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The purpose of this paper is to explore the reliability and sensitivity analysis of a system with M primary units, W standby units, and one repair facility when switching to primary units may fail.

Failure times of primary and standby units are assumed to have exponential distributions, and repair times of the failed units are also assumed to have an exponential distribution. Different failure rates and switching failure probabilities are given depending on the readiness states of standby units, designated hot, warm, or cold. The Laplace transform technique is used to transform a set of ordinary differential equations to a set of equations. After finding the solution, we can obtain the desired measures in the time domain by using the inverse Laplace transform.

Expressions for system reliability and mean time to failure (MTTF) are derived. Sensitivity analysis of the system reliability and the MTTF with respect to system parameters are investigated.

This paper presents the first time that a contour of the MTTF with respect to standby states has been obtained, which is quite useful for the decision makers.

The internet of things and just-in-time are the embryonic model of innovation for the state-of-the-art design of the service system. This paper aims to develop a fault-tolerant machining system with active and standby redundancy. The availability of the fault-tolerant redundant repairable system is a key concern in the successful deployment of the service system.

In this paper, the authors cogitate a fault-tolerant redundant repairable system of finite working units along with warm standby unit provisioning. Working unit and standby unit are susceptible to random failures, which interrupt the quality-of-service. The system is also prone to common cause failure, which tends its catastrophe. The instantaneous repair of failed unit guarantees the increase in the availability of the unit/system. The time-to-repair by the single service facility for the failed unit follows the arbitrary distribution. For increasing the practicability of the studied model, the authors have also incorporated real-time machining practices such as imperfect coverage of the failure of units, switching failure of standby unit, common cause failure, reboot delay, switch over delay, etc.

For deriving the explicit expression for steady-state probabilities of the system, the authors use a supplementary variable technique for which the only required input is the Laplace–Stieltjes transform (LST) of the repair time distribution.

For complex and multi-parameters distribution of repair time, derivation of performance measures is not possible. The authors prefer numerical simulation because of its importance in the application for real-time uses.

The stepwise recursive procedure, illustrative examples, and numerical results have been presented for the diverse category of repair time distribution: exponential (M), n-stage Erlang (Ern), deterministic (D), uniform (U(a,b)), n-stage generalized Erlang (GE_[n]) and hyperexponential (HE_[n]).

Concluding remarks and future scopes have also been included. The studied fault-tolerant redundant repairable system is suitable for reliability analysis of a computer system, communication system, manufacturing system, software reliability, service system, etc.

As per the survey in literature, no previous published paper is presented with so wide range of repair time distribution in the machine repair problem. This paper is valuable for system design for reliability analysis of the fault-tolerant redundant repairable.

The purpose of this paper is to examine the cost/benefit (C/B) analysis of four configurations for a repairable system with two primary components/units and one standby.

The four configurations are set to the status of the detection and switching failure of standby, as well as the possible reboot of failed units. The time to failure for each of the primary and standby is assumed to follow an exponential distribution. The time to repair and the time to reboot is assumed to have a k‐stage Erlang distribution. The paper develops the explicit expressions of the mean time to failure (or MTTF) and the steady‐state availability (or A) for four various configurations and performed some comparative analysis. Based on the C/B criterion, comparisons are made for specific values of distribution parameters and of the costs of the units. The four various configurations for a repairable system are ranked by using MTTF, A and C/B, where B is either MTTF or A.

Although it is uncertain which configuration is the optimal one among the four ones, the paper provides much comparative information to manager and manufacturers. Managers can use these results to choose the best configuration according to the used data of parameters and selections of the weight of MTTF or Cost/MTTF.

This paper shows a comparative analysis for a two‐unit online repairable system with one standby under four different configurations. It is the first discussion of comparable work on reliability and availability models for redundant repairable systems in which the units are characterized by detection, switching failure and reboot.

This paper aims to deal with the estimation of the empirical Bayesian exact confidence limits of reliability indexes of a cold standby series system with (n+k−1) units under the general progressive Type II censoring scheme.

Assuming that the lifetime of each unit in the system is identical and independent random variable with exponential distribution, the exact confidence limits of the reliability indexes are derived by using an empirical Bayes approach when an exponential prior distribution of the failure rate parameter is considered. The accuracy of these confidence limits is examined in terms of their coverage probabilities by means of Monte-Carlo simulations.

The simulation results show that accuracy of exact confidence limits of reliability indexes of a cold standby series system is efficient. Therefore, this approach is good enough to use for reliability practitioners in order to improve the system reliability.

When items are costly, the general progressive Type II censoring scheme is used to reduce the total test time and the associated cost of an experiment. The proposed method provides the means to estimate the exact confidence limits of reliability indexes of the proposed cold standby series system under this scheme.

The application of the proposed technique will help the reliability engineers/managers/system engineers in various industrial and other setups where a cold standby series system is widely used.

The purpose of this paper is to analyze a parallel system consisting of n dependent components with lifetimes following Weibull distribution. FGM Copula in multivariate case is used to generate the reliability function of the original system. A reduction method is introduced to improve system reliability. Other methods of hot, cold and warm duplication are established to improve system reliability. An application is introduced to show the results and compare between different improvement methods.

In this paper, a study of a parallel system consisting of n dependent and non-identical components is introduced. Reliability function of the original system is derived by using the concepts of copula, subject to Weibull distribution. Reliability function of the original system is improved according to reduction, hot duplication, warm and cold duplication methods. Reliability equivalence factors are introduced to compare between different system designs. Numerical illustration and real-time data application are discussed to show the results obtained in this paper.

Copulas can be used to model the reliability of systems with dependent units.

This paper is original. Improvement of the reliability of dependent systems is not discussed in literature. Copula is a useful tool to analyze the reliability of dependent systems. The introduced model is considered as a generalization of the models discussed in literature.

The purpose of this paper is to propose a procedure to construct the membership functions for a one-unit repairable system, which has both active and standby redundancy. The coverage factor is the same for the operating and standby unit failure.

The α-cut approach is used to extract a family of conventional crisp intervals from the fuzzy repairable system for the desired system characteristics. This can be determined with a set of non-linear parametric programing using the membership functions.

When system characteristics are governed by the membership functions, more information is provided to use by management. On the other hand, fuzzy theory is applied for the redundant system; therefore, the results are more useful for designers and practitioners.

Different from other studies, the authors’ model provides more accurate estimation compared to uncertain environments based on fuzzy theory. The research would help managers and manufactures to make a better decision in order to have the optimal maintenance strategy based on the desired mean time to failure and availability of the systems.

Profit analysis of a two non‐identical unit cold standby system model with mutual changeover of the units is carried out in this paper. With mutual changeover of the unit, the operating unit, after functioning for some random amount of time, becomes standby to take rest, and the standby unit becomes operative. The failure and repair times of each unit are jointly distributed as bivariate exponential (BVE) with different parameters. Various measures of system effectiveness useful to system engineers and designers are obtained by using the regenerative point technique. Behaviour of the mean time to system failure (MTSF) and availability have also been studied graphically.

The purpose of this paper is to present a sensitivity analysis of fault-tolerant redundant repairable computing systems with imperfect coverage, reboot and recovery process.

In this investigation, the authors consider the computing system having a finite number of identical working units functioning simultaneously with the provision of standby units. Working and standby units are prone to random failure in nature and are administered by unreliable software, which is also likely to unpredictable failure. The redundant repairable computing system is modeled as a Markovian machine interference problem with exponentially distributed failure rates and service rates. To excerpt the failed unit from the computing system, the system either opts randomized reboot process or leads to recovery delay.

Transient-state probabilities have been determined with which the authors develop various reliability measures, namely reliability/availability, mean time to failure, failure frequency, and so on, and queueing characteristics, namely expected number of failed units, the throughput of the system and so on, for the predictive purpose. To spectacle the practicability of the developed model, a numerical simulation, sensitivity analysis and so on for different parameters have also been done, and the results are summarized in the tables and graphs. The transient results are helpful to analyze the developing model of the system before having the stability of the system. The derived measures give direct insights into parametric decision-making.

The conclusion has been drawn, and future scope is remarked. The present research study would help system analyst and system designer to make a better choice/decision in order to have the economical design and strategy based on the desired mean time to failure, reliability/availability of the systems and other queueing characteristics.

Different from previous investigations, this studied model provides a more accurate assessment of the computing system compared to uncertain environments based on sensitivity analysis.

The present paper analyzed a model consisting of one unit with a warm standby unit where the main unit has three states: up, degraded and down.

The semi-Markov model under the regenerative method is used to construct the mathematical model for the system.

The effectiveness measures of the system are discussed such as availability, reliability, steady-state availability and mean time to system failure. The life and repair times of the system units are assumed to be discrete follow discrete Weibull distribution. Also, the parameters of the discrete Weibull distribution are assumed to be fuzzy with bell-shaped membership function. An application is introduced to show the results obtained for the system and the profit of the presented model.

Rarely papers in literature treated the topic of the discrete-time semi-Markov process using a regenerative point technique.

Presents the analysis of a two‐unit cold standby system in which the standby unit takes a random amount of time for operation whenever the operative unit fails. Each unit is first repaired by the assistant repairman and is then taken up for post‐repair if necessary. The failure and repair times of each unit are assumed to be correlated and their joint density is taken as bivariate exponential. Uses regenerative point technique to obtain various reliability characteristics of interest. Studies the behaviour of steady‐state availability through graphs. Verifies earlier results.