In many physical systems, reliability evaluation, such as ones encountered in telecommunications, the design of integrated circuits, microwave relay stations, oil pipeline systems, vacuum systems in accelerators, computer ring networks, and spacecraft relay stations, have had applied consecutivek-out-of-nsystem models. These systems are characterized as logical connections among the components of the systems placed in lines or circles. In literature, a great deal of attention has been paid to the study of the reliability evaluation of consecutivek-out-of-nsystems. In this paper, we propose a new method to compute the reliability of consecutivek-out-of-n:F systems, withnlinearly and circularly arranged components. The proposed method provides a simple way for determining the system failure probability. Also, we write R-Project codes based on our proposed method to compute the reliability of the linear and circular systems which have a great number of components.

In a repairable consecutive $C(k,n:F)$ system, after the system operates for a certain time, some components may fail, some failed components may be repaired and the state of the system may change. The models developed in the existing literature usually assume that the state of the system varies over time depending on the values of n and k and the state of the system is known. Since the system reliability will vary over time, it is of great interest to analyse the time-dependent system reliability. In this paper, we develop a novel and simple method that utilizes the eigenvalues of the transition rate matrix of the system for the computation of time-dependent system reliability when the system state is known. In addition, the transition performance probabilities of the system from a known state to the possible states are also analysed. Computational results are presented to illustrate the applicability and accuracy of the proposed method.

The main motivation of this paper is to compute a reliability for a closed recurring water supply system with n water pumps in a thermo-electric plant by modelling to a repairable circular consecutive-2-out-of-n:F system. In a thermo-electric plant system, let us have n water pumps for pumping the water and steam expelled from a turbine to a cooling tower. These pumps are installed around the system and each pump must be powerful enough to pump water and steam to at least the next two consecutive pumps. If any two or more consecutive pumps in the system are failed, the system is failed. For this system, it is important to determine the reliability. First, we developed mathematical formulations for the state transition probabilities in the system by using the definition of generalized transition probability and the concept of critical component under the assumption that pumps have unequal failure rates. Then, using these formulations we derived the state transition probability matrix of the system. Finally, a special model is given to calculate the system reliability.

Operation principle of the engineering systems occupies an important role in the reliability theory. In most of the studies, the reliability function of the system is obtained analytically according to the structure of the system. Also in such studies the mean operating time of the system is calculated. However, the reliability function of some systems, such as repairable system, cannot be easily obtained analytically. In this case, forming Laplace-Stieltjes transform of the system can provide a solution to the problem. In this paper, we have designed a system which consists of two components that can be repairable with the aging property. Firstly, the Laplace-Stieltjes transform of the system is formed. Later, the mean operating time of the system is calculated by means of Laplace-Stieltjes transform. The system’s repair policy is evaluated depending on the geometric process. This property provides the aging of the system. We also provide special systems with different marginal lifetime distributions to illustrate the theoretical results in this study