Journal of Quality and Reliability Engineering

Volume 2015, Article ID 212303, 20 pages

http://dx.doi.org/10.1155/2015/212303

## Consecutive-Type Reliability Systems: An Overview and Some Applications

Department of Statistics and Insurance Science, University of Piraeus, 80 Karaoli and Dimitriou Street, 18534 Piraeus, Greece

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.

#### Abstract

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 -out-of-: 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

A linear (circular) consecutive -out-of-: system consists of components which are linearly (circularly) arranged and the system fails if and only if at least consecutive components fail. The most popular applications of these systems pertain to computer network, telecommunication, pipeline network modeling, engineering, or integrated circuitry design. The consecutive -out-of-: system has been subject of substantial research interest for many years and a lot of generalizations have been suggested in order to accommodate more flexible operation principles. For a detailed presentation of the consecutive -out-of-: systems and some generalizations, the interested reader is referred to the excellent monograph of Kuo and Zuo [1] or the work of Chang et al. [2].

Let be the lifetime of a reliability system with components and its components’ lifetimes. If we assume that are exchangeable (and therefore identically distributed but not necessarily independent), the signature of the system is defined as the probability vector with where are the order statistics of the sample . It can be easily verified that, in the exchangeable case, the signature of a reliability system depends only on its structure and not on the specific underlying distribution of . In other words, is the proportion of permutations, among the equally likely permutations of , which result in a minimal cut set failing upon the occurrence of .

The signature of the system, which was first introduced by Samaniego [3], is closely related to many well-known reliability characteristics, a fact turning it to a very useful tool for studying coherent systems and their ageing properties. For example, the reliability polynomial of a structure can be easily expressed in terms of its signature. More precisely, Samaniego [3] proved that for any coherent system with independent and identical components which have absolutely continuous cumulative density functions, system’s reliability can be expressed as Navarro and Rychlik [4] proved that the above representation also holds true when the lifetimes have an absolutely continuous exchangeable distribution (this property had been mentioned earlier by Kochar et al. [5]). Moreover, Eryilmaz and Bayramoglu [6] used the system signature in order to evaluate the extreme residual lifetimes of the remaining components after the complete failure of the system.

Both reliability function and signature of a structure can be evaluated based on either recursive formulas or explicit expressions. In some cases, where neither of the aforementioned methods can be established for a specific system, approximating and limiting results are available. In the present review article, the family of consecutive-type reliability systems is under investigation. More specifically, Section 2 offers an up-to-date presentation of almost all generalizations of the well-known consecutive -out-of-: system that have been proposed and studied in the literature. For each structure, the corresponding fundamental work and selected results are displayed either briefly or in detail. In Section 3, applications of these reliability systems in several fields are described, while Section 4 presents a full-detailed diagram which connects the aforementioned structures by giving the information under which conditions a system can be treated as a special case of another one.

#### 2. Family of Consecutive-Type Reliability Systems

In this section, we study in detail almost all generalizations of the well-known consecutive -out-of-: systems that have been proposed in the literature till now. For each reliability structure that is included in the family of consecutive-type systems, the general operational structure is described, while several important and some recent relevant results are displayed.

##### 2.1. -Consecutive--out-of-: Systems

An -consecutive--out-of-: system consists of components and fails if and only if there exist at least nonoverlapping runs of consecutive failed components. This system was first introduced by Griffith [7] and since then it has attracted a considerable research attraction. In the sequel, we present the main results appearing in the literature, such as recurrence relations and closed formulas for the evaluation of the reliability function and signature vector of -consecutive--out-of-: systems. It is worth mentioning that the aforementioned system generalizes the well-known consecutive--out-of-: system (for ), while for the -consecutive--out-of-: system reduces to an ordinary -out-of-: system.

###### 2.1.1. Recursive Schemes for the Reliability of -Consecutive--out-of-: Systems

The following theorems provide recurrence for the calculation of reliability of an -consecutive--out-of-: system.

Theorem 1 (Papastavridis [8]). *Let denote the reliability function of a linear -consecutive--out-of-: system, where is the reliability of its th component. Then satisfies the following recurrence relation:**where and .**In order to launch the above recurrence scheme, a set of initial of conditions would be necessary. Observing that for the following ensues**while **we have at hand the set of initial values needed to evaluate reliability of the system.**Note that for the i.i.d case (e.g., ), the recurrence of Theorem 1 reduces to the following form: *

*Theorem 2 (Alevizos et al. [9]). Let denote the reliability function of a circular -consecutive--out-of-: system, where is the reliability (unreliability) of its th component. Then satisfies the following recurrence: where and denotes the reliability of a linear -consecutive--out-of-: subsystem with components .*

To launch the above recursive scheme, a set of initial of conditions would be necessary. Observing that the following ensue(i), for ,(ii), for ,(iii), for ,

we have at hand the set of initial values needed to evaluate reliability of the system.

Note that for the i.i.d case (e.g., ), the recurrence of Theorem 2 reduces to the following form:It is worth mentioning that the computational complexity of recurrence included in Theorem 1 is equal to , while the corresponding one for equations of Theorem 2 equals .

*2.1.2. Exact Formulas for the Reliability of -Consecutive--out-of-: Systems*

*2.1.2. Exact Formulas for the Reliability of -Consecutive--out-of-: Systems*

*The following theorems offer closed expressions for the evaluation of reliability of an -consecutive--out-of-: system.*

*Theorem 3 (Papastavridis [8]). Let denote the reliability function of a linear -consecutive--out-of-: system, where is the common reliability of its i.i.d. components. Then is given as follows: where , denotes the greatest integer lower bound of and can be expressed as *

*Theorem 4 (Makri and Philippou [10]). Let denote the reliability function of a linear -consecutive--out-of-: system composed by i.i.d. components, where is their common reliability. Then is given as follows: where is the well-known multinomial coefficient, while the inner summation is over all nonnegative integers such that .*

*Theorem 5 (Makri and Philippou [10]). Let denote the reliability function of a circular -consecutive--out-of-: system, where is the common reliability (unreliability) of its components. Then, can be expressed as follows: (i) where is the well-known multinomial coefficient, while the inner summation is overall nonnegative integers such that . Moreover, in the above expression denotes the greatest integer in , while is the Kronecker delta function. (ii) Consider where *

*2.1.3. Approximations for the Reliability of -Consecutive--out-of-: Systems*

*2.1.3. Approximations for the Reliability of -Consecutive--out-of-: Systems**The following theorems offer some limiting results for the evaluation of reliability of an -consecutive--out-of-: system.*

*Theorem 6 (Papastavridis [8]). Let be the common failure distribution of the components of an -consecutive--out-of-: system, where , are positive constants. Then the following ensueswhere is the time of first failure of the system.*

*Theorem 7 (Godbole [11]). Let , be the failure probability of the th component of an -consecutive--out-of-: system. Then the reliability of the system satisfies the following inequality:where and .*

*2.1.4. Signature Vector of -Consecutive--out-of-: Systems*

*2.1.4. Signature Vector of -Consecutive--out-of-: Systems*

*Let denote the common reliability of the components of an -consecutive--out-of-: system. The following theorem offers a generating function approach of the aforementioned system.*

*Theorem 8. Let and be the signature and the reliability function of an -consecutive--out-of-: system, respectively. Then (a)the double generating function of is given by (Eryilmaz et al. [12]),(b)the generating function of is given as follows: (Koutras [13]).*

The next theorem offers expressions for the evaluation of the signature vector of an -consecutive--out-of-: system.

*Theorem 9 (Eryilmaz et al. [12]). Let be the signature vector of an -consecutive--out-of-: system. Then the following ensues(a)the quantities satisfy the recurrence relation: for and .(b)the quantities can be expressed as for and if , where*

*2.1.5. Additional Results for -Consecutive--out-of-: Systems*

*2.1.5. Additional Results for -Consecutive--out-of-: Systems**Beyond the results mentioned in the previous subsections, additional studies have appeared in the literature for the -consecutive--out-of-: system. Eryilmaz [14] derived explicit expressions for the component importance measures for the aforementioned structure consisting of exchangeable components. More specifically, Eryilmaz [14] studied in detail the well-known Birnbaum and Barlow-Proschan importance measures for a -consecutive--out-of-: system. Furthermore, Ghoraf [15] offered recursive formulas for calculating the reliability function of the circular case of the aforementioned structure couching on the corresponding recurrence equations for the linear one.*

*2.2. -within-Consecutive--out-of-: Systems*

*2.2. -within-Consecutive--out-of-: Systems**An -within-consecutive--out-of-: system consists of components and fails if and only if there exist consecutive components which include among them at least failed components. This system was first introduced by Griffith [7], but its mathematical modelling has been done earlier by Greenberg [16] and Saperstein [17]. In the sequel, we present the main results appearing in the literature, such as recurrence relations and closed formulas for the evaluation of the reliability function and signature vector of -within-consecutive--out-of-: systems. It is worth mentioning that the aforementioned system generalizes the well-known consecutive--out-of-: system (for ), while for the -within-consecutive--out-of-: system reduces to an ordinary -out-of-: system.*

*2.2.1. Recursive Schemes for the Reliability of -within-Consecutive--out-of-: Systems*

*2.2.1. Recursive Schemes for the Reliability of -within-Consecutive--out-of-: Systems**The following theorems provide recurrence for the calculation of reliability of an -within-consecutive--out-of-: system.*

*Theorem 10 (Sfakianakis et al. [18]). Let denote the reliability function of a linear -within-consecutive--out-of-: system, where is the common reliability of its components. Then, for , , satisfies the following recurrence relation:where , if .*

*Theorem 11 (Eryilmaz [19]). Let denote the reliability function of a linear -within-consecutive--out-of-: system, where is the common reliability of its components. Then, for , satisfies the following recurrence relation:where is the reliability of -within-consecutive-()-out-of-2(): system with components , while denotes the lifetime of -out-of-: subsystem of components with the lifetimes , .*

*Theorem 12 (Koutras [13]). Let denote the reliability function of a linear r-within-consecutive--out-of-: system, where is the common reliability of its components. Then, satisfies the following recurrence relation:*

*2.2.2. Exact Formulas for the Reliability of -within-Consecutive--out-of-: Systems*

*2.2.2. Exact Formulas for the Reliability of -within-Consecutive--out-of-: Systems**The following Theorem offers closed expressions for the evaluation of reliability of an -within-consecutive--out-of-: system.*

*Theorem 13 (Sfakianakis et al. [18]). Let ( denote the reliability function of a linear (circular) 2-within-consecutive--out-of-: system, where is the common reliability of its components. Then the following recurrences ensue: (i) where , (ii) where .*

*2.2.3. Approximations for the Reliability of -within-Consecutive--out-of-: Systems*

*2.2.3. Approximations for the Reliability of -within-Consecutive--out-of-: Systems**The following theorems offer some limiting results for the evaluation of reliability of an -within-consecutive--out-of-: system.*

*Theorem 14 (Eryilmaz et al. [20]). Let denote the reliability function of a linear -within-consecutive--out-of-: system. Then, for , satisfies the following inequalities: (i) where is the th smallest lifetime among , , and (ii) Consider where and Let us next denote by the event where there are at least failed components from to , for , while , , are defined as follows: *

*Theorem 15 (Sfakianakis et al. [18]). Let denote the reliability function of a linear -within-consecutive--out-of-: system, where p is the common reliability of its components. Then satisfies the following inequalities: (i) where (ii) whereFor the next Theorem, the following definitions are necessary:(i) denotes the event that the linear -within-consecutive--out-of-: system consisting of the components is good ().(ii) denotes the event that the th component fails and there are at least failures among components , ().(iii) denotes the event that there are at most failures among components , ().(iv) denotes the event that, for , there is no failure among components , for .*

*Theorem 16 (Papastavridis and Koutras [21]). Let denote the reliability function of a linear -within-consecutive--out-of-: system. Then satisfies the following inequalities:where is the unreliability of the th component, while .*

*2.2.4. Signature Vector of -within-Consecutive--out-of-: Systems*

*2.2.4. Signature Vector of -within-Consecutive--out-of-: Systems*

*Let denote the common reliability of the components of an -within-consecutive--out-of-: system. The following theorem offers a generating function approach of the aforementioned system.*

*Theorem 17. Let and be the signature and the reliability function of an 2-within-consecutive--out-of-: system, respectively. Then (a)the double generating function of is given by (Triantafyllou and Koutras [22]),(b)the generating function of is given as follows: (Koutras [13]).The next theorem offers expressions for the evaluation of the signature vector of an 2-within-consecutive--out-of-: system.*

*Theorem 18 (Triantafyllou and Koutras [22]). Let be the signature vector of an 2-within-consecutive--out-of-: system. Then the following ensues:(i)The quantities , , where , satisfy the recurrence relation: for and .(ii)The quantities can be expressed as It is worth mentioning that well-performed simulations study of an -within-consecutive--out-of-: system has been developed by Eryilmaz et al. [20]. Moreover, Kan et al. [23] studied the circular case of the aforementioned structure and offered a new approximation for its reliability.*

*2.3. Systems*

*2.3. Systems*

*An system involves two common failure criteria. More specifically, it consists of components (ordered in a line or a circle) and fails if and only if there exist at least failed components or at least consecutive failed components. It is worth of mentioning that the configuration of an system was first introduced by Tung [24] as an application to a complex infrared detecting system and since then it has attracted considerable research attention. In the sequel, we present the main results for systems appearing in the literature, such as recurrence relations and closed formulas for the evaluation of the reliability function and signature vector. It is worth mentioning that the aforementioned system generalizes the well-known consecutive--out-of-: system (for ), while for the () system reduces to an ordinary -out-of-: system.*

*2.3.1. Recursive Schemes for the Reliability of Systems*

*2.3.1. Recursive Schemes for the Reliability of Systems*

*The following theorems provide recurrence for the calculation of reliability of an system.*

*Theorem 19 (Zuo et al. [25]). Let be the event that the subsystem fails (the subsystem consists of components , , ), while denotes the corresponding failure probability . Then the unreliability function of the system satisfies the following recurrence relation:where is the reliability (unreliability) of the th component.*

*In order to launch the aforementioned recursive scheme, the following set of initial conditions is necessary:The complexity for calculating using the above recurrence relation is equal to . The next theorem provides an alternative recursive scheme for the evaluation of the reliability function of the system.*

*Theorem 20 (Triantafyllou and Koutras [26]). The reliability function of an system with i.i.d. components satisfies the following recurrence relation: where denotes the common reliability of its components.*

*In order to launch the recursive scheme established above, an adequate number of initial conditions is necessary. These conditions are given as followswhere is the reliability of a -out-of: : system; that is, .*

*2.3.2. Exact Formulas for the Reliability of Systems*

*2.3.2. Exact Formulas for the Reliability of Systems*

*Let us first consider an system, where ; we recall that for the case the system coincides with the well-known -out-of-: system and its reliability properties have been extensively studied in the past. Chang et al. [27] established a Markov chain representation of the system, which leads to the computation of the reliability function of the aforementioned structure. More specifically, for the system with , let us define the state space for process aswhere indicates a working state in which the system consisting of components has failed components, its last components have failed, and the th component is working. State indicates the system fails. Then is a Markov chain with transition matrix of the formwhere and .*

*As proved in Koutras [13], the reliability of a structure can be expressed as where is the transition probability matrix associated to the structure andTherefore, applying the above expression obtained for the transition matrix of an system, one may easily calculate the reliability function of the aforementioned structure.*

*Let us next consider the following probabilities:where is the state of th component , for . The following theorem offers a closed expression for the evaluation of reliability of an system.*

*Theorem 21 (Eryilmaz [28]). The reliability function of an system with exchangeable components can be expressed as follows: whereIn the sequel, we present results for an system with Markov dependent components. Let denote the states of the Markov dependent components with transition probabilitieswith and initial probabilities , . If we define as the following theorem provides an expression for the evaluation of reliability of systems.*

*Theorem 22 (Demir [29]). The reliability function of an system with Markov dependent components can be expressed as follows: *

*2.3.3. Signature Vector of Systems*

*2.3.3. Signature Vector of Systems**Let denote the common reliability of the components of an system. The following theorem offers a generating function approach of the aforementioned system.*

*Theorem 23 (Triantafyllou and Koutras [26]). Let be the signature of an system, respectively. Then the double generating function of is given by whereTheorem 24 offers recursive relations for the evaluation of the signature vector of an system.*

*Theorem 24 (Triantafyllou [30]). The coordinates of the signature vector of an system satisfy the following recurrence relation: Let be the lifetime of a system whose components’ lifetimes are and the order statistics associated with them. It is easy to observe that the lifetime of an system can be represented as a function of the lifetimes of consecutive -out-of-: and -out-of-: systems. Generally speaking, the lifetimes of systems involving two common failure criteria can be expressed as either where denotes the lifetime associated with the system different from -out-of- structure but having the same components’ lifetimes . The next theorem presents an alternative way of computing the signature of an system.*

*Theorem 25 (Eryilmaz and Zuo [31]). Let be the signature of the system with lifetime . Then the signatures of the systems with lifetimes and are given, respectively, as It is noteworthy that the dual system of an structure has been studied by Cui et al. [32], while a generalization of these systems, named systems, with weighted components has been introduced by Eryilmaz and Aksoy [33]. Gera [34] studied reliability systems with two working criteria and presented some results associated with the well-known qualification tests, while Kamalja [35] derived expressions for the Birnbaum importance measures for both structures.*

*2.4. Reliability Systems with Weighted Components*

*2.4. Reliability Systems with Weighted Components*

*The idea that all components in a reliability structure are not created equal is seemingly an obvious concept. In other words, it is not quaint to assume that different components may have different failure probabilities. Generally speaking, a reliability system which consists of weighted components, for example, each component carries its own positive weight, fails if and only if the total weight of the failed components exceeds a specific benchmark. In the sequel, the literature on weighted reliability structures is briefly reviewed. Consider a system with components and suppose that the th component is associated with a weight , . Then the system is still working if and only if the sum of weights of the failed components is less than (or equal to) a certain threshold .*

*2.4.1. Weighted -out-of- Systems*

*2.4.1. Weighted -out-of- Systems*

*A weighted -out-of-: system consists of components, each with its own positive weight (total system weight equal to ), such that the system works (fails) if and only if the total weight of the working (failed) components is at least . It is noteworthy that the reliability of a weighted -out-of-: system is the complement of the unreliability of a weighted -out-of-: system. It goes without saying that the -out-of-: system is a special case of the corresponding weighted -out-of-: system wherein the weight of each component equals 1. The next theorem offers an efficient algorithm for the evaluation of the reliability of the weighted -out-of-: system.*

*Theorem 26 (Wu and Chen [36]). Let be the reliability of the weighted -out-of-: system, while is the weight of the th component. Then if we denote by the reliability (unreliability) of the th component, the reliability of the structure satisfies the following recurrence relation:The following theorem provides a similar result concerning the dual structure.*

*Theorem 27 (Chen and Yang [37]). Let be the reliability of the weighted -out-of-: system, while is the weight of the th component. Then if we denote by the reliability (unreliability) of the th component, the reliability of the structure satisfies the following recurrence relation:An extension of the aforementioned one-stage weighted -out-of- model has been proposed by Chen and Yang [37]. More specifically, Chen and Yang [37] considered a two-stage weighted -out-of- system, which consists of subsystems. Each subsystem has a (one-stage) weighted -out-of- structure, which is called the second level structure. The interrelationship between the subsystems follows a certain coherent structure, which is called the first-level structure. Generally speaking, a two-stage weighted -out-of- system can be decomposed into two hierarchical levels: the first (higher) level can be of any coherent structure and the second (lower) level has a weighted -out-of- structure.*

*An additional generalization of the well-known weighted -out-of- system has been introduced by Eryilmaz [38]. More specifically, Eryilmaz [38] assumed that the system has a performance level above if there are at least working components and the sum of the weights of all working components is above . Among others, Eryilmaz [38] deduced recursive relations for the calculation of the system state probabilities, while a detailed simulation study has been taken into play in order to observe the time spent by the system in state or above. Finally, Li and Zuo [39] studied the multistate weighted -out-of- systems, where each component may be in more than 2 states and therefore its contribution to the system’s weight can be differentiated analogously.*

*2.4.2. Consecutive Weighted -out-of- Systems*

*2.4.2. Consecutive Weighted -out-of- Systems*

*A weighted consecutive -out-of-: system consists of components, each with its own positive weight (total system weight equal to ), such that the system fails if and only if the total weight of the failed components is at least . It goes without saying that the consecutive -out-of-: system is a special case of the corresponding weighted consecutive -out-of-: system wherein the weight of each component equals 1. Efficient algorithms for the evaluation of the reliability of the linear weighted consecutive -out-of-: system have appeared in the literature (for more details see Kuo and Zuo [1]), while Samaniego and Shaked [40] extended the idea of weighted components, by giving to the components weights that can take any positive value (not necessary integer-valued).*

*2.5. Systems*

*2.5. Systems*

*Cui and Xie [41] introduced a generalized -out-of- system, denoted by . Such a system consists of modules ordered in a line or a circle, while the th module is composed of components in parallel. In other words, the system fails if and only if there are at least failed components or at least consecutive failed modules. It goes without saying that, for , an system reduces to a simple while for coincides with the well-known consecutive -out-of-: system and for with the ordinary -out-of-: structure. The following theorem offers recursive relations for the evaluation of the reliability function of a linear and circular system, respectively.*

*Theorem 28 (Cui and Xie [41]). (i) For a linear system, the reliability function satisfies the following recurrence: where is the probability that, in module , there are failed components.*

(ii) For a circular system, the reliability function satisfies the following recurrence:It is worth mentioning that the system can be obtained by adding more components in parallel to the basic components of an structure. Since the aforementioned system involves multiple failure criteria, Cui and Xie [41] studied the Birnbaum importance with respect to the th failure criterion as an importance measure computed only under this failure criterion. Firstly, for a reliability system with failure criteria, they assumed that, for any one of , the Birnbaum importance is different, while, for any one of failure criteria, the Birnbaum importance is the same. Moreover, Cui and Xie [41] denoted by works under failure criteria and works under failure criteria the specific disjoint events and they proved that the Birnbaum importance of the reliability system depends upon these failure criteria. Finally, note that the well-known series and parallel reliability systems are included in the more general systems’ family as special cases (for and and , resp.).

*2.6. and Systems*

*2.6. and Systems*

*Guo et al. [42] generalized the idea of the system, studied by Chang et al. [27] and Zuo et al. [25]. More specifically, they introduced a reliability structure, named the : system, which consists of components ordered in a line or circle and fails if and only if there exist at least failed components or at least consecutive failed components among components . It is noteworthy that, for the circular case, component index should have modulo property, that is, components and indicate the same one, while the following requirements should be satisfied:(i), if ,(ii), if .*

*
Otherwise, the consecutive failure criterion can be removed. It goes without saying that when , , or , the : system becomes the well-known structure. Furthermore, Guo et al. [42] mentioned an additional justification of the aforementioned reliability systems. More precisely, they introduced the : system, which consists of components ordered in a line or circle, and fails if and only if there exist at least failed components and at least consecutive failed components among components . Finally, Guo et al. [42] considered the dual structures of the : and : systems, taking into account the argumentation applied by Cui et al. [32].*

*The main result of the aforementioned paper is the employment of a two-stage Markov chain in order to give the system reliability in the form of product of matrices. Generally speaking, the finite Markov chain imbedding technique is to embed a Markov chain defined on the state space and the discrete index space into a given system, while the system fails if there exists , , such that , for all .*

*The unique justification is to divide the discrete index space and the nonabsorbing state space into two disjoined parts. For example, if we consider the : (: ) system with component reliabilities , , by rearranging their components without changing the system reliability, we can transform it into the linear : (: ) system with component reliabilities (), where , and , satisfy the following relations: The rearrangement is as follows by a transform:where the first line (old-component order) denotes the components order before rearrangement, the second line (New-component order) denotes the components order after rearrangement, that is, we move to the end of the line that section to which the consecutive criterion applies so that the new system consists of components in an ordered line. Thus, the study of the aforementioned reliability structures is covered by the linear case *