Consecutive 𝑘-within-𝑚-out-of-𝑛:F System with Nonidentical Components

Abstract

As a generalisation of consecutive 𝑘-out-of-𝑛:F and 𝑘-out-of-𝑛:F system models, a consecutive 𝑘-within-𝑚-out-of-𝑛:F system consists of 𝑛 linearly ordered components and fails if and only if there are 𝑚 consecutive components which include among them at least 𝑘 failed components. In this paper, we study the survival function of a consecutive 𝑘-within-𝑚-out-of-𝑛:F system consisting of independent but nonidentical components. We obtain exact expressions for the survival function when 2𝑚≥𝑛. A detailed analysis for consecutive 2-within-𝑚-out-of-𝑛:F systems is presented and the asymptotic behaviour of hazard rate of these systems is investigated using mixture representations.

1. Introduction

It is a well-accepted fact that all components in an engineered system are not created equal. This in turn implies that different components may have different survival probabilities. The study of systems consisting of nonidentical components is a difficult task especially when the system has a complex structure. Some recent contributions on systems with independent but nonidentical components appear in Navarro [1], Zhao et al. [2], Kochar and Xu [3], and Navarro et al. [4].

Consecutive type systems have been extensively studied in the literature. One of the most widely studied consecutive type system is a linear consecutive 𝑘-out-of-𝑛:F system which consists of 𝑛 linearly ordered components and fails if and only if at least 𝑘 consecutive components fail. This type of systems is potentially useful for modeling transportation and transmission systems. Much of the previous research has concentrated on the optimal design or reliability computation of such systems. There are several papers which study the dynamic reliability properties of consecutive 𝑘-out-of-𝑛 systems. Boland and Samaniego [5] obtained some stochastic ordering results on lifetimes of consecutive 𝑘-out-of-𝑛 systems consisting of independent components. Triantafyllou and Koutras [6] studied the lifetime distribution of consecutive 𝑘-out-of-𝑛:F systems consisting of independent and identical components. A review of recent developments on consecutive 𝑘-out-of-𝑛 and related systems is presented in Eryilmaz [7].

In this paper we study the dynamic reliability of consecutive 𝑘-within-𝑚-out-of-𝑛:F systems consisting of independent but nonidentical (inid) components. A consecutive 𝑘-within-𝑚-out-of-𝑛:F system consists of 𝑛 linearly ordered components and fails if and only if there are 𝑚 consecutive components which include among them at least 𝑘-failed components (1<𝑘≤𝑚≤𝑛). There are numerous applications for such systems in practice, for example, quality control, inspection procedures, radar detection, transportation, and transmission systems (see, e.g., Chang et al. [8]). A consecutive 𝑘-within-𝑚-out-of-𝑛:F system involves consecutive 𝑘-out-of-𝑛:F and 𝑘-out-of-𝑛:F (a system which fails if and only if at least 𝑘 components fail) systems for 𝑚=𝑘 and 𝑚=𝑛, respectively. The dynamic reliability properties of consecutive 𝑘-within-𝑚-out-of-𝑛:F systems with identical components have been studied in several papers (Papastavridis [9], Iyer [10], Eryilmaz et al. [11], Eryilmaz and Kan [12], Triantafyllou and Koutras [13]).

Consecutive 𝑘-within-𝑚-out-of-𝑛:F system can be represented as a series system of 𝑛−𝑚+1-dependent 𝑘-out-of-𝑚:F systems. That is, the lifetime of this system can be expressed as𝑇_{𝑘,𝑚∶𝑛}=min𝑇^[1∶𝑚]_𝑘∶𝑚,𝑇^[2∶𝑚+1]_𝑘∶𝑚,…,𝑇^{[𝑛−𝑚+1∶𝑛]}_𝑘∶𝑚, (1.1) where 𝑇^{[𝑖∶𝑖+𝑚−1]}_𝑘∶𝑚 shows the lifetime of 𝑘-out-of-𝑚:F subsystem of components with the lifetimes 𝑇_𝑖,𝑇_𝑖+1,…,𝑇_{𝑖+𝑚−1},1≤𝑖≤𝑛−𝑚+1.

The evaluation of the survival function associated with 𝑇_{𝑘,𝑚∶𝑛} is of special importance for understanding the dynamic behaviour of the system since the reliability characteristics such as hazard rate and mean residual life function can be obtained from this function. In the present paper, we obtain expressions for the survival functions of consecutive 𝑘-within-𝑚-out-of-𝑛:F systems for 2𝑚≥𝑛 when the components are independent but not necessarily identically distributed. In Section 2, a detailed analysis for consecutive 2-within-𝑚-out-of-𝑛:F systems is presented. Section 3 contains results for 2𝑚≥𝑛.

In the following, we provide the notations that will be used throughout the paper. 𝑛 is the number of components; 𝑇_𝑖 is the lifetime of component 𝑖;   𝑋_𝑖(𝑡) is the state of component 𝑖 at time 𝑡: 𝑋_𝑖(𝑡)=1(0) if 𝑇_𝑖≤𝑡(𝑇_𝑖>𝑡);𝑇_{𝑘,𝑚∶𝑛} is the lifetime of consecutive 𝑘-within-𝑚-out-of-𝑛:F system; 𝑇^{[𝑎∶𝑏]}_{𝑘∶𝑏−𝑎+1} is 𝑘th smallest among 𝑇_𝑎,𝑇_𝑎+1,…,𝑇_𝑏; 𝑅_{𝑘,𝑚∶𝑛}(𝑡)=𝑃{𝑇_{𝑘,𝑚∶𝑛}>𝑡} is the survival function of consecutive 𝑘-within-𝑚-out-of-𝑛:F system; ℎ_{𝑘,𝑚∶𝑛}(𝑡) is the hazard rate of consecutive 𝑘-within-𝑚-out-of-𝑛:F system.

Throughout the paper the components are assumed to be independent and the survival function associated with the 𝑖th component is 𝐹_𝑖(𝑡)=𝑃{𝑇_𝑖>𝑡}=1−𝐹_𝑖(𝑡),𝑖=1,2,…,𝑛.

2. Results for Consecutive 2-within-𝑚-out-of-𝑛:F Systems

If 𝑇₁,𝑇₂,…,𝑇_𝑛 represent the lifetimes of 𝑛 components in a coherent system, then the system lifetime can be represented as𝑇=max_{1≤𝑗≤𝑠}min_{𝑖∈𝑃𝑗}𝑇_𝑖, (2.1)where 𝑃₁,𝑃₂,…,𝑃_𝑠 are the minimal path sets. If 𝑇₁,𝑇₂,…,𝑇_𝑛 are independent, then the system survival function can be computed from the series survival functions as𝑆(𝑡)=𝑃{𝑇>𝑡}=_{{}𝐴⊆1,2,…,𝑠}(−1)^|𝐴|+1_{𝑖∈𝑃𝐴}𝐹_𝑖(𝑡), (2.2) where 𝐹_𝑖(𝑡)=𝑃{𝑇_𝑖>𝑡}, 𝑖=1,2,…,𝑛, and 𝑃_𝐴=⋃_𝑗∈𝐴𝑃_𝑗.

The hazard rate associated with the subset 𝐴 of (2.2) isℎ_𝐴(𝑡)=_{𝑖∈𝑃𝐴}𝑟_𝑖(𝑡), (2.3)

where 𝑟_𝑖(𝑡) is the hazard rate associated with 𝐹_𝑖(𝑡).

Example 2.1. Let 𝑛=4,𝑚=3, and 𝑘=2. Then the path sets of consecutive 2-within-3-out-of-4:F system are {1,2,3,4}, {2,3,4}, {1,3,4}, {1,2,4}, {1,2,3}, and {2,3}. The minimal path sets are 𝑃₁={2,3}, 𝑃₂={1,2,4}, and 𝑃₃={1,3,4}. Therefore 𝑇_2,3∶4=maxmin𝑇₂,𝑇₃,min𝑇₁,𝑇₂,𝑇₄,min𝑇₁,𝑇₃,𝑇₄, (2.4) and hence the survival function of consecutive 2-within-3-out-of-4:F system is 𝑅_2,3∶4(𝑡)=𝐹₂(𝑡)𝐹₃(𝑡)+𝐹₁(𝑡)𝐹₂(𝑡)𝐹₄(𝑡)+𝐹₁(𝑡)𝐹₃(𝑡)𝐹₄(𝑡)−2𝐹₁(𝑡)𝐹₂(𝑡)𝐹₃(𝑡)𝐹₄(𝑡) (2.5) which is a mixture of series survival functions with the weight vector (1,1,1,−2). The hazard rates of each element of (2.5) are ℎ₁(𝑡)=𝑟₂(𝑡)+𝑟₃(𝑡),ℎ₂(𝑡)=𝑟₁(𝑡)+𝑟₂(𝑡)+𝑟₄(𝑡),ℎ₃(𝑡)=𝑟₁(𝑡)+𝑟₃(𝑡)+𝑟₄(𝑡),ℎ₄(𝑡)=𝑟₁(𝑡)+𝑟₂(𝑡)+𝑟₃(𝑡)+𝑟₄(𝑡). (2.6)

Lemma 2.2. For 2𝑚≥𝑛 the consecutive 2-within-𝑚-out-of-𝑛:F system has 𝑛+₂^{𝑛−𝑚+1}+1 path sets, and these sets are {},{𝑖}𝐶=1,2,…,𝑛𝐶⧵,𝑖=1,2,…,𝑛,𝐶⧵𝑖,𝑚+𝑖+𝑗,𝑖=1,2,…,𝑛−𝑚,𝑗=0,1,…,𝑛−𝑖−𝑚. (2.7)

Proof. For 2𝑚≥𝑛, the consecutive 2-within-𝑚-out-of-𝑛:F system works if and only if there is no failed component or there is only one failed component or there are at most two failed components separated by at least 𝑚−1 working components. Thus the proof is complete.

The following results are direct consequences of Lemma 2.2.

Lemma 2.3. For 2𝑚≥𝑛, the consecutive 2-within-𝑚-out-of-𝑛:F system has 2𝑛−𝑚+1 minimal path sets with 𝑛−2 elements. These minimal path sets are 𝑙₁^(𝑠),…,𝑙^(𝑠)_𝑛−2≡{}⧵,2.1,2,…,𝑛𝑖,𝑚+𝑖+𝑗𝑖=1,2,…,𝑛−𝑚,𝑗=0,1,…,𝑛−𝑖−𝑚,𝑠=1,2,…,𝑛−𝑚+1 (2.8)

Lemma 2.4. Let 𝑇₁,𝑇₂,…,𝑇_𝑛 be inid lifetimes of components with 𝐹_𝑖(𝑡)=𝑃{𝑇_𝑖≤𝑡},𝑖=1,2,…,𝑛. For 2𝑚≥𝑛, 𝑅_{2,𝑚∶𝑛}(𝑡)=^𝑛_𝑗=1𝐹_𝑗(𝑡)+^𝑛_𝑖=1𝐹_𝑖(𝑡)^𝑛_{𝑗=1𝑗≠𝑖}𝐹_𝑗(𝑡)+^𝑛−𝑚_𝑖=1^{𝑛−𝑖−𝑚}_𝑗=0𝐹_𝑖(𝑡)𝐹_{𝑚+𝑖+𝑗}(𝑡)^𝑛_{𝑙=1𝑙≠𝑖,𝑚+𝑖+𝑗}𝐹_𝑙(𝑡). (2.9)

Theorem 2.5. [1] Let 𝑆 be a survival function such that 𝑆(𝑡)=^𝑛_𝑖=1𝜔_𝑖𝑆_𝑖(𝑡), (2.10) for all 𝑡≥0, where 𝜔₁,…,𝜔_𝑛 are real numbers such that ∑^𝑛_𝑖=1𝜔_𝑖=1. Let ℎ_𝑖(𝑡) be the failure rate function corresponding to 𝑆_𝑖(𝑡),𝑖=1,…,𝑛. If lim_𝑡→∞ℎinf^𝑖(𝑡)ℎ₁(𝑡)>1,lim_𝑡→∞ℎsup^𝑖(𝑡)ℎ₁(𝑡)<∞, (2.11) for 𝑖=2,3,…,𝑛, then lim_𝑡→∞(ℎ(𝑡)/ℎ₁(𝑡))=1, where ℎ(𝑡) is the failure rate function corresponding to 𝑆(𝑡).

In the following, one will study the limiting behaviour of the hazard rate of a consecutive 2-within-𝑚-out-of-𝑛:F system.

Theorem 2.6. Let 𝑇₁,𝑇₂,…,𝑇_𝑛 be independent and the hazard rate of 𝑇_𝑖 is 𝑟_𝑖(𝑡). For 2𝑚≥𝑛, if lim_𝑡→∞𝑟_𝑖(𝑡)=𝜆_𝑖 and 𝜆₁>𝜆₂>⋯>𝜆_𝑛, then lim_𝑡→∞ℎ_{2,𝑚∶𝑛}(𝑡)=min𝜆_𝑙1(1)+⋯+𝜆_{𝑙(1)𝑛−2},𝜆_𝑙1(2)+⋯+𝜆_{𝑙(2)𝑛−2},…,𝜆_𝑙1(𝑠)+⋯+𝜆_{𝑙(𝑠)𝑛−2}, (2.12) where 𝑠=₂^{𝑛−𝑚+1}.

Proof. From Lemma 2.3, the minimum number of elements in the minimal path sets of consecutive 2-within-𝑚-out-of-𝑛 system is 𝑛−2 (for 2𝑚≥𝑛) and the total number of these minimal path sets is 2𝑛−𝑚+1. Thus the proof follows from Theorem 2.5 and the conditions of Theorem 2.6.

Example 2.7. Let 𝑛=4,𝑚=3, and 𝑘=2. Suppose that lim_𝑡→∞𝑟_𝑖(𝑡)=𝜆_𝑖,𝑖=1,2,3,4 and 𝜆₁>𝜆₂>𝜆₃>𝜆₄. Then using the hazard rates given in (2.6), we have lim_𝑡→∞ℎinf²(𝑡)ℎ₁(𝑡)=𝜆₁+𝜆₂+𝜆₄𝜆₂+𝜆₃>1,lim_𝑡→∞ℎinf³(𝑡)ℎ₁(𝑡)=𝜆₁+𝜆₃+𝜆₄𝜆₂+𝜆₃>1,lim_𝑡→∞ℎinf⁴(𝑡)ℎ₁(𝑡)=𝜆₁+𝜆₂+𝜆₃+𝜆₄𝜆₂+𝜆₃>1,lim_𝑡→∞ℎsup^𝑖(𝑡)ℎ₁(𝑡)<∞, (2.13) for 𝑖=2,3,4. Therefore lim_𝑡→∞ℎ_2,3∶4(𝑡)=𝜆₂+𝜆₃. (2.14)

3. General Results

The reliability of consecutive 𝑘-within-𝑚-out-of-𝑛:F system is closely related to the discrete scan statistic defined by𝑆_𝑛,𝑚(𝑡)=max^{𝑖+𝑚−1}_𝑗=𝑖𝑋_𝑗(𝑡)∶1≤𝑖≤𝑛−𝑚+1. (3.1)

A consecutive 𝑘-within-𝑚-out-of-𝑛:F system survives at time 𝑡 if and only if less than 𝑘 components are failed among any consecutive 𝑚 components. Thus its survival function can be expressed as𝑅_{𝑘,𝑚∶𝑛}(𝑡)=𝑃𝑇_{𝑘,𝑚∶𝑛}>𝑡=𝑃𝑆_𝑛,𝑚(𝑡)<𝑘, (3.2)

or equivalently,𝑅_{𝑘,𝑚∶𝑛}(𝑡)=𝑃𝑇^[1∶𝑚]_𝑘∶𝑚>𝑡,𝑇^[2∶𝑚+1]_𝑘∶𝑚>𝑡,…,𝑇^{[𝑛−𝑚+1∶𝑛]}_𝑘∶𝑚>𝑡, (3.3)

for 𝑡≥0 and 1≤𝑘≤𝑚≤𝑛.

The proof of the following result is easy and hence is omitted.

Lemma 3.1. For 𝑘=1 and 𝑏−𝑎≥0, 𝑃𝑇^{[𝑎∶𝑏]}_{𝑘∶𝑏−𝑎+1}>𝑡=^𝑏_𝑖=𝑎𝐹_𝑖(𝑡), (3.4) for 𝑘>1 and 𝑏−𝑎=𝑘−1, 𝑃𝑇^{[𝑎∶𝑏]}_{𝑘∶𝑏−𝑎+1}>𝑡=1−^𝑏_𝑖=𝑎𝐹_𝑖(𝑡), (3.5) and for 𝑘>1 and 𝑏−𝑎≥𝑘, 𝑃𝑇^{[𝑎∶𝑏]}_{𝑘∶𝑏−𝑎+1}>𝑡=𝑃𝑇^{[𝑎∶𝑏−1]}_{𝑘−1∶𝑏−𝑎}>𝑡𝐹_𝑏(𝑡)+𝑃𝑇^{[𝑎∶𝑏−1]}_{𝑘∶𝑏−𝑎}>𝑡𝐹_𝑏(𝑡). (3.6)

Theorem 3.2. For 2𝑚≥𝑛, 𝑅_{𝑘,𝑚∶𝑛}(𝑡)=^{min(𝑛−𝑚,𝑘−1)}_𝑠=0𝑃𝑇^{[𝑛−𝑚+1∶𝑚]}_{𝑘−𝑠∶2𝑚−𝑛}>𝑡𝑅^∗_{()𝑠+1,𝑛−𝑚∶2𝑛−𝑚}(𝑡)−𝑅^∗_{()𝑠,𝑛−𝑚∶2𝑛−𝑚}(𝑡), (3.7) where 𝑅^∗_{𝑠,𝑛−𝑚∶2(𝑛−𝑚)}(𝑡) is the reliability of consecutive 𝑠-within-(𝑛−𝑚)-out-of-2(𝑛−𝑚):F system with components 1,…,𝑛−𝑚,𝑚+1,…,𝑛.

Proof. By the definition of 𝑆_𝑛,𝑚(𝑡), 𝑃𝑆_𝑛,𝑚(𝑡)<𝑘=𝑃^𝑚_𝑗=1𝑋_𝑗(𝑡)<𝑘,^𝑚+1_𝑗=2𝑋_𝑗(𝑡)<𝑘,…,^𝑛_{𝑗=𝑛−𝑚+1}𝑋_𝑗(𝑡)<𝑘. (3.8)
For 2𝑚≥𝑛, 𝑃𝑆_𝑛,𝑚(𝑡)<𝑘=_{𝑥1,…,𝑥𝑛−𝑚,𝑥𝑚+1,…,𝑥𝑛∈{}0,1}𝑃^𝑚_{𝑖=𝑛−𝑚+1}𝑋_𝑖(𝑡)<𝑚^∗,𝑋₁(𝑡)=𝑥₁,…,𝑋_𝑛−𝑚(𝑡)=𝑥_𝑛−𝑚,𝑋_𝑚+1(𝑡)=𝑥_𝑚+1,…,𝑋_𝑛(𝑡)=𝑥_𝑛, (3.9) where 𝑚^∗=min𝑘−𝑥₁−⋯−𝑥_𝑛−𝑚,𝑘−𝑥₂−⋯−𝑥_𝑛−𝑚−𝑥_𝑚+1,…,𝑘−𝑥_𝑚+1−⋯−𝑥_𝑛=𝑘−max𝑥₁+⋯+𝑥_𝑛−𝑚,𝑥₂+⋯+𝑥_𝑛−𝑚+𝑥_𝑚+1,…,𝑥_𝑚+1+⋯+𝑥_𝑛. (3.10)If 𝑆^∗_{2(𝑛−𝑚),𝑛−𝑚}(𝑡) denotes the scan statistic based on 𝑋₁(𝑡),…,𝑋_𝑛−𝑚(𝑡),𝑋_𝑚+1(𝑡),…,𝑋_𝑛(𝑡), then 𝑃𝑆_𝑛,𝑚(𝑡)<𝑘=^{min(𝑛−𝑚,𝑘−1)}_𝑠=0𝑃^𝑚_{𝑖=𝑛−𝑚+1}𝑋_𝑖(𝑡)<𝑘−𝑠,𝑆^∗_{2()𝑛−𝑚,𝑛−𝑚}(𝑡)=𝑠. (3.11) Thus the proof is completed by the independence of ∑^𝑛_{𝑖=𝑛−𝑚+1}𝑋_𝑖(𝑡) and 𝑆^∗_{2(𝑛−𝑚),𝑛−𝑚}(𝑡) and 𝑃^𝑚_{𝑖=𝑛−𝑚+1}𝑋_𝑖(𝑡)<𝑘−𝑠=𝑃𝑇^{[𝑛−𝑚+1∶𝑚]}_{𝑘−𝑠∶2𝑚−𝑛},𝑃𝑆>𝑡_{∗2()𝑛−𝑚,𝑛−𝑚}(𝑡)=𝑠=𝑅^∗_{()𝑠+1,𝑛−𝑚∶2𝑛−𝑚}(𝑡)−𝑅^∗_{()𝑠,𝑛−𝑚∶2𝑛−𝑚}(𝑡). (3.12)

Theorem 2.6 can be extended to any consecutive 𝑘-within-𝑚-out-of-𝑛:F system in the following way.

Theorem 3.3. Let 𝑇₁,𝑇₂,…,𝑇_𝑛 be independent and the hazard rate of 𝑇_𝑖 is 𝑟_𝑖(𝑡). For 1<𝑘≤𝑚≤𝑛, if lim_𝑡→∞𝑟_𝑖(𝑡)=𝜆_𝑖 and 𝜆₁>𝜆₂>⋯>𝜆_𝑛, then lim_𝑡→∞ℎ_{𝑘,𝑚∶𝑛}(𝑡)=min𝜆_𝑙1(1)+⋯+𝜆_{𝑙(1)𝑛−𝑧(𝑛,𝑚,𝑘)},𝜆_𝑙1(2)+⋯+𝜆_{𝑙(2)𝑛−𝑧(𝑛,𝑚,𝑘)},…,𝜆_𝑙1(𝑠)+⋯+𝜆_{𝑙(𝑠)𝑛−𝑧(𝑛,𝑚,𝑘)}, (3.13) where 𝑠 is the number of minimal path sets with 𝑛−𝑧(𝑛,𝑚,𝑘) elements and 𝑧(𝑛,𝑚,𝑘) is the maximum number of failed components such that the system can still work.

The number 𝑧(𝑛,𝑚,𝑘) has been derived in Eryilmaz and Kan [12] as𝑧()=𝑛𝑛,𝑚,𝑘𝑛−𝑚(𝑚−𝑘+1)if𝑛𝑛−𝑚𝑚()𝑛<𝑘,𝑘−11+𝑚if𝑛𝑛−𝑚𝑚≥𝑘. (3.14)

Example 3.4. Let 𝑛=7,𝑚=3, and 𝑘=2. Then 𝑧(𝑛,𝑚,𝑘)=3 and there is only one minimal path set with 𝑛−𝑧(𝑛,𝑚,𝑘)=7−3=4 elements, that is, 𝑠=1 and the corresponding minimal path set is {2,3,5,6}. Thus under the conditions of Theorem 3.3, we have lim_𝑡→∞ℎ_2,3∶7(𝑡)=𝜆₂+𝜆₃+𝜆₅+𝜆₆. (3.15)

Acknowledgment

The author would like to express gratitude to the referee for a thorough review and valuable comments that led to some improvements in this article.