Reliability and Failure Probability Functions of the m-Consecutive-k-out-of- n: F Linear and Circular Systems

The m-consecutive-k-out-of-n: F linear and circular system consists of n sequentially connected components; the components are ordered on a line or a circle; it fails if there are at least m non-overlapping runs of consecutive-k failed components. This paper proposes the reliability and failure probability functions for both linearly and circularly m-consecutive-k-out-of-n: F systems. More precisely, the failure states of the system components are separated into two collections (the working and the failure collections); where each one is defined as a collection of finite mutual disjoint classes of the system states. Illustrative example is provided.


Introduction:
The reliability of engineering systems has become a pivotal issue through the design phase; the daily life of people depends on the satisfactory functioning of these systems. The engineers started with few components to design these systems, and by the time, the systems have become highly complex and sophisticated, and may consist of hundreds or perhaps thousands of components. In this context, the engineers developed theories for such systems, and applied the available results for all types of systems, including system reliability, optimal system design, component reliability importance, and reliability bounds.
The family of the consecutive systems had a wide area of interest for many engineers through the last few decades, the members of this family have been used to model and create optimal designs for many engineering systems, such as, the telecommunication networks, spacecraft relay stations, vacuum system of the electron accelerator, oil pipeline systems, microwave stations, photographing of a nuclear accelerator etc. where the "consecutive failures" is the basic condition for all these systems to shut down, which makes the system more reliable than the series system and less expense than that in the parallel system.
The first member in this family which appeared in literature is the consecutive-k-out-of-n: F system, Kontoleon (1) mentioned it firstly, the system consists of n connected components (arranged in a line (a circle)), and it fails if at least k consecutive components fail. Nashwan (2) proposed an algorithm to compute the sample space of all failure states of the components for the consecutive k-outof-n: F linear (circular) system, and classified them into a working and failure spaces, also Gökdere et al. (3) proposed a new method to compute the reliability of consecutive k-out-of-n: F linear and circular systems using combinatorial approach. Shingyochit and Yamarnoto (4), and Cai et al. (5) found an optimal component arrangement where n components are assigned to n positions to maximize the system reliability. The maintenance problem was discussed by Endharta et al. (6); they evaluated the maintenance policy by comparing the expected cost rate of the proposed policy with those of corrective maintenance and age policy maintenance. Cai et al. (7) discussed the maintenance problem through exchanging the position of components and improving the reliability of components.
Some researchers studied the combination with the consecutive k-out-of-n: F system models, Mohammadi et al. (8)  linear consecutive k-out-of-n: F system in which each component has a constant failure probability. Dui et al. (9) computed the marginal reliability importance and joint reliability importance in k-outof-n: F systems and consecutive k-out-of-n: F systems for some situations. The latest researches that discussed the various generalizations and applications of the system were conducted in (10)(11)(12)(13)(14)(15)(16).
Another important member of the consecutive systems family is the m-consecutive-kout-of-n: F linear (circular) system (L(C) (m,k,n) system), it is more complicated. Griffith (17) was firstly introduced the system, the system that consists of n connected components linearly (circularly), and fails if there are at least m nonoverlapping runs of consecutive k failed components. Actually, it is a generalization of the consecutive-k-out-of-n: F system when m=1, and for k=1 it will be m-out-of-n: system. Such system model is applied in the bank automatic payment system, infrared detecting system, inspection in production line and quality control, as well as the above-mentioned applications.
Papastavridis (18) and Ghoraf (19) studied the L(m,k,n) system and provided a recursive algorithm for non-equal components reliabilities, Papastavridis (18) proposed the exact failure probability function, while Makri and Philippou (20) used the multinomial coefficients to obtain the exact reliability of the L(C) (m,k,n) system. Godbole (21) established Poisson approximations for the reliability of the system, Eryilmaz et al. (22) studied the reliability of the system with exchangeable components, Eryilmaz in (23) obtained closed expression for the system signature, and Gharof (24) gave a recursive algorithm to compute the reliability of the system.
The repair problem was discussed by Tang (25), he studied the repairable L(m,k,n) system with l repairs, and when the exponential distribution represents the working and the repair time of each component, where the repair is perfect, Sheng and Gen (26) computed the reliability of the repairable L (m,k,n) system, they assumed that both working lifetime and repair lifetime of each component were also an exponentially distributed.
Agarwal and Mohan (27) used the Graphical Evaluation and Review Technique (GERT) analysis to compute the reliability of the L(C) (m,k,n) system for both independent and identically distributed components, and (k-1)-step Markov dependent components. Ghoraf (28) used Markov-dependent components to compute the reliability of the L(C) (m,k,n) system. The combination with the L(C) (m,k,n) system models attracted the attention of Mohan et al. (29), they studied the combination with the consecutive k-out-of-n: F system, and used the GERT Analysis and applied the model on a various complex systems such as infrared detecting and signal processing, and bank automatic payment systems, after that, Eryilmaz (30) derived the reliability of the system using a combinatorial equation for the number working states, Boushaba and Benyahia (31) computed the reliability and importance measures for the combination with consecutive k-out-of-n: F system with nonhomogeneous Markov-dependent components, while Gera (32) evaluated the reliability of the combination of L(C) (m 1 ,k 1 ,n) and L(C) (m 2 ,k 2 ,n) system. Further investigations on reliability, optimal system design, component reliability importance, applications, and generalization models of the L(C) (m,k,n) system are in (33)(34)(35)(36).
The main contribution of this paper is to compute the exact reliability function and the exact failure probability function of the L(C) (m,k,n) system. In fact, it developed the classification technique of Nashwan (2) to determine the working and failure states of the L(C) (m,k,n) system, which is the main stone to find the reliability function and failure probability function of the system, which is arranged in the paper as follows: The second section derives the working and the failure states of the C(m,k,n) system only, and determines the necessary conditions of its failure states. Thereafter, Section 3 withdraws these conditions on the linear type (L(m,k,n) system). Finally, the last section introduced a mathematical algorithm to find the reliability and the failure probability functions of the L(C) (m,k,n) system. Through all, this paper is assuming that (All components of the system are mutually statistically independent, and they are either "failed" or "working" states, as well as the system). Table 1 presents the signs and notations frequently use in this paper.
The same as   RX, but the symbol used to distinguish that the system is in the failure state.

The circular m-consecutive-k-out-of-n: system
The C(m,k,n) system consists of n components (connected sequentially in a circle), it fails when the event "at least k consecutive failed components" occurs disjointedly at least m times, and denotes the components indices by 1 n I ,    (for simply X=1348) means that, all components are in the working state except the components with indices the first, third , fourth, and the eighth components, more precisely X is a failed set, it has 2 disjointed subsets, each one consists of 2 consecutive indices (failed components), the first subset is the components with indices 1and 8, while the second subset is the components with indices 3 and 4). Nashwan (2) defined an equivalence relations for any circular system using the bijection function 11 : represents the C(m,k,n) system, such that mk j n , takes the rotation of the set X,

The linear m-consecutive-k-out-of-n: system
Consider the L(m,k,n) system, and connect the first and last components, treat the system as a C(m,k,n) system. Actually, this connection generates additional failure states, i.e. the failure collection (sets) of the L(m,k,n) system ,, It is worth mentioning that, the set X may be a failure set in the circular type, but it is a working set in the linear type, for example, in the C (2,3,9)
For j = 0, 1, 2,…,5 all states are in a working states  the failure collection of the L (2, 3,9) are the 2,3,9 C  without all underlined elements.

Open Access
Baghdad Science Journal This paper aimed to compute the exact reliability and failure probability functions for the m-consecutive-k-out-of-n: F linear and circular system. The linear system is treated as a special case of the circular one. An algorithm is suggested to classify the states of the m-consecutive-k-out-ofn: F linear and circular system into a working and a failure collections, moreover these collections are consisting of a finite pairwise disjointed classes, where the reliability and the failure probability functions of the m-consecutive-k-out-of-n: F linear and circular system are the summations of reliabilities of these working and failure collections respectively.