreliability) 𝐶 the recurrence of Theorem 2 reduces to the following form: of its components. Then, 𝑅𝑚,𝑘,𝑛(𝑝) can be expressed as follows: 𝐶 𝐿 𝐶 𝑅𝑚,𝑘,𝑛 (𝑝) = 𝑝𝑅𝑚,𝑘,𝑛−1 (𝑝) − 𝑝 𝑚,𝑘,𝑛−− 𝑞𝑅 1 (𝑝) (i)
𝑚 𝑠𝑘−1 𝑚−1 − ∑ ∑ 𝑝2𝑞𝑠𝑘 (𝑅𝐿 (𝑝) 𝐶 𝑚−𝑠+1,𝑘,𝑛−𝑠𝑘−2 (8) 𝑅𝑚,𝑘,𝑛 (𝑝) = ∑ 𝑀𝑥,𝑐 (𝑞;𝑘,𝑛), (13) 𝑠=1 𝑖=0 𝑥=0 𝐿 −𝑅𝑚−𝑠,𝑘,𝑛−𝑠𝑘−2 (𝑝)) , 𝑛 ≥ 𝑘𝑚 + 2. where It is worth mentioning that the computational complexity of recurrence included in Theorem 1 is equal to 𝑂(𝑛𝑚),whilethe 𝑀 (𝑞;𝑘,𝑛)=𝑝( −𝑝)𝑛−1 3 𝑥,𝑐 1 corresponding one for equations of Theorem 2 equals 𝑂(𝑛𝑘𝑚 ). 𝑘 𝑥1 +𝑥2 +⋅⋅⋅+𝑥𝑘 +𝑥 𝑚 ⋅ ∑ 𝑖 ∑ ( ) 2.1.2. Exact Formulas for the Reliability of -Consecutive- 𝑖=1 𝑥1,𝑥2,...,𝑥𝑘,𝑥 𝑘-out-of-𝑛: 𝐹 Systems. The following theorems offer closed 𝑥 +𝑥 +⋅⋅⋅+𝑥 expressions for the evaluation of reliability of an 𝑚- 𝑝 1 2 𝑘 𝑘 𝑛 𝐹 ⋅( ) consecutive- -out-of- : system. 1 −𝑝 (14) Theorem 3 𝑅𝐿 (𝑝) 𝑘 (Papastavridis [8]). Let 𝑚,𝑘,𝑛 denote the relia- 𝑛−1 𝑥 bility function of a linear 𝑚-consecutive-𝑘-out-of-𝑛: 𝐹 system, +𝑘𝑝𝑞 ∑ ∑ 𝑥1 +𝑥2 +⋅⋅⋅+𝑥𝑘 + 1 where 𝑝 is the common reliability of its i.i.d. components. Then 𝑖=1 𝐿 𝑅 (𝑝) 𝑥 +𝑥 +⋅⋅⋅+𝑥 𝑚,𝑘,𝑛 is given as follows: 𝑥 +𝑥 +⋅⋅⋅+𝑥 +𝑥 𝑝 1 2 𝑘 ×( 1 2 𝑘 )( ) ∗ 𝑠 𝑛 𝑠+𝑛−𝑖 𝑥1,𝑥2,...,𝑥𝑘,𝑥 𝑞 𝐿 𝑅 (𝑝) = ∑ ∑ ( )𝑁(𝑖−𝑠𝑘,𝑛−𝑖+1) 𝑚,𝑘,𝑛 𝑠 𝑛 𝑠=𝑚 𝑖=𝑠𝑘 (9) +𝑞 𝛿𝑥,[𝑛/𝑘] 𝑖 𝑛−𝑖 ⋅(1 −𝑝) 𝑝 , is the well-known multinomial coefficient, while ∗ where 𝑛≥𝑘𝑚, 𝑠 (𝑥) denotes the greatest integer lower bound the inner summation is overall nonnegative integers 𝑥 𝑁(𝑖, 𝑗) 𝑘 of and can be expressed as 𝑥1,𝑥2,...,𝑥𝑘 such that ∑𝑗=1 𝑗𝑥𝑗 = 𝑛−𝑖−𝑘𝑥.Moreover, [𝑥] 𝑗 𝑗 𝑖+𝑗− −𝑠𝑘 in the above expression denotes the greatest integer 𝑠 1 𝑥 𝛿 𝑁 (𝑖,) 𝑗 = ∑ (−1) ( )( ) . (10) in ,while 𝑖,𝑗 is the Kronecker delta function. 𝑠 𝑖−𝑠𝑘 𝑠=0 (ii) Consider Theorem 4 𝑅𝐿 (𝑝) (Makri and Philippou [10]). Let 𝑚,𝑘,𝑛 denote 𝑚−1 the reliability function of a linear 𝑚-consecutive-𝑘-out-of-𝑛: 𝐹 𝐶 𝑅𝑚,𝑘,𝑛 (𝑝) = ∑ 𝐵𝑥,𝑐 (𝑞; 𝑘, 𝑛) , (15) system composed by i.i.d. components, where 𝑝 is their common 𝑥=0 𝐿 reliability. Then 𝑅𝑚,𝑘,𝑛(𝑝) is given as follows: 𝐿 where 𝑅𝑚,𝑘,𝑛 (𝑝) 𝐵 (𝑞;𝑘,𝑛)=𝑞𝑛𝛿 𝑚−1 𝑘−1 𝑥 +𝑥 +⋅⋅⋅+𝑥 +𝑥 𝑥,𝑐 𝑥,[𝑛/𝑘] = ∑ ∑ ∑ ( 1 2 𝑘 ) [(𝑛−𝑘𝑥−1−𝑦)/𝑘] 𝑥 ,𝑥 ,...,𝑥 𝑥 ,𝑥 ,...,𝑥 ,𝑥 (11) 𝑛−𝑘𝑥−1 𝑥=0 𝑖=0 1 2 𝑘 1 2 𝑘 𝑛−𝑦−1 𝑦+1 𝑗 + ∑ 𝑞 𝑝 ∑ (−1) 𝑥 +𝑥 +⋅⋅⋅+𝑥 𝑛 𝑝 1 2 𝑘 𝑦=[(𝑛−𝑘𝑥−1)/𝑘] 𝑗=0 ⋅(1 −𝑝) ( ) , 1 −𝑝 𝑦+𝑥 𝑦+1 𝑛−𝑘𝑥−𝑗𝑘 ⋅[( )( )( ) where 𝑦 𝑗 𝑦+1 (16) 𝑎 𝑎! 𝑦+𝑥 𝑦 𝑛−𝑘𝑥−𝑗𝑘− ( )= (12) 1 𝑏 ,𝑏 ,...,𝑏 𝑏 !𝑏 !⋅⋅⋅𝑏! +𝑘( )( )( ) 1 2 𝑐 1 2 𝑐 𝑦+1 𝑗 𝑦 is the well-known multinomial coefficient, while the inner 𝑦+𝑥+1 𝑦 𝑛−𝑘𝑥−1 −(𝑗+1)𝑘 summation is over all nonnegative integers 𝑥1,𝑥2,...,𝑥𝑘 such −𝑘( )( )( )]. 𝑘 𝑦+ 𝑗 𝑦 that ∑𝑗=1 𝑗𝑥𝑗 =𝑛−𝑖−𝑘𝑥. 1 4 Journal of Quality and Reliability Engineering
2.1.3. Approximations for the Reliability of 𝑚-Consecutive- Theorem 9 (Eryilmaz et al. [12]). Let (𝑠1(𝑛, 𝑘, 𝑚),2 𝑠 (𝑛,𝑘,𝑚), 𝑘-out-of-𝑛: 𝐹 Systems. The following theorems offer some ...,𝑠𝑛(𝑛,𝑘,𝑚))be the signature vector of an 𝑚-consecutive-𝑘- limiting results for the evaluation of reliability of an 𝑚- out-of-𝑛: 𝐹 system. Then the following ensues consecutive-𝑘-out-of-𝑛: 𝐹 system. (a) the quantities 𝑠𝑖(𝑛,𝑘,𝑚)satisfy the recurrence relation: 𝑎 𝑎 Theorem 6 (Papastavridis [8]). Let 𝑞(𝑡) = (𝜆𝑡) +𝑂(𝑡) 𝑛 be the common failure distribution of the components of an 𝑖( )𝑠𝑖 (𝑛, 𝑘,) 𝑚 𝑚-consecutive-𝑘-out-of-𝑛: 𝐹 system, where 𝜆, 𝑎 are positive 𝑖 constants. Then the following ensues 𝑘 𝑛−𝑙 𝑚−1 𝑎𝑘𝑖 = ∑ (𝑖−𝑙+ ) ( )𝑠 (𝑛 − 𝑙,) 𝑘,𝑚 1/𝑘𝑎 𝑎𝑘 (𝜆𝑡) 1 𝑖−𝑙+1 (21) 𝑃(𝑛 𝑇 >𝑡)= [− (𝜆𝑡) ] ∑ , 𝑙=1 𝑖−𝑙+1 𝑛→∞lim 𝑛 exp (17) 𝑖=0 𝑖! 𝑚 𝑛−𝑘 + (𝑖−𝑘) ( )𝑠 (𝑛 − 𝑘, 𝑘, 𝑚− ) where 𝑇𝑛 is the time of first failure of the system. 𝑖−𝑘 1 𝑚−1 𝑖−𝑘
Theorem 7 (Godbole [11]). Let 𝑞𝑗 = 1 −𝑝𝑗, 𝑗=1, 2,...,𝑛 be the failure probability of the 𝑗th component of an 𝑚- for 𝑖=1, 2,...,𝑛and 𝑛≥𝑘+1. 𝑘 𝑛 𝐹 𝑅 (𝑝 ) consecutive- -out-of- : system. Then the reliability 𝑚,𝑘,𝑛 𝑗 (b) the quantities 𝑠𝑖(𝑛,𝑘,𝑚)can be expressed as of the system satisfies the following inequality: 𝑛 −1 [𝑖/𝑘] 𝑠+𝑛−𝑖 𝑚−1 𝑥 (−𝜆𝑛)𝜆 𝑘 𝑘 𝑠 (𝑛, 𝑘,) 𝑚 =( ) [ ∑ ( ) 𝑅 (𝑝 )− ∑ 𝑛 ≤( 𝑘+ +𝑛𝑞 )𝑞 , 𝑖 𝑖 𝑠 𝑚,𝑘,𝑛 𝑗 exp 𝑥! 2 2 (18) 𝑠=𝑚 𝑥=0 𝑛−𝑖+1 𝑞= 𝑞 𝜆 = ∑𝑛 𝑝 𝑞 ⋅⋅⋅𝑞 ⋅𝐶(𝑖−𝑠𝑘,𝑘,𝑛−𝑖+ ) − where max𝑗≥1 𝑗 and 𝑛 𝑗=𝑘+1 𝑗−𝑘 𝑗−𝑘+1 𝑗. 1 𝑖 (22) [(𝑖−1)/𝑘] 𝑠+𝑛−𝑖+1 2.1.4. Signature Vector of 𝑚-Consecutive-𝑘-out-of-𝑛: 𝐹 Systems. ⋅ ∑ ( ) Let p denote the common reliability of the components of an 𝑠=𝑚 𝑠 𝑚-consecutive-𝑘-out-of-𝑛: 𝐹 system. The following theorem offers a generating function approach of the aforementioned ⋅𝐶(𝑖−𝑠𝑘− ,𝑘,𝑛−𝑖+ )] system. 1 2
Theorem 8. Let (𝑠1(𝑛, 𝑘, 𝑚),2 𝑠 (𝑛,𝑘,𝑚),...,𝑠𝑛(𝑛,𝑘,𝑚)) and 𝑚𝑘≤𝑖≤𝑛 𝑠 (𝑛,𝑘,𝑚)= 𝑖<𝑚𝑘 𝑅𝑚,𝑘,𝑛(𝑝) be the signature and the reliability function of an 𝑚- for and 𝑖 0 if ,where consecutive-𝑘-out-of-𝑛: 𝐹 system, respectively. Then 𝑗 𝑗 𝑖+𝑗− −𝑠𝑘 𝑠 1 𝑖 ( 𝑛 ) 𝑠 (𝑛,𝑘,𝑚) 𝐶 (𝑖, 𝑘, 𝑗)= ∑ (−1) ( )( ). (23) (a) the double generating function of 𝑖 𝑖 is 𝑠 𝑖−𝑠𝑘 given by 𝑠=0
∞ 𝑛 𝑛 2.1.5. Additional Results for 𝑚-Consecutive-𝑘-out-of-𝑛: 𝐹 Sys- ∑ ∑ 𝑖( )𝑠 (𝑛, 𝑘,) 𝑚 𝑡𝑖𝑥𝑛 𝑖 tems. Beyond the results mentioned in the previous sub- 𝑛=1 𝑖=1 𝑖 sections, additional studies have appeared in the literature 𝑘𝑚 𝑚−1 𝑘 (19) 𝑚 𝑘 𝑛 𝐹 𝑚 (𝑡𝑥) (𝑡𝑥 − 1) {𝑘−𝑘𝑡𝑥+𝑡𝑥[(𝑡𝑥) − 1]} for the -consecutive- -out-of- : system. Eryilmaz [14] = derived explicit expressions for the component importance 𝑘 𝑚+1 {𝑥 [1 +𝑡−(𝑡𝑥) ]−1} measures for the aforementioned structure consisting of exchangeable components. More specifically, Eryilmaz [14] (Eryilmaz et al. [12]), studied in detail the well-known Birnbaum and Barlow- 𝑚 𝑘 𝑅 (𝑝) Proschan importance measures for a -consecutive- -out- (b) the generating function of 𝑚,𝑘,𝑛 is given as follows: of-𝑛: 𝐹 system. Furthermore, Ghoraf [15]offeredrecursive ∞ formulas for calculating the reliability function of the circular 𝑛 𝑟 (𝑧) = ∑ 𝑅𝑛,𝑚,𝑘 (𝑝) 𝑧 case of the aforementioned structure couching on the corre- 𝑛=0 sponding recurrence equations for the linear one. (20) (𝑞𝑧)𝑚𝑘 = 1 − 2.2. 𝑟-within-Consecutive-𝑘-out-of-𝑛: 𝐹 Systems. An 𝑟- −𝑧 𝑘−1 𝑗 𝑚 𝑘 𝑛 𝐹 𝑛 1 (1 −𝑧) (1 −𝑝𝑧∑𝑗=0 (𝑞𝑧) ) within-consecutive- -out-of- : system consists of components and fails if and only if there exist 𝑘 consecutive 𝑟 (Koutras [13]). components which include among them at least failed components. This system was first introduced by Griffith The next theorem offers expressions for the evaluation ofthe [7], but its mathematical modelling has been done earlier signature vector of an 𝑚-consecutive-𝑘-out-of-𝑛: 𝐹 system. by Greenberg [16] and Saperstein [17]. In the sequel, we Journal of Quality and Reliability Engineering 5
𝐿 𝐶 present the main results appearing in the literature, such as Theorem 13 (Sfakianakis et al. [18]). Let 𝑅2,𝑘,𝑛(𝑝) (𝑅2,𝑘,𝑛(𝑝)) recurrence relations and closed formulas for the evaluation denote the reliability function of a linear (circular) 2-within- of the reliability function and signature vector of 𝑟-within- consecutive-𝑘-out-of-𝑛: 𝐹 system, where 𝑝 is the common consecutive-𝑘-out-of-𝑛: 𝐹 systems. It is worth mentioning reliability of its components. Then the following recurrences that the aforementioned system generalizes the well-known ensue: consecutive-𝑘-out-of-𝑛: 𝐹 system (for 𝑟<𝑘), while for 𝑘=𝑛 the 𝑟-within-consecutive-𝑘-out-of-𝑛: 𝐹 system reduces to an (i) ordinary 𝑟-out-of-𝑛: 𝐹 system. 𝑚 𝑛−(𝑗− ) 𝑘− 𝐿 1 ( 1) 𝑗 𝑛−𝑗 𝑅 (𝑝) = ∑ ( )(1 −𝑝) 𝑝 , 2,𝑘,𝑛 𝑗 (27) 2.2.1. Recursive Schemes for the Reliability of 𝑟-within-Consec- 𝑗=0 utive-𝑘-out-of-𝑛: 𝐹 Systems. The following theorems provide recurrence for the calculation of reliability of an 𝑟-within- where 𝑚=[(𝑛+𝑘−1)/𝑘], 𝑘 𝑛 𝐹 consecutive- -out-of- : system. (ii) Theorem 10 𝑅𝐿 (𝑝) 𝐶 (Sfakianakis et al. [18]). Let 𝑟,𝑘,𝑛 denote the 𝑅2,𝑘,𝑛 (𝑝) reliability function of a linear 𝑟-within-consecutive-𝑘-out-of-𝑛: 𝐹 𝑝 𝑠 𝑛−𝑗(𝑘− ) system, where is the common reliability of its components. 𝑛 1 𝑗 𝑛−𝑗 (28) 𝐿 = ∑ ( )( −𝑝) 𝑝 , Then, for 𝑛 = 𝑟 + 𝜆,, 𝜆≤𝑟 𝑅 (𝑝) satisfies the following 1 𝑟,𝑘,𝑛 𝑗=0 𝑛−𝑗(𝑘−1) 𝑗 recurrence relation: 𝑅𝐿 (𝑝) 𝑟,𝑘,𝑛 where 𝑚=[𝑛/𝑘]. 𝑘 𝑘−𝜆 (24) 𝐿 𝑘−𝜆−𝑟+𝑥 𝑟−𝑥 = ∑ 𝑅𝑥,𝜆,2𝜆 (𝑝) ( )𝑝 (1 −𝑝) , 2.2.3. Approximations for the Reliability of r-within-Consecu- 𝑥=1 𝑟−𝑥 tive-𝑘-out-of-𝑛: 𝐹 Systems. The following theorems offer some limiting results for the evaluation of reliability of an r- 𝑅𝐿 (𝑝) = 𝑥>𝜆 where 𝑥,𝜆,2𝜆 1,if . within-consecutive-𝑘-out-of-𝑛: 𝐹 system. Theorem 11 𝑅𝐿 (𝑝) 𝐿 (Eryilmaz [19]). Let 𝑟,𝑘,𝑛 denote the reliability Theorem 14 (Eryilmaz et al. [20]). Let 𝑅𝑟,𝑘,𝑛(𝑡) = 𝑃(𝑇𝑟,𝑘,𝑛 >𝑡) function of a linear 𝑟-within-consecutive-𝑘-out-of-𝑛: 𝐹 system, denote the reliability function of a linear 𝑟-within-consecutive- 𝑝 𝐿 where is the common reliability of its components. Then, for 𝑘-out-of-𝑛: 𝐹 system. Then, for 1 ≤𝑟≤𝑘≤𝑛, 𝑅𝑟,𝑘,𝑛(𝑡) satisfies 𝑛≤ 𝑘 𝑅𝐿 (𝑝) 2 , 𝑟,𝑘,𝑛 satisfies the following recurrence relation: the following inequalities: min(𝑛−𝑘,𝑟−1) 𝐿 [𝑛−𝑘+1:𝑘] (i) 𝑅𝑟,𝑘,𝑛 (𝑡) = ∑ 𝑃(𝑇𝑟−𝑠:2𝑘−𝑛 >𝑡) 𝑠=0 𝐿 (1) (25) 𝑅𝑟,𝑘,𝑛 (𝑡) ≥ 1 − (𝑛−𝑘+1) 𝑃(𝑇𝑟:𝑘 ≤𝑡) ∗ ∗ (29) ⋅[𝑅𝑠+1,𝑛−𝑘:2(𝑛−𝑘) (𝑡) −𝑅𝑠,𝑛−𝑘:2(𝑛−𝑘) (𝑡)], (1) (2) + (𝑛−𝑘) 𝑃(𝑇𝑟:𝑘 ≤𝑡,𝑇𝑟:𝑘 ≤𝑡), ∗ where 𝑅𝑠,𝑛−𝑘:2(𝑛−𝑘)(𝑡) is the reliability of 𝑠-within-consecutive- (𝑛−𝑚)-out-of-2(𝑛−𝑚): 𝐹 system with components 1, 2,...,𝑛− (𝑗) where 𝑇 is the 𝑟th smallest lifetime among 𝑇𝑗,𝑇𝑗+1, 𝑚, 𝑚 + ,...,𝑛 𝑇[𝑖:𝑖+𝑚−1] 𝑘 𝑟:𝑘 1 ,while 𝑘:𝑚 denotes the lifetime of - ...,𝑇𝑗+𝑘−1,𝑟≤𝑘, 1 ≤𝑗≤𝑛−𝑘+1 and out-of-𝑚: 𝐹 subsystem of components with the lifetimes (1) 𝑇𝑖,𝑇𝑖+1,...,𝑇𝑖+𝑚−1, 1 ≤𝑖≤𝑛−𝑚+1. 𝑃(𝑇𝑟:𝑘 ≤𝑡) 𝐿 Theorem 12 (Koutras [13]). Let 𝑅𝑟,𝑘,𝑛(𝑝) denote the reliability 𝑘 𝑘 𝑟−𝑠 𝑟−𝑠 (30) 𝑖 function of a linear r-within-consecutive-𝑘-out-of-𝑛: 𝐹 system, = ∑ ( ) ∑ (−1) ( )𝑃(𝑇1 ≤𝑡,...,𝑇𝑠+𝑖 ≤𝑡). 𝑠 𝑖 where 𝑝 is the common reliability of its components. Then, 𝑠=𝑟 𝑖=0 𝐿 𝑅𝑟,𝑘,𝑛(𝑝) satisfies the following recurrence relation: (ii) Consider 𝑅𝐿 (𝑝) 𝑟,𝑘,𝑛 𝐿 𝑅𝑟,𝑘,𝑛 (𝑡) {1,𝑓𝑜𝑟𝑛=0, { (26) 𝑟−1 𝑘 𝑘 ℎ ℎ (31) = 𝑝𝑅 (𝑝) + 𝑛−𝑞𝑝 1,𝑓𝑜𝑟≤𝑛≤𝑘− , { 𝑟,𝑘,𝑛−1 1 1 ≤ ∑ ( )⋅⋅⋅( )𝑓(ℎ𝑘−∑ 𝑗𝑖, ∑ 𝑗𝑖), { 𝑗 𝑗 𝑘−1 𝑗1,𝑗2,...,𝑗ℎ=0 1 ℎ 𝑖=1 𝑖=1 {𝑝𝑅𝑟,𝑘,𝑛−1 (𝑝) + 𝑞𝑝 𝑅𝑟,𝑘,𝑛−𝑘 (𝑝) , 𝑓𝑜𝑟 𝑛≥𝑘. where ℎ=[𝑛/𝑘]and 2.2.2. Exact Formulas for the Reliability of 𝑟-within-Consecu- tive-𝑘-out-of-𝑛: 𝐹 Systems. The following Theorem offers 𝑎 𝑎 𝑖 closed expressions for the evaluation of reliability of an 𝑟- 𝑓 (𝑎,) 𝑏 = ∑ (−1) ( )𝑃(𝑇1 ≤𝑡,...,𝑇𝑏+𝑖 ≤𝑡). (32) 𝑖 within-consecutive-𝑘-out-of-𝑛: 𝐹 system. 𝑖=0 6 Journal of Quality and Reliability Engineering
Letusnextdenoteby𝐴𝑖 the event where there are at (iii) 𝐵𝑖 denotes the event that there are at most 𝑟−1 failures least 𝑟 failed components from 𝑖 to 𝑖+𝑘−1,for𝑖= among components 𝑖−𝑘+1,𝑖−𝑘+2,...,𝑖−1,(𝑖= 1, 2,...,𝑛−𝑘+1,while𝑆1, 𝑆2, 𝑆3 are defined as follows: 𝑘, 𝑘 + 1,...,𝑛).
𝑛−𝑘+1 (iv) 𝐶𝑖 denotes the event that, for 𝑓=min(𝑖 − 𝑘, 𝑘 −𝑟+ 1), (𝑖−𝑘)−𝑓+ ,(𝑖− 𝑆1 = ∑ 𝑃(𝐴𝑖), there is no failure among components 1 𝑖=1 𝑘) − 𝑓 + 2,...,𝑖−𝑘,for𝑖=𝑘+1,𝑘+2,...,𝑛. 𝑆 = ∑ 𝑃(𝐴 𝐴 ), 𝐿 2 𝑖 𝑗 (33) Theorem 16 (Papastavridis and Koutras [21]). Let 𝑅𝑟,𝑘,𝑛(𝑡) 1≤𝑖<𝑗≤𝑛−𝑘+1 denote the reliability function of a linear 𝑟-within-consecutive- 𝐿 𝑘-out-of-𝑛: F system. Then 𝑅 (𝑡) satisfies the following 𝑆 = ∑ 𝑃(𝐴 𝐴 𝐴 ). 𝑟,𝑘,𝑛 3 𝑖 𝑗 ] inequalities: 𝑖<𝑗<]≤𝑛−𝑘+1 𝑛 𝑃(𝑍 ) ∏ 𝑃(𝑋) Theorem 15 𝑅𝐿 (𝑝) 𝑘 𝑖 (Sfakianakis et al. [18]). Let 𝑟,𝑘,𝑛 denote the 𝑖=𝑘+1 reliability function of a linear 𝑟-within-consecutive-𝑘-out-of-𝑛: 𝐹 system, where p is the common reliability of its components. ≤𝑅𝐿 (𝑡) 𝐿 𝑟,𝑘,𝑛 (38) Then 𝑅𝑟,𝑘,𝑛(𝑝) satisfies the following inequalities: 𝑛 (i) ≤𝑃(𝑍𝑘) ∏ [1 −𝛾𝑖𝑃(𝑋𝑖)+𝑞𝑖𝛾𝑖𝑃(𝐵𝑖 )] , 𝑖=𝑘+1 𝐿 𝑅𝑟,𝑘,𝑛 (𝑝) ≥1 𝑎 𝑆1 −𝑎2𝑆2 +𝑎3𝑆3, (34) where 𝑞𝑖 is the unreliability of the 𝑖th component, while 𝛾𝑖 = 𝑃(𝐶𝑖)/𝑃(𝐵𝑖). where 2 (𝑛−𝑘+1) +𝑡−1 2.2.4. Signature Vector of 𝑟-within-Consecutive-𝑘-out-of-𝑛: 𝐹 𝑎1 = , (𝑛−𝑘+1)(𝑡+1) Systems. Let 𝑝 denote the common reliability of the compo- nents of an 𝑟-within-consecutive-𝑘-out-of-𝑛: 𝐹 system. The 2 (𝑛−𝑘+2𝑡−1) 𝑎2 = , following theorem offers a generating function approach of (𝑛−𝑘+1) 𝑡 (𝑡+1) the aforementioned system. (35) 6 𝑎 = , Theorem 17. Let (𝑠1(𝑛),2 𝑠 (𝑛),...,𝑠𝑛(𝑛)) and 𝑅2,𝑘,𝑛(𝑝) be 3 (𝑛−𝑘+ ) 𝑡 (𝑡+ ) 1 1 the signature and the reliability function of an 2-within- ((𝑛−𝑘− ) 𝑆 − 𝑆 ) consecutive-𝑘-out-of-𝑛: 𝐹 system, respectively. Then 𝑡=[2 1 2 3 3 ], (𝑛−𝑘) 𝑆 − 𝑆 𝑛 1 2 2 (a) the double generating function of 𝑖 ( 𝑖 ) 𝑠𝑖(𝑛) is given by
∞ 𝑛 𝑛 (ii) ∑ ∑ 𝑖( )𝑠 (𝑛) 𝑡𝑖𝑥𝑛 𝑖 𝑖 𝐿 𝑛=1 𝑖=1 𝑅𝑟,𝑘,𝑛 (𝑝) ≤ min (1,𝑆1 −𝑏2𝑆2 +𝑏3𝑆3), (36) 2 2 𝑘 𝑘+1 2𝑘 (39) 𝑡 𝑥[2𝑥−2𝑥 + ((𝑘−1) 𝑡−2) 𝑥 + (2 −𝑘𝑡) 𝑥 +𝑡𝑥 ] = 2 2 where (𝑥−1) (1 −𝑥−𝑡𝑥𝑘) 2 (2𝑡−1) 𝑏2 = , 𝑡 (𝑡+1) (Triantafyllou and Koutras [22]), 𝑅 (𝑝) 6 (b) the generating function of 2,𝑘,𝑛 is given as follows: 𝑏3 = , (37) 𝑡 (𝑡+1) 𝑘−2 𝑗 1 +𝑞𝑧∑𝑗=0 (𝑝𝑧) 𝑅 (𝑧; 𝑝)= (40) 3𝑆3 𝑘−1 𝑘 𝑡=[ ]+2. 1 −𝑝𝑧−𝑞𝑝 𝑧 𝑆2 (Koutras [13]). For the next Theorem, the following definitions are necessary: The next theorem offers expressions for the evaluation of (i) 𝑍𝑖 denotes the event that the linear 𝑟-within-consecu- the signature vector of an 2-within-consecutive-𝑘-out-of-𝑛: 𝐹 tive-𝑘-out-of-𝑖: 𝐹 system consisting of the components system. 1, 2,...,𝑖is good (𝑖=𝑘,𝑘+1,...,𝑛).
(ii) 𝑋𝑖 denotes the event that the 𝑖th component fails and Theorem 18 (Triantafyllou and Koutras [22]). Let (𝑠1(𝑛), there are at least 𝑟−1 failures among components 𝑖− 𝑠2(𝑛),...,𝑠𝑛(𝑛)) be the signature vector of an 2-within- 𝑘+1,𝑖−𝑘+2,...,𝑖−1,(𝑖=𝑘,𝑘+1,...,𝑛). consecutive-𝑘-out-of-𝑛: 𝐹 system. Then the following ensues: Journal of Quality and Reliability Engineering 7
(i) The quantities 𝑞𝑖(𝑛) = (𝑛)𝑖𝑠𝑖(𝑛), 𝑖 = 1, 2,...,𝑛,where where 𝑝𝑎(𝑞𝑎) is the reliability (unreliability) of the 𝑎th compo- (𝑛)𝑖 = 𝑛(𝑛 − 1)⋅⋅⋅(𝑛− 𝑖 + 1), satisfy the recurrence nent. relation: In order to launch the aforementioned recursive scheme, 𝑞 (𝑛+ ) = 𝑞 (𝑛) + 𝑞 (𝑛− ) − 𝑞 (𝑛− ) 𝑖+1 1 4 𝑖+1 4 𝑖+1 2 6 𝑖+1 1 the following set of initial conditions is necessary: −𝑞 (𝑛− ) +𝑖( (𝑞 (𝑛−𝑘+ ))−𝑞 (𝑛−𝑘− ) 𝑖+1 3 2 𝑖 1 𝑖 2 𝑄 (𝑖, 𝑗, 𝑘)= 0, if 𝑖 ⋅(𝑞𝑖 (𝑛−2𝑘+1) −𝑞𝑖 (𝑛−2𝑘) −𝑞𝑖 (𝑛−2𝑘−1))) 𝑝0 = 1. for 𝑖=0, 1,...,𝑛−1 and 𝑛≥2𝑘+2. The complexity for calculating 𝑄(𝑛, 𝑓,𝑘) using the above 𝑂(𝑛𝑓) (ii) The quantities 𝑠𝑖(𝑛) can be expressed as recurrence relation is equal to . The next theorem provides an alternative recursive scheme for the evaluation 𝑛−(𝑘−1)(𝑖−2) 𝑛−(𝑘−1)(𝑖−1) (𝑛, 𝑓,𝑘) (𝑛−𝑖+1) ( )−𝑖( ) of the reliability function of the system. 𝑠 (𝑛) = 𝑖−1 𝑖 . (42) 𝑖 𝑖 ( 𝑛 ) 𝑖 Theorem 20 (Triantafyllou and Koutras [26]). The reliability function 𝑅𝑛 of an (𝑛, 𝑓, 2) system with i.i.d. components satisfies It is worth mentioning that well-performed simulations study the following recurrence relation: of an 𝑟-within-consecutive-𝑘-out-of-𝑛: 𝐹 system has been developed by Eryilmaz et al. [20]. Moreover, Kan et al. [23] 𝑓 𝑚−1 𝑓−1 𝑚(1 −𝑝)+(𝑓+1)𝑝 studied the circular case of the aforementioned structure and 𝑅 = ∑ (−𝑝) ( )𝑓 𝑛 𝑚− 𝑚(𝑓−𝑚+ ) offered a new approximation for its reliability. 𝑚=1 1 1 (45) 𝑓 ⋅𝑅𝑛−𝑚 +(−𝑝) 𝑅𝑛−𝑓−1, 2.3. (𝑛, 𝑓,𝑘) Systems. An (𝑛, 𝑓,𝑘) system involves two com- mon failure criteria. More specifically, it consists of 𝑛 com- where 𝑝 denotes the common reliability of its components. ponents (ordered in a line or a circle) and fails if and only if there exist at least 𝑓 failed components or at least 𝑘 In order to launch the recursive scheme established above, consecutive failed components. It is worth of mentioning an adequate number of initial conditions is necessary. These that the configuration of an (𝑛, 𝑓,𝑘) system was first intro- conditions are given as follows duced by Tung [24] as an application to a complex infrared 𝑅 = , 𝑛< (𝑓, ), detecting system and since then it has attracted considerable 𝑛 1 if min 2 research attention. In the sequel, we present the main results 𝑅𝑛 = 0, if 𝑓=0or𝑛=0, (46) for (𝑛, 𝑓,𝑘) systems appearing in the literature, such as recurrence relations and closed formulas for the evaluation 𝑅𝑛 =𝑅𝑓,𝑛:𝐹 , if 𝑓≤2, of the reliability function and signature vector. It is worth mentioning that the aforementioned system generalizes the where 𝑅𝑓,𝑛:𝐹 is the reliability of a 𝑓-out-of: 𝑛: 𝐹 system; that 𝑘 𝑛 𝐹 𝑓>𝑘 well-known consecutive- -out-of- : system (for ), 𝑅 =∑𝑓−1 ( 𝑛 )𝑝𝑛−𝑗( −𝑝)𝑗 while for 𝑓≤𝑘the (𝑛, 𝑓,𝑘) system reduces to an ordinary is, 𝑓,𝑛:𝐹 𝑗=0 𝑗 1 . 𝑓-out-of-𝑛: 𝐹 system. 2.3.2. Exact Formulas for the Reliability of (𝑛, 𝑓,𝑘) Systems. 2.3.1. Recursive Schemes for the Reliability of (𝑛, 𝑓,𝑘) Systems. Letusfirstconsideran(𝑛, 𝑓,𝑘) system, where 𝑓>𝑘;werecall The following theorems provide recurrence for the calcula- that for the case 𝑓≤𝑘the (𝑛, 𝑓,𝑘) system coincides with the tion of reliability of an (𝑛, 𝑓,𝑘) system. well-known 𝑓-out-of-𝑛: 𝐹 system and its reliability properties have been extensively studied in the past. Chang et al. [27] Theorem 19 (Zuo et al. [25]). Let 𝐴(𝑖, 𝑗, 𝑘) be the event established a Markov chain representation of the (𝑛, 𝑓,𝑘) that the (𝑖,𝑗,𝑘) subsystem fails (the subsystem consists of system, which leads to the computation of the reliability components 1, 2,...,𝑖, 𝑖≥𝑗≥0, 𝑖≥𝑘), while 𝑄(𝑖, 𝑗, 𝑘) function of the aforementioned structure. More specifically, denotes the corresponding failure probability 𝑃(𝐴(𝑖, 𝑗, 𝑘)).Then for the (𝑛, 𝑓,𝑘) system with 𝑓>𝑘, let us define the state space the unreliability function of the (𝑖,𝑗,𝑘) system satisfies the for process {𝑌(𝑡), 𝑡= 0, 1,...}as following recurrence relation: 𝑆 = {(𝑖, 𝑗): 0 ≤𝑖≤𝑘−1, 𝑖≤𝑗≤𝑓−1}∪{𝑠𝑁}, (47) 𝑄 (𝑖, 𝑗, 𝑘) (𝑖, 𝑗) =𝑝𝑄(𝑖− ,𝑗,𝑘)+𝑞𝑄(𝑖− ,𝑗− ,𝑘) where indicates a working state in which the system con- 𝑖 1 𝑖 1 1 , ,...,𝑡 𝑗 (43) sisting of components 1 2 has failed components, its 𝑖 last 𝑖−1 components have failed, and the (𝑡−𝑖)th component is 𝑠 ( , ,...,𝑡) +[1 −𝑄(𝑖−𝑘−1,𝑗−𝑘,𝑘)]𝑝𝑖−𝑘 ∏ 𝑞𝑙, working. State 𝑁 indicates the system 1 2 fails. Then 𝑙=𝑖−𝑘+1 {𝑌(𝑡)} is a Markov chain with transition matrix of the form 8 Journal of Quality and Reliability Engineering (1) (1) (1) 𝐴𝑓×𝑓 𝐵𝑓×(𝑓−1) 00𝐶𝑓×1 𝐴(2) 0𝐵(2) 0𝐶(2) ( (𝑓−1)×𝑓 (𝑓−1)×(𝑓−2) (𝑓−1)×1 ) ( ) Λ (𝑛) = ( . ) , 𝑡 ( . ) (48) (𝑘) (𝑘) (𝑘) 𝐴(𝑓−𝑘+1)×𝑓 00𝐵(𝑓−𝑘+1)×(𝑓−𝑘) 𝐶(𝑓−𝑘+1)×1 00 0 0 1 ( )𝑁×𝑁 where Theorem 21 (Eryilmaz [28]). The reliability function 𝑅𝑛 of an (𝑛, 𝑓,𝑘) system with exchangeable components can be ⏟⏟⏟⏟⏟⏟⏟⏟⏟0 ⋅⋅⋅0 𝑝 expressed as follows: 𝑖−1 (𝑖) 𝑓−1 𝑛−𝑖 𝐴(𝑓−𝑖+1)×𝑓 =( d ), 𝑛−𝑖 𝑅 = ∑ ∑ 𝑁 (𝑖, 𝑘, 𝑛)(− )𝑗 ( )𝜃(0) , 𝑝 𝑛 1 𝑖+𝑗 (53) 𝑖=0 𝑗=0 𝑗 𝑝 where d 𝑁 (𝑖, 𝑘, 𝑛) 𝐵(𝑖) =( ), (49) (𝑓−𝑖+1)×(𝑓−𝑖) 𝑝 min([𝑖/𝑘],𝑛−𝑖+1) 𝑛−𝑖+ 𝑛−𝑗𝑘 𝑗 1 (54) 0 = ∑ (−1) ( )( ). 𝑗=0 𝑗 𝑛−𝑖 𝑖=1, 2,...,𝑘−1, In the sequel, we present results for an (𝑛, 𝑓,𝑘) system with (𝑖) Markov dependent components. Let 𝑋1,𝑋2,...,𝑋𝑛 denote the 𝐶 =(0 ⋅⋅⋅0 𝑞) ,𝑖=1, 2,...,𝑘−1 (𝑓−𝑖+1)×1 states of the Markov dependent components with transition probabilities and 𝑁=(2𝑓−𝑘+1)𝑘/2 + 1. 𝑃(𝑋 = |𝑋 = )=𝑝 , As proved in Koutras [13], the reliability 𝑅𝑛 of a structure 𝑖 0 𝑖−1 0 00 canbeexpressedas 𝑃(𝑋𝑖 = 1 |𝑋𝑖−1 = 0)=𝑝01, (55) 𝑅 = 𝜋 Λ𝑛u = − 𝜋 Λ𝑛e , 𝑛 0 1 0 𝑁 (50) 𝑃(𝑋𝑖 = 0 |𝑋𝑖−1 = 1)=𝑝10, 𝑃(𝑋 = |𝑋 = )=𝑝 , where Λ is the transition probability matrix associated to the 𝑖 1 𝑖−1 1 11 structure and with 1 ≤𝑖≤𝑛and initial probabilities 𝑝1 = 𝑃(𝑋1 = 1), 𝑝 = 𝑃(𝑋 = ) 𝑔(𝑛, 𝑟, 𝑙) 0 1 0 .Ifwedefine as 𝜋0 = (1, 0, 0,...,0) , 1 1 𝑛−𝑙−1 𝑙−𝑟 𝑔 (𝑛, 𝑟, 𝑙) = ∑ ∑ ( )(𝑝 ) u = (1, 1,...,1, 0) , (51) 𝑟−𝑡−𝑠 00 𝑡=0 𝑠=0 (56) e𝑁 = (0, 0,...,0, 1) . 𝑟−𝑠 𝑟−𝑡 𝑛−𝑙−𝑟+𝑡+𝑠−1 ⋅𝑝01 𝑝10 (𝑝11) 𝑝1−𝑡, Therefore, applying the above expression obtained for the the following theorem provides an expression for the evaluation transition matrix of an (𝑛, 𝑓,𝑘) system,onemayeasilycalcu- of reliability of (𝑛, 𝑓,𝑘) systems. late the reliability function of the aforementioned structure. Theorem 22 𝑅 Let us next consider the following probabilities: (Demir [29]). The reliability function 𝑛 of an (𝑛, 𝑓,𝑘) system with Markov dependent components can be (0) expressed as follows: 𝜃𝑚 =𝑃(𝑋1 =⋅⋅⋅=𝑋𝑚 = 0), 𝑓−1 min(𝑚,𝑛−𝑚) (1) 𝑛−1 𝜃𝑚 =𝑃(𝑋1 =⋅⋅⋅=𝑋𝑚 = 1), (52) 𝑅𝑛 =𝑝1𝑝11 + ∑ ∑ 𝑁(𝑟,𝑓,𝑚)𝑔(𝑛, 𝑟,) 𝑚 . (57) 𝑚=1 𝑟=1 for 𝑚≥1, 2.3.3. Signature Vector of (𝑛, 𝑓,𝑘) Systems. Let p denote the where 𝑋𝑖 is the state of 𝑖th component (𝑋𝑖 ∈{0, 1}),for𝑖= common reliability of the components of an (𝑛, 𝑓,𝑘) system. 1, 2,...,𝑛. The following theorem offers a closed expression The following theorem offers a generating function approach for the evaluation of reliability of an (𝑛, 𝑓,𝑘) system. of the aforementioned system. Journal of Quality and Reliability Engineering 9 𝑛−𝑗− Theorem 23 (Triantafyllou and Koutras [26]). Let (𝑠1(𝑛), 6 ⋅( )𝑠𝑖−2 (𝑛 − 𝑗 − 6)−2 (𝑖−1) 𝑠2(𝑛),...,𝑠𝑛(𝑛)) be the signature of an (𝑛, 𝑓, 2) system, respec- 𝑖− 𝑛 2 tively. Then the double generating function of 𝑖 ( 𝑖 ) 𝑠𝑖(𝑛) is given by 𝑛−𝑗−6 ⋅( )𝑠𝑖−1 (𝑛 − 𝑗 − 6)−(𝑖−2) ∞ 𝑛 𝑛 𝑖−1 𝑖 𝑛 ∑ ∑ 𝑖( )𝑠𝑖 (𝑛) 𝑡 𝑥 𝑛=1 𝑖=1 𝑖 𝑛−𝑗−7 (58) ⋅( )𝑠𝑖−2 (𝑛 − 𝑗 − 7)−(𝑖−3) 𝑑 𝑖−2 = , 2 2 2 2 𝑓+1 𝑥 (1 −𝑥−𝑡𝑥) (1 −𝑥−𝑡𝑥 ) (1 −𝑥) 𝑛−𝑗−7 ⋅( )𝑠𝑖−3 (𝑛 − 𝑗 − 7))= 0. 𝑖− where 3 (60) 4 𝑓+1 6 3 2 𝑓 𝑑=−𝑡 (1 −𝑥) 𝑥 −𝑓(1 −𝑥) (− (𝑡𝑥 ) + (1 Let 𝑇 bethelifetimeofasystemwhosecomponents’lifetimes 𝑓 𝑓 are 𝑇1,𝑇2,...,𝑇𝑛 and 𝑇1:𝑛 <𝑇2:𝑛 <⋅⋅⋅<𝑇𝑛:𝑛 the order −𝑥) (𝑡𝑥2) )+𝑡3𝑥4 (− (𝑡𝑥2) ( 𝑥+𝑓− )+( 2 2 1 statistics associated with them. It is easy to observe that the lifetime of an (𝑛, 𝑓,𝑘) system can be represented as a function 𝑓 2 2 2 𝑓 𝑘 𝑛 𝐹 𝑘 𝑛 𝐹 −𝑥) 𝑥(𝑥 − 1)) + 𝑡 (𝑥−1) ((1 −𝑥) (𝑡𝑥 ) of the lifetimes of consecutive -out-of- : and -out-of- : systems. Generally speaking, the lifetimes of systems involving 2 2 𝑓 2 two common failure criteria can be expressed as either ⋅((𝑓−1)𝑥 +𝑥+𝑓−2)−(𝑡𝑥 ) (2𝑓𝑥 + 2𝑥 (59) 𝑇∗ = min (𝑆,𝑎:𝑛 𝑇 ), 2 2 2 𝑓 +𝑓−2)) + (𝑡𝑥) (− (1 −𝑥) (𝑡𝑥 ) (𝑥+𝑓−3) ∗ (61) or 𝑇 = max (𝑆,𝑎:𝑛 𝑇 ), 𝑓+1 2 + (𝑥−1) (−2 (1 −𝑥) 𝑥 where 𝑆 denotes the lifetime associated with the system different from 𝑎-out-of-𝑛 structurebuthavingthesamecomponents’life- 2 𝑓 2 times 𝑇1,𝑇2,...,𝑇𝑛. The next theorem presents an alternative −(𝑡𝑥 ) (4 (𝑥−1) +𝑓(𝑥 + 2)))) . way of computing the signature of an (𝑛, 𝑓,𝑘) system. Theorem 24 offersrecursiverelationsfortheevaluationofthe Theorem 25 (Eryilmaz and Zuo [31]). Let (𝑠1,𝑠2,...,𝑠𝑛) be signature vector of an (𝑛, 𝑓, 2) system. the signature of the system with lifetime 𝑆. Then the signatures ∗ of the systems with lifetimes 𝑇∗ and 𝑇 are given, respectively, Theorem 24 𝑠 (𝑛) (Triantafyllou [30]). The coordinates 𝑖 of the as signature vector of an (𝑛, 𝑓, 2) system satisfy the following recurrence relation: 𝑛 p∗ =(𝑠1,𝑠2,...,𝑠𝑎−1, ∑ 𝑠𝑖, 0,...,0), 𝑓+1 𝑓+1 𝑛−𝑗−2 𝑖=𝑎 ∑ ( ) (− )𝑗 (𝑖 ( )𝑠 (𝑛 − 𝑗 − ) (62) 1 𝑖 2 𝑎 𝑗=0 𝑗 𝑖 ∗ p =(0,...,0, ∑ 𝑠𝑖,𝑠𝑎+1,...,𝑠𝑛). 𝑖=𝑎 𝑛−𝑗−3 − (𝑖−1) ( )𝑠𝑖−1 (𝑛 − 𝑗 − 3) 𝑖−1 It is noteworthy that the dual system of an (𝑛, 𝑓,𝑘) structure has been studied by Cui et al. [32], while a generalization 𝑛−𝑗−3 of these systems, named (𝑛, 𝑓,𝑘) systems, with weighted com- − 3𝑖( )𝑠𝑖 (𝑛 − 𝑗 − 3) 𝑖 ponents has been introduced by Eryilmaz and Aksoy [33]. Gera [34] studied reliability systems with two working criteria 𝑛−𝑗−4 and presented some results associated with the well-known + 3𝑖( )𝑠𝑖 (𝑛 − 𝑗 − 4)+2 (𝑖−2) 𝑖 qualification tests, while Kamalja [35] derived expressions for the Birnbaum importance measures for both structures. 𝑛−𝑗−5 ⋅( )𝑠𝑖−2 (𝑛 − 𝑗 − 5)𝑠𝑖−2 (𝑛 − 𝑗 − 5) 𝑖− 2.4. Reliability Systems with Weighted Components. The idea 2 thatallcomponentsinareliabilitystructurearenotcreated 𝑛−𝑗−5 equal is seemingly an obvious concept. In other words, it is + 3 (𝑖−1) ( )𝑠𝑖−1 (𝑛 − 𝑗 − 5) not quaint to assume that different components may have 𝑖−1 different failure probabilities. Generally speaking, a reliability 𝑛−𝑗− system which consists of weighted components, for example, 5 each component carries its own positive weight, fails if and −𝑖( )𝑠𝑖 (𝑛 − 𝑗 − 5)−(𝑖−2) 𝑖 only if the total weight of the failed components exceeds 10 Journal of Quality and Reliability Engineering aspecificbenchmark.Inthesequel,theliteratureonweighted An additional generalization of the well-known weighted reliability structures is briefly reviewed. Consider a system 𝑘-out-of-𝑛 system has been introduced by Eryilmaz [38]. with 𝑛 components and suppose that the 𝑖th component is More specifically, Eryilmaz38 [ ] assumed that the system associated with a weight 𝑤𝑖 > 0, 𝑖=1, 2,...,𝑛.Thenthe has a performance level above 𝑐 if there are at least 𝑘 system is still working if and only if the sum of weights of the working components and the sum of the weights of all failed components is less than (or equal to) a certain threshold working components is above 𝑐.Amongothers,Eryilmaz 𝑇> 0. [38] deduced recursive relations for the calculation of the system state probabilities, while a detailed simulation study 2.4.1. Weighted 𝑘-out-of-𝑛 Systems. Aweighted𝑘-out-of-𝑛: has been taken into play in order to observe the time spent 𝐺(𝐹) 𝑛 system consists of components, each with its own by the system in state 𝑐 or above. Finally, Li and Zuo [39] 𝑤 > 𝑤 positive weight 𝑖 0 (total system weight equal to ), such studied the multistate weighted 𝑘-out-of-𝑛 systems, where that the system works (fails) if and only if the total weight of each component may be in more than 2 states and therefore the working (failed) components is at least 𝑘.Itisnoteworthy 𝑘 𝑛 𝐺 its contribution to the system’s weight can be differentiated that the reliability of a weighted -out-of- : system is the analogously. complement of the unreliability of a weighted (𝑛 − 𝑘 + 1)- out-of-𝑛: 𝐹 system. It goes without saying that the 𝑘-out-of-𝑛: 𝐺(𝐹) system is a special case of the corresponding weighted 𝑘- 2.4.2. Consecutive Weighted 𝑘-out-of-𝑛 Systems. Aweighted out-of-𝑛: 𝐺(𝐹) system wherein the weight of each component consecutive 𝑘-out-of-𝑛: 𝐹 system consists of 𝑛 components, equals 1. The next theorem offers an efficient algorithm for each with its own positive weight 𝑤𝑖 > 0 (total system weight theevaluationofthereliabilityoftheweighted𝑘-out-of-𝑛: 𝐺 equal to 𝑤), such that the system fails if and only if the total system. weight of the failed components is at least 𝑘.Itgoeswithout saying that the consecutive 𝑘-out-of-𝑛: 𝐹 system is a special Theorem 26 𝑅(𝑖, 𝑗) (Wu and Chen [36]). Let be the reliability case of the corresponding weighted consecutive 𝑘-out-of-𝑛: of the weighted 𝑗-out-of-𝑖: 𝐺 system, while 𝑤𝑖 > 0 is the 𝐹 𝑖 𝑝 (𝑞 ) system wherein the weight of each component equals 1. weight of the th component. Then if we denote by 𝑖 𝑖 the Efficient algorithms for the evaluation of the reliability of reliability (unreliability) of the 𝑖th component, the reliability of the linear weighted consecutive 𝑘-out-of-𝑛: 𝐹 system have the structure satisfies the following recurrence relation: appeared in the literature (for more details see Kuo and Zuo 𝑅 (𝑖, 𝑗) [1]), while Samaniego and Shaked [40] extended the idea of weighted components, by giving to the components weights 𝑝 𝑅(𝑖− ,𝑗−𝑤)+𝑞𝑅(𝑖− ,𝑗), 𝑗−𝑤 ≥ , { 𝑖 1 𝑖 𝑖 1 if 𝑖 0 (63) thatcantakeanypositivevalue(notnecessaryinteger- = { 𝑝 +𝑞𝑅(𝑖− ,𝑗), valued). { 𝑖 𝑖 1 otherwise. The following theorem provides a similar result concerning the 2.5. ((𝑛1,𝑛2,...,𝑛𝑁), 𝑓,𝑘) Systems. Cui and Xie [41] dual structure. introduced a generalized 𝑘-out-of-𝑛 system, denoted ((𝑛 ,𝑛 ,...,𝑛 ), 𝑓,𝑘) 𝑁 Theorem 27 (Chen and Yang [37]). Let 𝑅(𝑖, 𝑗) be the relia- by 1 2 𝑁 . Such a system consists of 𝑖 bility of the weighted 𝑗-out-of-𝑖: 𝐹 system, while 𝑤𝑖 > 0 is the modules ordered in a line or a circle, while the th module 𝑛 ≥ weight of the 𝑖th component. Then if we denote by 𝑝𝑖(𝑞𝑖) the is composed of 𝑖 1componentsinparallel.Inother ((𝑛 ,𝑛 ,...,𝑛 ), 𝑓,𝑘) reliability (unreliability) of the 𝑖th component, the reliability of words, the 1 2 𝑁 system fails if and only the structure satisfies the following recurrence relation: if there are at least 𝑓 failed components or at least 𝑘 𝑅 (𝑖, 𝑗) consecutivefailedmodules.Itgoeswithoutsayingthat,for 𝑛1 =𝑛2 = ⋅⋅⋅ = 𝑛𝑁 = 1, an ((𝑛1,𝑛2,...,𝑛𝑁), 𝑓,𝑘) system 𝑁 {0, if 𝑖≤0,𝑗≥0, reduces to a simple (𝑁, 𝑓,𝑘) while for 𝑓=∑𝑖=1 𝑛𝑖 coincides { (64) = , 𝑖> ,𝑗= , with the well-known consecutive 𝑘-out-of-𝑁: 𝐹 system {1 if 0 0 { and for 𝑁=1withtheordinary𝑓-out-of-𝑛: 𝐹 structure. ( −𝑝)𝑅(𝑖−𝑤 ,𝑗− )+𝑝𝑅 (𝑖, 𝑗− ), . { 1 𝑖 𝑗 1 𝑗 1 otherwise The following theorem offers recursive relations for the An extension of the aforementioned one-stage weighted 𝑘- evaluation of the reliability function of a linear and circular out-of-𝑛 model has been proposed by Chen and Yang [37]. ((𝑛1,𝑛2,...,𝑛𝑁), 𝑓,𝑘) system, respectively. More specifically, Chen and Yang [37]consideredatwo- Theorem 28 ((𝑛 ,𝑛 ,..., stage weighted 𝑘-out-of-𝑛 system, which consists of 𝑚 sub- (Cui and Xie [41]). (i) For a linear 1 2 𝑛 ), 𝑓,𝑘) 𝑅 ((𝑛 ,𝑛 ,..., systems. Each subsystem has a (one-stage) weighted 𝑘-out- 𝑁 system, the reliability function 𝐿 1 2 𝑛 ), 𝑓,𝑘) of-𝑛 structure, which is called the second level structure. The 𝑁 satisfies the following recurrence: interrelationship between the 𝑚 subsystems follows a certain coherent structure, which is called the first-level structure. 𝑛𝑁 Generally speaking, a two-stage weighted 𝑘-out-of-𝑛 system can 𝑅𝐿 ((𝑛1,𝑛2,...,𝑛𝑁),𝑓,𝑘)=∑ 𝐴 (𝑁,) 𝑠 be decomposed into two hierarchical levels: the first (higher) 𝑠=0 level can be of any coherent structure and the second (lower) level has a weighted 𝑘-out-of-𝑛 structure. ⋅𝑅𝐿 ((𝑛1,𝑛2,...,𝑛𝑁−1),𝑓−𝑠,𝑘) Journal of Quality and Reliability Engineering 11 𝑛 𝑛 −1 𝑁 𝑖 𝑁−𝑘 failed components or at least 𝑘 consecutivefailedcomponents −( ∏ ∏ ( −𝑝 )) ∑ 𝐴 (𝑁−𝑘,𝑠) 1 𝑖𝑗 among components 𝑖, 𝑖 + 1,...,𝑗−1,𝑗.Itisnoteworthythat, 𝑗=1 𝑠=0 𝑖=𝑁−𝑘+1 for the circular case, component index should have modulo 𝑁 𝑛 property, that is, components 𝑖 and 𝑛+𝑖indicate the same ⋅𝑅𝐿 ((𝑛1,𝑛2,...,𝑛𝑁−𝑘−1),𝑓− ∑ 𝑛𝑖 −𝑠,𝑘), one, while the following requirements should be satisfied: 𝑖=𝑁−𝑘+1 (i) 𝑗−𝑖+1 ≥𝑘,if𝑗>𝑖, (65) (ii) 𝑛+𝑗−𝑖+1 ≥𝑘,if𝑗<𝑖. where 𝐴(𝑗, 𝑠) is the probability that, in module 𝑗,thereare𝑠 𝑘 failed components. Otherwise, the consecutive failure criterion can be 𝑖= 𝑗=𝑛 (ii) For a circular ((𝑛1,𝑛2,...,𝑛𝑁), 𝑓,𝑘) system, the relia- removed. It goes without saying that when 1, ,or 𝑖−𝑗= (𝑛, 𝑓,𝑘(𝑖, 𝑗)) 𝐹 bility function 𝑅𝐶((𝑛1,𝑛2,...,𝑛𝑁), 𝑓,𝑘) satisfies the following 1, the : system becomes the well-known (𝑛, 𝑓,𝑘) recurrence: structure. Furthermore, Guo et al. [42]mentioned an additional justification of the aforementioned reliability 𝑅 ((𝑛 ,𝑛 ,...,𝑛 ),𝑓,𝑘) 𝐶 1 2 𝑁 systems. More precisely, they introduced the ⟨𝑛,𝑓,𝑘(𝑖,𝑗)⟩: 𝐹 𝑛 𝑠 −1 system, which consists of components ordered in a line 𝑁 𝑓 = ∑ 𝐴 (𝑁,) 𝑠 𝑅 ((𝑛 ,𝑛 ,...,𝑛 ),𝑓−𝑠,𝑘) or circle, and fails if and only if there exist at least failed 𝐿 1 2 𝑁−1 𝑘 𝑛=0 components and at least consecutive failed components among components 𝑖, 𝑖 + 1,...,𝑗 − 1,𝑗.Finally,Guoet 𝑛 𝑁 al. [42] considered the dual structures of the (𝑛, 𝑓,𝑘(𝑖,: 𝑗)) + ∏ ( −𝑝 )𝑅 ((𝑛 ,𝑛 ,...,𝑛 ),𝑓−𝑛 ,𝑘) 1 𝑁𝑖 𝐶 1 2 𝑁−1 𝑁 𝐹 and ⟨𝑛, 𝑓,𝑘(𝑖,: 𝑗)⟩ 𝐹 systems, taking into account the 𝑖=1 argumentation applied by Cui et al. [32]. 𝑛 −1 𝑘 𝑛𝑁−𝑘+𝑙−1−1 𝑗 The main result of the aforementioned paper is the (66) − ∑ ∑ ∑ 𝐴(𝑁−𝑘+𝑖−1,𝑠1) 𝐴 (𝑖,2 𝑠 ) employment of a two-stage Markov chain in order to give 𝑖=1 𝑠1=0 𝑠2=0 the system reliability in the form of product of matrices. Generally speaking, the finite Markov chain imbedding {𝑌(𝑡)} ×𝑅 ((𝑛 ,...,𝑛 ),𝑓−𝑠 −𝑠 technique is to embed a Markov chain definedonthe 𝐿 𝑖+1 𝑁−𝑘+𝑖−2 1 2 state space 𝑆={1, 2,...,𝑁} and the discrete index space 𝑇={1, 2,...,𝑛} into a given system, while the system fails 𝑡 ≤𝑡 ≤𝑛 𝑌(𝑡) =𝑁 𝑁+𝑖−1 𝑁+𝑖−1 𝑛𝑖 if there exists 0,1 0 ,suchthat ,forall 𝑡 ≤𝑡≤𝑛 − ∑ 𝑛𝑗,𝑘)( ∏ ∏ (1 −𝑝𝑖𝑗 )) . 0 . 𝑗=𝑁−𝑘+𝑖 𝑙=𝑁−𝑘+𝑖 𝑗=1 The unique justification is to divide the discrete index space and the nonabsorbing state space into two disjoined It is worth mentioning that the ((𝑛1,𝑛2,...,𝑛𝑁), 𝑓,𝑘) system parts. For example, if we consider the (𝑛, 𝑓,𝑘(𝑖,: 𝑗)) 𝐹 can be obtained by adding more components in parallel (⟨𝑛, 𝑓,𝑘(𝑖,: 𝑗)⟩ 𝐹) system with component reliabilities 𝑝𝑡 (𝑞𝑡 = (𝑛, 𝑓,𝑘) to the basic components of an structure. Since the 1 −𝑝𝑡), 𝑡=1, 2,...,𝑛,byrearrangingtheircomponents aforementioned system involves multiple failure criteria, Cui without changing the system reliability, we can transform it and Xie [41] studied the Birnbaum importance with respect to into the linear (𝑛, 𝑓,𝑘(𝑛 − 𝑗 :+ 𝑖,𝑛)) 𝐹 (⟨𝑛,𝑓,𝑘(𝑛−𝑗+𝑖,𝑛)⟩: 𝑘 the th failure criterion as an importance measure computed 𝐹) system with component reliabilities 𝑢𝑡 (𝜐𝑡 = 1 −𝑢𝑡), where only under this failure criterion. Firstly, for a reliability system 𝑝𝑡, 𝑞𝑡 and 𝑢𝑡, 𝜐𝑡 satisfy the following relations: 𝑆 with 𝑚≥1 failure criteria, they assumed that, for any one of 𝑗(𝑗≤𝑚), the Birnbaum importance is different, while, for any (𝑢𝑡,𝜐𝑡) one of 𝑚−𝑗failure criteria, the Birnbaum importance is the (𝑝 ,𝑞), ≤𝑡≤𝑖− , same. Moreover, Cui and Xie [41]denotedby𝐴1 ={𝑆works { 𝑡 𝑡 if 1 1 { (67) under failure criteria 1, 2,...,𝑗} and 𝐴2 ={𝑆works under = {(𝑝𝑡−𝑖+𝑗+1,𝑞𝑡−𝑖+𝑗+1), if 𝑖≤𝑡≤𝑛−𝑗+𝑖−1, failure criteria 𝑗+1,...,𝑚}the specific disjoint events and they { 𝑆 { proved that the Birnbaum importance of the reliability system {(𝑝𝑡−𝑛+𝑗,𝑞𝑡−𝑛+𝑗), if 𝑛−𝑗+𝑖≤𝑡≤𝑛. depends upon these 𝑗 failure criteria. Finally, note that the well- known series and parallel reliability systems are included in the The rearrangement is as follows by a transform: more general ((𝑛1,𝑛2,...,𝑛𝑁), 𝑓,𝑘) systems’ family as special cases (for 𝑓≥max{𝑛𝑖} and 𝑘=1 and {𝑁=𝑘𝑎𝑛𝑑𝑓= Old-component order (1, 2,...,𝑖−1,𝑖,𝑖 𝑁 ∑𝑖=1 𝑛𝑖},resp.). + 1,...,𝑗,𝑗+1,𝑗+2,...,𝑛), (𝑛, 𝑓,𝑘(𝑖, 𝑗)) ⟨𝑛, 𝑓,𝑘(𝑖, 𝑗)⟩ (68) 2.6. and Systems. Guo et al. [42] New-component order (1, 2,...,𝑖−1,𝑗+1,𝑗 generalized the idea of the (𝑛, 𝑓,𝑘) system, studied by Chang et al. [27]andZuoetal.[25]. More specifically, they + 2,...,𝑛,𝑖,𝑖+1,...,𝑗), introduced a reliability structure, named the (𝑛, 𝑓,𝑘(𝑖,: 𝑗)) 𝐹 system, which consists of 𝑛 components ordered in a where the first line (old-component order) denotes the line or circle and fails if and only if there exist at least 𝑓 components order before rearrangement, the second line 12 Journal of Quality and Reliability Engineering (New-component order) denotes the components order after × 𝐹 ((𝑚1 −𝑠 −𝑠2 −⋅⋅⋅−𝑠𝑗)𝑘,𝑟−𝑠1 −⋅⋅⋅−𝑠𝑗 rearrangement, that is, we move to the end of the line that section to which the consecutive criterion applies so that the +𝑡). new system consists of components 1, 2,,...,𝑖−1,𝑗+1,𝑗+ (70) 2,...,𝑛,𝑖,𝑖+1,...,𝑗inanorderedline.Thus,thestudyofthe aforementioned reliability structures is covered by the linear It is clear from the above result that the unreliability 𝐹(𝑛, 𝑟) is case (𝑛,𝑓,𝑘(𝑛−𝑗+𝑖,𝑛)): 𝐹 (⟨𝑛, 𝑓,𝑘(𝑖,: 𝑗)⟩ 𝐹)systemwith independent of the order 𝑞1,𝑞2,...,𝑞𝑘 when 𝑛 is a multiple of 𝑢 component reliabilities 𝑡. 𝑘.For𝑛=𝑚𝑘+1,itfollowsthat ItisworthnotingthatGuoetal.[42] presented a detailed study for the evaluation of the reliability function of the 𝐹 (𝑚𝑘 + 1,𝑟) =𝐹(𝑚𝑘,) 𝑟 (𝑛, 𝑓,𝑘(𝑖,: 𝑗)) 𝐹 (⟨𝑛, 𝑓,𝑘(𝑖,: 𝑗)⟩ 𝐹)systems,byemployinga two-stageMarkovchainprocedureandillustratingnumerical 1 (71) +𝑝 ∑ ∑ 𝑄𝑠𝐹 ((𝑚−𝑠) 𝑘,𝑟−𝑠+𝑡) examples. 1 𝑡=0 𝑠 2.7. Consecutive 𝑘-out-of-𝑛: 𝐹 Systems with Cycle 𝑘. An and therefore we deduce that the quantity 𝐹(𝑛, 𝑟) = 𝐹(𝑚𝑘+1,𝑟) 𝑟 consecutive 𝑘-out-of-𝑛: 𝐹 system consists of 𝑛 linearly is maximized by maximizing 𝑝1 or, generally speaking, the arranged components and the system fails if and only if at 𝐹(𝑛, 𝑟) is maximized (minimized) when 𝑞1 ≤𝑞2 ≤⋅⋅⋅≤𝑞𝑘 least 𝑟 nonoverlapping runs of 𝑘 components fail. Boland and (𝑞1 ≥𝑞2 ≥ ⋅⋅⋅ ≥ 𝑞𝑘). It is worth mentioning that Boland Papastavridis [43] studied the case where there are 𝑘 distinct and Papastavridis [43] provided additionally interesting appli- components with failure probabilities 𝑞𝑖,𝑖 = 1, 2,...,𝑘 and cations of the 𝑟 consecutive 𝑘-out-of-𝑛: 𝐹 system with cycle where the failure probability of the 𝑗th component is 𝑞𝑖 (𝑗 = 𝑘 for the arrangement of sport competitions and inspections 𝑚𝑘+𝑖 (1 ≤𝑖≤𝑘)). In other words, they focused on the special procedures in quality control. case of an 𝑟 consecutive 𝑘-out-of-𝑛: 𝐹 system where there is a “cyclical” pattern in failure probabilities of the components. 2.8. 𝑚-Consecutive 𝑘-out-of-𝑛: 𝐹 Systems with Overlapping More precisely, Boland and Papastavridis [43] were interested Runs. Agarwal and Mohan [44]introducedandstudied 𝑞 ,𝑞 ,...,𝑞 in how the order of 1 2 𝑘 affects the failure probability the 𝑚-consecutive-𝑘-out-of-𝑛: 𝐹 system with overlapping 𝑟 𝑘 𝑛 𝐹 of this new structure, named consecutive -out-of- : runs. This reliability structure consists of 𝑛 linearly ordered 𝑘 system with cycle . components and fails if and only if there exist at least 𝑚 𝐹(𝑛, 𝑟) Let us next denote by the failure probability of overlapping sequences of 𝑘 consecutive failed components 𝑟 𝑘 𝑛 𝐹 𝑘 an consecutive -out-of- : system with cycle , while (𝑛≥𝑚+𝑘−1). It is straightforward that for 𝑚= 𝑝 (𝑞 ) 𝑖 𝑖 𝑖 is the reliability (unreliability) of the th component. 1 the aforementioned system coincides with an ordinary The following theorems offer recursive expressions for the consecutive-𝑘-out-of-𝑛: 𝐹 structure. Graphical Evaluation unreliability of the aforementioned structure. and Review Technique (GERT) analysis is used for reliability 𝑘−1 Theorem 29 𝑟 evaluation of the system for both i.i.d. and ( )-step (Boland and Papastavridis [43]). Consider an Markov dependent components, in a unified manner. More consecutive 𝑘-out-of-𝑛: 𝐹 system with cycle 𝑘,where𝑛=𝑚𝑘+𝑖, ≤𝑖≤𝑘 𝑚≥ 𝑟 𝑟=𝑚+ specifically, Agarwal and Mohan [44]provedthefollowing 1 and 0.Thenforanyinteger (except for 1 results: and 𝑖=𝑘), the unreliability function of the system satisfies the following recurrence: (i) For a 𝑚-consecutive-𝑘-out-of-𝑛: 𝐹 system with over- lapping runs composed by i.i.d components, the 𝐹 (𝑚𝑘+𝑖,𝑟) generating function of the waiting time for the occur- rence of the system failure is given as =𝐹(𝑚𝑘 + 𝑖− 1,𝑟) (69) 𝑚−1 𝑚− 1 𝑚 1 𝑡 𝑠 𝑊0,𝑚+𝑘−1 (0,𝑧) = ∑ ( ) + ∑ ∑ (−1) 𝑝𝑖𝑄 𝐹 ((𝑚−𝑠) 𝑘+𝑖−1,𝑟−𝑠+𝑡) , 𝑗=0 𝑗 𝑡=0 𝑠=1 (72) 𝑞𝑚+𝑘+𝑗(𝑘−1)−1𝑝𝑗𝑧𝑚+𝑘+𝑘𝑗−1 𝑘 ⋅ . where 𝑄=∏ 𝑞𝑖. 𝑗+1 𝑖=1 (1 −𝑝𝑧−𝑞𝑝𝑧2 −𝑞2𝑝𝑧3 −⋅⋅⋅−𝑞𝑘−1𝑝𝑧𝑘) Theorem 30 (Boland and Papastavridis [43]). Consider an 𝑟 consecutive 𝑘-out-of-𝑛: 𝐹 system with cycle 𝑘,where1 ≤𝑖≤𝑘 Hence the reliability function of the above structure and 𝑚≥0. Then the unreliability function of the system can be expressed as satisfies the following recurrence: 𝑛 𝑅 (𝑛) = − ∑ 𝜉 (𝑢) , 𝑚0 1 (73) 𝐹 (𝑚𝑘+𝑖,𝑟) =𝐹(𝑚𝑘,) 𝑟 𝑢=𝑚+𝑘−1 𝑗 𝑢 𝑡 𝑠1+𝑠2+⋅⋅⋅+𝑠𝑗 𝜉(𝑢) 𝑧 + ∑ ∑ ∑ (−1) ( )𝜎𝑗 (𝑖) 𝑄 where is the coefficient of in the power series 𝑡 𝑗>0 𝑡∈(−∞,∞) 𝑠1,𝑠2,...,𝑠𝑗 expansion of the generating function 𝑊0,𝑚+𝑘−1(0,𝑧). Journal of Quality and Reliability Engineering 13 (ii) For a 𝑚-consecutive-𝑘-out-of-𝑛: 𝐹 system with over- respectively (for more details, see Boland [46]andSamaniego lapping runs composed by (𝑘 − 1)-step Markov [3]),itisgoeswithoutsayingthatTheorem 31 leads immedi- dependent components, the generating function of atelytothedesiredresults. the waiting time for the occurrence of the system failure is given as 2.9. Combined 𝑚-Consecutive 𝑘-out-of-𝑛: 𝐹 and Consecutive- 𝑘 𝑛 𝐹 𝑚−1 𝑚− 𝑐-out-of- : Systems. Mohan et al. [47]proposedand 𝑘−1 1 𝑊 (0,𝑧) = ∑ ( ) studied a new model, named the combined 𝑚-consecutive- 0,𝑚+𝑘−1 𝑗 𝑗=0 𝑘-out-of-𝑛: 𝐹 and consecutive-𝑘𝑐-out-of-𝑛: 𝐹 systems. This (74) reliability structure consists of 𝑛 linearly ordered components 𝑘−1 𝑗+1 𝑚 𝑗 𝑗 (𝑞 𝑞 ⋅⋅⋅𝑞 𝑧 ) (𝑞 𝑧) 𝑝 𝑧 and fails if and only if there exist at least 𝑘𝑐 consecutive ⋅ 0 1 𝑘−2 𝑘−1 𝑘−1 . 𝑗+1 failed components or at least 𝑚 nonoverlapping runs of ( −𝑝 𝑧−𝑞 𝑝 𝑧2 −⋅⋅⋅−𝑞 𝑞 ⋅⋅⋅𝑞 𝑝 𝑧𝑘) 1 0 0 1 0 1 𝑘−2 𝑘−1 𝑘 consecutivefailedcomponents,where𝑘𝑐 <𝑚𝑘.By assuming that the components are i.i.d., Mohan et al. [47] Hence, the reliability function of the above structure can be employed the well-known GERT analysis to evaluate the expressed as reliability function and presented interesting applications 𝑛 of the aforementioned structure in several areas, such as 𝑅𝑘−1 (𝑛) = − ∑ 𝜉 (𝑢) , 𝑚0 1 (75) signal processing or bank automatic payment systems. More 𝑢=𝑚+𝑘−1 specifically, the failure probability of the aforementioned 𝑢 where 𝜉(𝑢) is the coefficient of 𝑧 in the power series systemisexplainedasatreestructure,whilethedirectionof 𝑘−1 the calculating procedure is from the root node to a leaf node expansion of the generating function 𝑊 (0,𝑧). 0,𝑚+𝑘−1 via intermediate ones. The main steps of the algorithm can be In addition, Eryilmaz [45] derived an explicit combina- briefly described as follows: torial expression for the number of its path sets including a specified number of working components. More specifically, (i) At step 1, the failure probability at root node 𝑥0 is Eryilmaz [45] proved the following result. determined as 𝑛 Theorem 31 𝑟 (𝑛) (Eryilmaz [45]). The number of path sets 𝑖 of 𝐹(𝑥 ) = ∑ 𝜉 (𝑢) , 0 0 (79) an 𝑚-consecutive-𝑘-out-of-𝑛: 𝐹 system with overlapping runs 𝑢=𝑘𝑐 including 𝑖 working components can be expressed as follows: 𝜉(𝑢) 𝑧𝑢 𝑟𝑖 (𝑛) =𝑁(𝑖+1,𝑘+1,𝑛+1) where is the coefficient of in the power series expansion of the generating function of the waiting 𝑚−1 min(𝑖+1,𝑥) 𝑖+1 𝑥−1 time for the occurrence of the system failure at root + ∑ ∑ ( )( ) (76) 𝑥 𝑎 𝑎− node 0. 𝑥=1 𝑎=1 1 𝑗 (ii) At step 2, the failure probability at node (𝑥0)𝑡 is ⋅𝑁(𝑖−𝑎+1,𝑘+1,𝑛−𝑥−𝑎𝑘+1) , deduced as follows: 𝑛−𝑧 ≤𝑖<𝑛 𝑟 (𝑛) = (𝑧 =𝑛− −[(𝑛−𝑚−𝑘+ )/𝑘]) 𝑛 where 𝜙 , 𝑛 1 𝜙 1 1 𝐹𝑗(𝑥 ) = ∑ 𝜉 (𝑢) , and 0 𝑡 (80) 𝑢=𝑘 +𝑘 ∑𝑖 𝑖 +𝑡 𝑁 (𝑎, 𝑏,) 𝑐 𝑐 𝑠=1 𝑠𝑗 𝜉(𝑢) 𝑧𝑢 min(𝑎,[(𝑐−𝑎)/(𝑏−1)]) 𝑎 𝑐−𝑗(𝑏− ) − where is the coefficient of in the power series 𝑗 1 1 (77) = ∑ (−1) ( )( ). expansion of the generating function of the waiting 𝑗=0 𝑗 𝑎−1 time for the occurrence of the system failure at root 𝑗 node (𝑥0)𝑡. Having at hand the above expression for the determination (iii) At step 3, the total failure probability of the system is of 𝑟𝑖(𝑛),onemayeasilycomputeimportantcharacteristics of the 𝑚-consecutive-𝑘-out-of-𝑛: 𝐹 system with overlapping calculated as runs,suchasitsreliabilityfunctionoritssignaturevector. 𝐹(𝑛,𝑘 ,𝑚𝑘)=𝐹(𝑥 ) + ∑ 𝐹𝑗(𝑥 ) . 𝑧 𝑐 0 0 𝑖 𝑡 (81) Indeed, let us first denote by 𝜙 the maximum number of failed all nodes generated components such that the system with structure function 𝜙 can 𝑝 still work successfully and by the common reliability of its It is noteworthy that the complexity of the proposed algo- components. Since the reliability and the signature of a coherent rithm for the calculation of the reliability of a combined structure can be expressed as 𝑚-consecutive-𝑘-out-of-𝑛: 𝐹 and consecutive-𝑘𝑐-out-of-𝑛: 𝐹 𝑛 systems has been determined to be equal to 𝑂([(𝑛𝑐 −𝑘 )/(𝑘 + 𝑖 𝑛−𝑖 3 𝑅= ∑ 𝑟𝑖 (𝑛) 𝑝 (1 −𝑝) , 1)] ). For an up-to-date overview of GERT analysis results of 𝑖=𝑛−𝑧 𝜙 consecutive-𝑘 systems, the interested reader is referred to Sen (78) et al. [48]. 𝑟𝑛−𝑖+1 (𝑛) 𝑟𝑛−𝑖 (𝑛) 𝑠𝑖 (𝑛) = 𝑛 − 𝑛 , Moreover, Eryilmaz [49] derived an explicit expression ( 𝑛−𝑖+1 ) ( 𝑛−𝑖 ) for the number of its path sets including a specified number 14 Journal of Quality and Reliability Engineering of working components. More specifically, Eryilmaz [49] less than 𝑑 consecutive working components between any of proved the following result. two failed ones. It goes without saying that, for 𝑑=0, the constrained (𝑘, 𝑑)-out-of-𝑛: 𝐹 system reduces to the ordinary Theorem 32 (Eryilmaz [49]). The number of path sets 𝑟𝑖(𝑛) of 𝑘-out-of-𝑛: 𝐹 system. 𝑚 𝑘 𝑛 𝐹 𝑘 [𝑑,∞) acombined -consecutive- -out-of- : and consecutive- 𝑐- Let Ω denote the set of all binary sequences of length 𝑛 𝑖 𝑛 out-of- : F systems with i.i.d. components including working 𝑛 where the runs of successes between two consecutive fail- ones can be expressed as follows: ures have length at least 𝑑.Thenthereliabilityofconstrained (𝑘, 𝑑)-out-of-𝑛: 𝐹 system is given as follows: 𝑟𝑖 (𝑛) 𝐹 [𝑑,∞) (0) 𝑅𝑘,𝑑,𝑛 =𝑃{X ∈Ω𝑛 ,𝑆𝑛 <𝑘} , (85) 𝑚−1 min(𝑖+1,[(𝑛+1)/2]) 𝑖+1 (82) = ∑ ∑ ( )𝐶(𝑠,𝑘,𝑘𝑐,𝑛−𝑖,𝑥), where X =(𝑋1,𝑋2,...,𝑋𝑛) denotes the states of components 𝑥=0 𝑠=1 𝑠 (𝑋𝑖 = 1ifthe𝑖th component is operating, and 𝑋𝑖 = 0, (0) where 𝑧min ≤𝑖<𝑛,𝑟𝑛(𝑛) = 1 (𝑧min indicates the minimum otherwise), while 𝑆𝑛 represents the total number of failed number of failed components at the moment of system failure) components. The following theorem offers expressions for and the quantities 𝐶(𝑠, 𝑘,𝑐 𝑘 ,𝑙,𝑥) can be calculated via the the evaluation of the reliability function of the constrained following recurrence: (𝑘, 𝑑)-out-of-𝑛: 𝐹 system with i.i.d. and Markov dependent components, respectively. 𝐶(𝑠,𝑘,𝑘𝑐,𝑙,𝑥) Theorem 33 (Eryilmaz and Zuo [50]). (i) For a constrained 𝑘−1 (𝑘, 𝑑) 𝑛 𝐹 𝑛 = ∑ 𝐶(𝑠− ,𝑘,𝑘 ,𝑙−𝑎,𝑥) -out-of- : system consisting of independent and 1 𝑐 𝑝 𝑎=1 (83) identical components with common reliability ,thereliability function is given as follows: 𝑘 −1 𝑐 𝑎 + ∑ 𝐶(𝑠− ,𝑘,𝑘 ,𝑙−𝑎,𝑥−[ ]) , 𝑅𝐹 1 𝑐 𝑘 𝑘,𝑑,𝑛 𝑎=𝑘 min(𝑘−1,[(𝑛+𝑑)/(𝑑+1)]) 𝑛−𝑑(𝑙− ) (86) 1 𝑛−𝑙 𝑙 with = ∑ ( )𝑝 (1 −𝑝) . 𝑙=0 𝑙 𝐶(1,𝑘,𝑘𝑐,𝑙,𝑥) (ii) For a constrained (𝑘, 𝑑)-out-of-𝑛: 𝐹 system consisting of { , <𝑙<𝑘,𝑥= , 𝑛 Markov dependent components with transition probabilities {1 if 0 0 { (84) 𝑝𝑟𝑠 = 𝑃(𝑋𝑖 =𝑠|𝑋𝑖−1 = 𝑟), 𝑟,0 𝑠∈{ , 1}, 𝑖 = 2, 3,...,𝑛,and = {1, if 𝑘𝑥≤𝑙 Theorem 34 (Eryilmaz and Zuo [50]). (i) For a constrained It is noteworthy that Eryilmaz and Zuo [50]provedexplicit consecutive (𝑘, 𝑑)-out-of-𝑛: 𝐺 system consisting of 𝑛 indepen- formulas for the calculation of the signature of both constrained dent and identical components with common reliability 𝑝,the (k, d)-out-of-𝑛 systems they considered. reliability function is given as follows: [(𝑛+𝑑)/(𝑑+1)] 𝑛−𝑑(𝑙−1) 2.11. Combined 𝑚𝑓 -Consecutive 𝑘𝑐𝑓 -out-of-𝑛 and 𝑚𝑓 -Con- 𝑅𝐺 = ∑ ( )𝑝𝑛−𝑙 ( −𝑝)𝑙 1 1 2 𝑘,𝑑,𝑛 1 secutive-𝑘𝑐𝑓 -out-of-𝑛 Systems. The common 𝑚-consecutive 𝑙=0 𝑙 2 𝑘𝑐-out-of-𝑛: 𝐹 system consists of 𝑛 linearly ordered com- 𝑛 𝑚 − ∑ 𝐴 (𝑘− ,𝑘−𝑑− ,...,𝑘−𝑑− ,𝑘 (89) ponents and fails if and only if there exist at least 𝑙+1,𝑛−𝑙 1 1 1 𝑘 𝑙=0 nonoverlapping runs of 𝑐 consecutive failed components. Gera [51] considered a more general reliability model, 𝑛−𝑙 𝑙 − 1) 𝑝 (1 −𝑝) , named the combined 𝑚𝑓 -consecutive 𝑘𝑐𝑓 -out-of-𝑛 and 𝑚𝑓 - 𝑘 𝑛1 1 2 consecutive- 𝑐𝑓2 -out-of- systems. The new structure con- where sists of 𝑛 components and fails if and only if there exist 𝑚 𝑘 at least 𝑓1 nonoverlapping runs of 𝑐𝑓1 consecutive failed 𝐴𝑛,𝑘 (𝑢1,𝑢2,...,𝑢𝑛) 𝑚 𝑘 components or at least 𝑓2 nonoverlapping runs of 𝑐𝑓2 𝑛+𝑘−𝑠−1 consecutive failed ones. Gera [51]shedlightonthecases =( ) 𝑚 ≥𝑚 ,𝑘 ≤𝑘 𝑘 where 𝑓1 𝑓2 𝑐𝑓1 𝑐𝑓2 , and additionally 𝑐𝑓1 is not an 𝑛−1 (90) 𝑘 𝑘 /𝑘 integer divisor of 𝑐𝑓2 ;thatis, 𝑐𝑓2 𝑐𝑓1 is not integer-valued. 𝑛 𝑛+𝑘−𝑠−𝑢 −𝑢 −𝑟− It goes without saying that otherwise the aforementioned 𝑟 𝑖1 𝑖2 1 𝑚 𝑘 𝑛 + ∑ (−1) ∑ ( ). system reduces to an ordinary -consecutive 𝑐-out-of- 𝑟=1 𝑛−1 structure. 𝑍 (𝑘, 𝑑) 𝑛 𝐺 Let denote the total number of components such that (ii) For a constrained consecutive -out-of- : system thesystemfailsduetothespecificrequirementsasmen- consisting of 𝑛 Markov dependent components with transition tioned, while 𝑋𝑧 represents the state of the 𝑧th component probabilities (𝑋𝑧 = 1 (0) for success (failure)). In order to determine the 𝑚 𝑘 reliability function of combined 𝑓1 -consecutive 𝑐𝑓1 -out-of- 𝑝𝑟𝑠 =𝑃(𝑋𝑖 =𝑠|𝑋𝑖−1 =𝑟), 𝑛 and 𝑚𝑓 -consecutive-𝑘𝑐𝑓 -out-of-𝑛 systems, it is necessary (91) 2 2 𝑟, 𝑠 ∈ {0, 1} ,𝑖=2, 3,...,𝑛 to evaluate the following probabilities: and initial probabilities 𝑝0 = 𝑃(𝑋1 = 0), 𝑝1 = 𝑃(𝑋1 = 1),the reliability is given as 𝑃 (𝑚 ,𝑚 ,𝑧) 0 𝑓1 𝑓2 𝑅𝐺 =𝑅𝐹 − ∑ 𝐴 (𝑘− ,𝑘−𝑑− ,...,𝑘 𝑘,𝑑,𝑛 𝑛+1,𝑑,𝑛 𝑙+1,𝑛−𝑙 2 1 =𝑃( 𝑚 𝑘 , 𝑙 exactly 𝑓1 times strings of length 𝑐𝑓1 𝑛−2𝑙−1 𝑙 𝑙 −𝑑−1,𝑘−2) 𝑝1𝑝11 𝑝10𝑝01 − ∑ 𝐴𝑙,𝑛−𝑙 (𝑘−2,𝑘 𝑚 𝑘 ,𝑋 = ), exactly 𝑓2 times strings of length 𝑐𝑓2 𝑧 0 𝑙 (93) 𝑛−2𝑙 𝑙 𝑙−1 𝑃1 (𝑚𝑓 ,𝑚𝑓 ,𝑧) −𝑑−1,...,𝑘−𝑑−1) 𝑝1𝑝11 𝑝10𝑝01 1 2 (92) =𝑃( 𝑚 𝑘 , − ∑ 𝐴𝑙,𝑛−𝑙 (𝑘−𝑑−1,...,𝑘−𝑑−1,𝑘−2) 𝑝0 exactly 𝑓1 times strings of length 𝑐𝑓1 𝑙 𝑚 𝑘 ,𝑋 = ). 𝑛−2𝑙 𝑙−1 𝑙 exactly 𝑓2 times strings of length 𝑐𝑓2 𝑧 1 ⋅𝑝11 𝑝10 𝑝01 − ∑ 𝐴𝑙−1,𝑛−𝑙 (𝑘−𝑑−1,...,𝑘−𝑑 𝑙 𝑛−2𝑙+1 𝑙−1 𝑙−1 − 1) 𝑝0𝑝11 𝑝10 𝑝01 . Gera [51] managed to express the above quantities as follows: 𝑚 𝑚 𝑎max 𝑓1 𝑓2 𝑎 𝑎 𝑎 𝑃0 (𝑚𝑓 ,𝑚𝑓 ,𝑧)= ∑ ∑ ∑ {𝑞 𝛿[[ ]−𝑟]⋅𝛿[[ ]−𝑡]𝑃1 (𝑚𝑓 −𝑟,𝑚𝑓 −𝑡,𝑧−𝑎)} 1 2 𝑘 𝑘 1 2 𝑎=1 𝑟=0 𝑡=0 𝑐𝑓1 𝑐𝑓2 𝑧 +𝑞 {𝑢 [𝑧𝑥 −𝑚 ]−𝑢[𝑧−𝑚𝑛]} , (94) 𝑠−1 𝑃 (𝑚 ,𝑚 ,𝑧)= ∑ 𝑝𝑏𝑃 (𝑚 ,𝑚 ,𝑧−𝑏), 1 𝑓1 𝑓2 0 𝑓1 𝑓2 𝑏=1 16 Journal of Quality and Reliability Engineering where 𝑢[⋅], 𝛿[⋅] are the unit step and delta function, respec- elements containing at least 𝑘 failed components is less than tively, and 𝑝(𝑞) is the common reliability (unreliability) of the 𝑚 components. For the evaluation of the reliability function components, while of the aforementioned structure, Levitin [54]employed the universal moment generating function technique and {𝑧−1,𝑚𝑥 ≤𝑧<𝑚𝑛, deduced a recursive algorithm that leads to the exact value 𝑎max = { (95) 𝑚 − ,𝑚≤𝑧. of system’s reliability. { 𝑛 1 𝑛 The following theorem gives the reliability of any subsystem 2.14. Linear 𝑚-Consecutive-𝑘, 𝑙-out-of-𝑛: 𝐹 Systems. Eryilmaz including 𝑧 elements of a combined 𝑚𝑓 -consecutive 𝑘𝑐𝑓 -out- and Mahmoud [55] proposed a new reliability model that 𝑛 𝑚 𝑘 𝑛1 1 𝑚 𝑘 𝑛 𝐹 of- and 𝑓2 -consecutive- 𝑐𝑓2 -out-of- systems. generalizes the linear -consecutive- -out-of- : system to the case of 𝑙-overlapping runs. More specifically, the linear Theorem 35 (Gera [51]). For combined 𝑚𝑓 -consecutive 𝑘𝑐𝑓 - 𝑚-consecutive 𝑘, 𝑙-out-of-𝑛: 𝐹 system consists of 𝑛 linearly 𝑛 𝑚 𝑘 𝑛 1 1 out-of- and 𝑓2 -consecutive- 𝑐𝑓2 -out-of- systems consisting ordered components and fails if and only if there are at least of 𝑛 independent and identical components with common 𝑚𝑙-overlapping runs of 𝑘 consecutive failed components (𝑛≥ reliability 𝑝, the reliability function of any subsystem including 𝑚(𝑘 − 𝑙) + 𝑙,). 𝑙<𝑘 The number of 𝑙-overlapping runs of 𝑧 elements is given as follows: length 𝑘, each of which may have a part of length at most 𝑙 overlapping with the previous run of length 𝑘.Theinclusion 𝑚 −1 𝑚 −1 𝑓1 𝑓2 of the new system parameter 𝑙 provides flexibility for wider 𝑃 (𝑍>𝑧) = ∑ ∑ (𝑃 (𝑖, 𝑗, 𝑧) +𝑃 (𝑖, 𝑗, 𝑧)) . 0 1 (96) application of the model. 𝑖=0 𝑗=0 Moreover, Eryilmaz and Mahmoud [55] derived an explicit expression for the number of its path sets including a 2.12. Linear 𝑚-Consecutive 𝑘-out-of-𝑟-from-𝑛: 𝐹 Systems. specified number of working components. More specifically, Levitin and Dai [52] proposed a new reliability model they proved the following result. that generalizes the linear 𝑟-within-consecutive-𝑘-out-of-𝑛: 𝐹 system (or consecutive-𝑘-out-of-𝑟-from-𝑛: 𝐹 system) to Theorem 36 (Eryilmaz and Mahmoud [55]). The number of the case of 𝑚 consecutive overlapping runs of 𝑟 elements. path sets 𝑟𝑖(𝑛) of a linear 𝑚-consecutive 𝑘, 𝑙-out-of-𝑛: 𝐹 system More specifically, the linear 𝑚-consecutive 𝑘-out-of-𝑟-from- including 𝑖 working components can be expressed as follows: 𝑛: 𝐹 system consists of 𝑛 linearly ordered 𝑠-independent components and fails if and only if in each one of at least 𝑚 𝑟𝑖 (𝑛) =𝐶(𝑛−𝑖;𝑖+1, 0;𝑘−1;𝑘−1) consecutive overlapping groups of 𝑟 consecutive elements at 𝑘 least of them fail. 𝑚−1 min(𝑖+1) 𝑖+ 𝑅 1 𝑛−1 For the evaluation of the reliability function of the + ∑ ∑ ( )( )×𝐶(𝑛−𝑖−𝑎𝑙 (98) aforementioned structure, Levitin and Dai [52]noticedthat 𝑠=1 𝑎=1 𝑎 𝑎−1 the following equality ensues − (𝑘−𝑙) 𝑠; 𝑎, 𝑖 −𝑎+ 1; 𝑘−𝑙−1;𝑘−1) , {𝑛−𝑟−𝑚+2 ℎ+𝑚−1 𝑠+𝑟−1 [ ] 𝑅=𝑃{ ∑ ∏ 1 ( ∑ (1 −𝑋𝑗)≥𝑘) 𝑛−𝑧 ≤𝑖≤𝑛 𝑧 { ℎ=1 [ 𝑠=ℎ 𝑗=𝑠 ] for 𝜙 ,where 𝜙 denotes the maximum (97) number of failed components such that the system can still work } successfully, and the quantities 𝐶(𝛽; 𝑎, 𝑟−𝑎;1 𝑚 −1,𝑚2 −1) can = 0} be calculated via the following formula (see, e.g., Makri et al. } [56]): and developed an efficient algorithm for calculating the func- 𝐶 (𝛽; 𝑎, 𝑟 −𝑎;𝑚 − ,𝑚 − ) tion 𝑅, by employing a procedure based on the universal 𝑧- 1 1 2 1 transform (for more details about the universal moment gen- [𝑎/𝑚 ] [(𝛽−𝑚 𝑗 )/𝑚 ] 1 1 1 2 𝑎 𝑟−𝑎 𝑗 +𝑗 erating function approach, the interested reader is referred to = ∑ ∑ (− ) 1 2 ( )( ) 𝑚 1 Ushakov [53]). It is noticeable that the linear -consecutive 𝑗 =0 𝑗 =0 𝑗1 𝑗2 (99) 𝑘-out-of-𝑟-from-𝑛: 𝐹 system meets interesting applications in 1 2 different areas, such as heating systems’ modeling. 𝛽−𝑚1𝑗1 −𝑚2𝑗2 +𝑟−1 ⋅( ). 𝑟− 2.13. Linear 𝑚-Gap-Consecutive 𝑘-out-of-𝑟-from-𝑛: 𝐹 Sys- 1 tems. Levitin [54] introduced a new reliability model that generalizes the linear 𝑟-within-consecutive-𝑘-out-of-𝑛: 𝐹 Having at hand the above expression for 𝑟𝑖(𝑛) and applying system (or consecutive-𝑘-out-of-𝑟-from-𝑛: 𝐹 system). More the well-known equalities for the evaluation of reliability and specifically, the linear 𝑚-gap-consecutive 𝑘-out-of-𝑟-from- signature of a coherent system in terms of 𝑟𝑖(𝑛) (see, e.g., 𝑛: 𝐹 system consists of 𝑛 linearly ordered, identical, and TriantafyllouandKoutras[57]),onemayeasilycomputethese statistically independent components and fails if and only important characteristics of the linear m-consecutive k,l-out- if the gap between any pair of groups of 𝑟 consecutive of-n: F system. Journal of Quality and Reliability Engineering 17 2.15. Sparsely Connected Consecutive-𝑘 Systems. Zhao et al. (i) the linear consecutive 𝑘-out-of-𝑛: 𝐹 system with [58] introduced generalized consecutive-𝑘 systems with sparse 𝑑,whichconsistsof𝑛 linearly ordered com- sparse 𝑑. The consecutive failures with sparse 𝑑 are defined as ponents and fails if and only if there exist at least 𝑘 follows. When components are ordered in a line, two adjacent consecutive failures with sparse 𝑑.Fortheaforemen- failed components have no failed components between them tioned structure, Zhao et al. [58]employedthewell- and the number of working components between the two knownMarkovChainImbeddingtechniqueinorder failed components is equal to 𝑑; then the two failed compo- to achieve expressions for the reliability function of nents are called consecutive failures with sparse 𝑑.Zhaoetal. such systems. Moreover, Mohan et al. [59]provedthat [58]proposedthefollowingreliabilitymodels: the generating function of the waiting time for the occurrence of the system failure is given as 𝑘 𝑑+1 𝑘−1 𝑑+1 (𝑞𝑧) (1 −(𝑝𝑧) ) (1 −𝑧+𝑞𝑧(𝑝𝑧) ) 𝑊0,𝑚+𝑘−1 (0,𝑧) = . (100) 𝑘−1 𝑘 𝑑+1 𝑑+1 𝑘−1 (1 −𝑝𝑧)((1 −𝑧) (1 −𝑝𝑧) +(𝑞𝑧) (𝑝𝑧) (1 −(𝑝𝑧) ) ) Hence, the reliability function of the above structure referring to humans’ activity. For example, an emergency canbeexpressedas backup power supply in a hospital or a power production at 𝑛 a specific voltage at the microscale may be viewed as direct implementations of reliability modeling. Moreover, double- 𝑅 (𝑛,) 𝑘 𝑑 = 1 − ∑ 𝜉 (𝑢) , (101) 𝑢=𝑘 loop computer networks, such as the daisy chain or the brained ring, can be studied through the lines of a reliability 𝑢 where 𝜉(𝑢) is the coefficient of 𝑧 in the power series structure. 𝑑 expansion of the generating function 𝑊0,𝑘(0,𝑧). Let us consider a television screen characterized by a certainnumberofpixelsandassumethatthevisualquality (ii) The 𝑚-consecutive 𝑘-out-of-𝑛: 𝐹 system with sparse is not acceptable if and only if one of the following ensues: 𝑑 consists of 𝑛 linearly ordered components and fails 𝑚 if and only if there exist at least nonoverlapping (i) there exist at least 𝑓 failed pixels, 𝑘 𝑑 runs of consecutive failures with sparse .Forthe 𝑘 aforementioned structure, Zhao et al. [58] employed (ii) there exist at least consecutive failed pixels. the well-known Markov Chain Imbedding technique It is clear that the aforementioned device can be studied as an in order to achieve expressions for the reliability (𝑛, 𝑓,𝑘) system. function of such systems. Moreover, Mohan et al. [59] In addition, it is of some interest that vacuum systems in proved that the generating function of the waiting accelerators or belt conveyors in open-cast mining operate time for the occurrence of the system failure is given within the framework of the well-known consecutive 𝑘- as out-of-𝑛 system. Among applications of 𝑘-out-of-𝑛 systems, 𝑑 𝑑 𝑚 the design of electronic circuits such as very large scale 𝑊0,𝑘𝑚 (0,𝑧) = (𝑊0,𝑘 (0,𝑧)) . (102) integrated (VLSI) and the automatic repairs of faults in an (iii) The linear (𝑛, 𝑓,𝑘): 𝐹 system with sparse 𝑑 consists online system would be the most conspicuous. This type of 𝑛 linearly ordered components and fails if and of structure consists of several parallel outputs channeled only if there exist at least 𝑓 total failures or at through a decision-making device that provides the required 𝑘 𝑑 system function as long as at least a predetermined number 𝑘 least consecutive failures with sparse .Forthe 𝑛 aforementioned structure, Zhao et al. [58] employed of parallel outputs are in agreement. the well-known Markov Chain Imbedding technique It is worth mentioning that Kalyan and Kumar [60] in order to achieve expressions for the reliability studied in detail the redundancy optimization in consecutive 𝑘-out-of-𝑛: 𝐹 systems. More specifically, for a consecutive function of such systems. Moreover, Mohan et al. 𝑘 𝑛 [59] proposed an algorithm for the evaluation of the -out-of- structure that has a fluctuation of demand on corresponding reliability function. the system performance, Kalyan and Kumar [60] considered extra resources in order to improve the system performance It is worth mentioning that, in case of 𝑑=0, the consecutive- and developed alternative methods for the maximization 𝑘 systems with sparse 𝑑 reduce to the ordinary consecutive of the structures’ reliability. It goes without saying that the 𝑘-out-of-𝑛: 𝐹 structures. reliability systems studied in the article of Kalyan and Kumar [60]canbeclassedundertheso-calledProteansystems. 3. Applications Zhang et al. [61] presented an application of the well- known consecutive 𝑘-out-of-𝑛: 𝐺 system to a railroad opera- Applications of reliability models can range from electrical tion. More specifically, they studied a railroad system consist- or engineering and mechanical field up to several contexts ing of 17 lines (numbered from 1 to 17), which was formulated 18 Journal of Quality and Reliability Engineering (n, f,k): F with sparse d Consecutive k-out-of-n: F (n1,n2,...,nN,f,k): F (n, f,k(i, j)): F d=0 with sparse d n =1 n=N i , d=0 i=1, j=n N f=∑ ni, n=N or i−j=1 N=1 (n, f,k): F i=1 m-consecutive k-out-of-n: F with cycle k f≤k f≥n r=1 k=n=f f-out-of-n: F Consecutive k-out-of-n: F m=1, kc k=1, m=f m-consecutive k-out-of-n: F m=1 and consecutive-kc-out-of-n: F Weighted f-out-of- n: F with weights wi k=n, r=f kc =mk d=0, f=k r=k m-consecutive k-out-of-n: F Constrained (k, d)-out-of-n m=1 r-within-consecutive k-out-of-n : F d=0 m m=1 m-consecutive k-out-of-n: F with overlapping runs m k Combined f1 -consecutive cf1 - n F m out-of- : and f2 -consecutive- m k r n k n F -consecutive -out-of- -from- cf2 -out-of- : m-consecutive k-out-of-n: F with sparse d Figure 1: Connection between members of the consecutive-type systems’ family. as a reliability study of a linear consecutive 𝑘-out-of-𝑛: 𝐺 Many safety-critical applications including nuclear power structure. plants are equipped with 𝑘-out-of-𝑛 or specific-voting-logic Zuo et al. [62] reported an application of 𝑘-out-of-𝑛: 𝐹 redundant safety signal generation systems for ensuring both and consecutive-𝑘-out-of-𝑛: 𝐹 systems in evaluation of the safety and economy. Kang and Kim [64]developedaquan- lifetime distribution of furnaces used in a petrochemical tification method for the evaluation of the unreliability of a company. More precisely, different scenarios were investi- reactor protection system (RPS), while several configurations gatedinmodelingthereliabilitybehaviorsofthefurnacesas of 𝑘-out-of-𝑛 models are investigated. a function of the reliabilities of the tubes used in the furnaces. Lu and Liu [63] used the well-known 𝑘-out-of-𝑛: 𝐺 struc- 4. How Members of the Consecutive-Type ture to quantify the reliability benefit of redundancy units Systems’ Family Are Connected in main circuit of prevailing static generators, called Static Synchronous Compensators (STATCOMs). The fundamental In the literature, there exist a lot of references that mark aim of applying a reliability study in the aforementioned down the connection between two well-known reliability operation is to maintain specific parameters of the electric systems. It is really often for a newly introduced reliability power system by using the variation of the outputs of the structure to coincide with an older one under some spe- so-called STATCOM and describe the effect of the redun- cific restrictions concerning their design parameters. In this dancy technique to the total device reliability performance. section, we attempt to harvest all information that refers to More specifically, Lu and Liu [63]providedtheprobabilis- relations among all members of the generalized consecutive- tic model of the 50 MVA STATCOM devices which were type systems’ family. brought into operation at a certain substation in Shanghai, Figure 1 displays for each system that is included in the China. consecutive-type systems’ family the way that is connected Journal of Quality and Reliability Engineering 19 to one or more of its members. It goes without saying that [13]M.V.Koutras,“OnaMarkovchainapproachforthestudyof both the 𝑘-out-of-𝑛 system and the consecutive 𝑘-out-of-𝑛 reliability structures,” Journal of Applied Probability,vol.33,no. system play a crucial role in the aforementioned group of 2, pp. 357–367, 1996. reliability structures. For example, based on Figure 1,onemay [14] S. 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