Reliability Modeling on Consecutive-<inline-formula> <tex-math notation="LaTeX">$k_r $</tex-math> </inline-formula>-out-of-<inline-formula> <tex-math notation="LaTeX">$n_r $</tex-math> </inline-formula>:F Linear Zigzag Structure and Circular Polygon Structure

Abstract

We propose the consecutive-k r -out-of-n r :F linear zigzag structure and circular polygon structure, which extend the consecutive-k-out-of-n:F system model. By employing the finite Markov chain imbedding approach, we have proven the equations for calculating the reliability of consecutive-k-out-of-n:F system with different patterns given the state of the first and(or) the last component(s). Based on those reliability equations, we provide the rules for obtaining the reliability of the linear zigzag structure and circular polygon structure. Numerical examples are presented as an illustration of our new system model and demonstrate the use of the equations and rules proposed in this paper.Index Terms-Birnbaum importance, circular polygon structure, consecutive-k-out-of-n:F system, finite Markov chain imbedding approach (FMCIA), joint reliability importance (JRI), linear zigzag structure.

Efficient reliability computation of consecutive-k-out-of-n: F systems with shared components

“…The reliabilities of SC=lin=con=k=n : F and SC=cir=con=k=n : F systems, are derived by employing the finite Markov chain imbedding approach for statistically independent and Markovdependent cases. These provide extensions of known results for the independent case given by Lin et al 16 and Yin and Cui 6 ; Unified and explicit reliability formulas are derived for the considered linear and circular systems by redefining the state space of the Markov chain, instead of listing all disjoint cases, which makes it to be computationally superior to exhaustive and sum of disjoint products methods; Table 1. Differences between the method developed here and those of Lin et al 16 and Yin and Cui.…”

“…Lin et al 16 Yin and Cui The method developed here is simpler and more efficient than those of Lin et al 16 and Yin and Cui, 6 which involve too many cases and the corresponding reliability formulas are far too complex;…”

“…15 Recently, many researchers discussed some systems considering shared components as the main characteristic. 6,[16][17][18][19][20] For example, Coolen-Maturi et al 19 introduced a joint survival signature for multiple systems with multiple types of components and with some components being shared by the systems. Considering several coherent systems with some shared components, with independent and identically distributed lifetimes, Zarezadeh et al 20 made use of joint signature to study the joint reliability of the two coherent systems and specifically obtained a pseudo-mixture representation.…”

“…17,18,[25][26][27] A survey of FMCIA is due to Cui et al 28 This method is highly useful as a tool for system reliability evaluation. The FMCIA is closely connected with the study of scans and runs, as seen in the books of Balakrishnan and Koutras, 29 Fu and Lou, 30 and Balakrishnan et al 31 Based on an engineering application, Lin et al 16 presented a consecutive-k r -out-of-n r : F linear zigzag structure as a system that consists of m subsystems, each of which is treated as a linear consecutive-k r -out-of-n r : F subsystem (r = 1, 2, . .…”

“…The shared components are railway platforms where railroad tracks enter and exit the station. Moreover, in real-life applications, there are such systems, for example, the highway system (street lights at intersections can be seen as shared components), the fire monitoring system, 16 and so on. Some systems with shared components can maximize resource utilization and reduce cost.…”