Analysis of R-out-of-N repairable systems: the case of phase-type distributions

2004 ◽  
Vol 36 (1) ◽  
pp. 116-138 ◽  
Author(s):  
Yonit Barron ◽  
Esther Frostig ◽  
Benny Levikson
Keyword(s):  
Repairable System ◽  
Repairable Systems ◽  
Regenerative Processes ◽  
Numerical Examples ◽  
Independent Components ◽  
Duality Results ◽  
Phase Type ◽  
Phase Type Distributions ◽  
Two Systems ◽  
Repair Facility

An R-out-of-N repairable system, consisting of N independent components, is operating if at least R components are functioning. The system fails whenever the number of good components decreases from R to R-1. A failed component is sent to a repair facility. After a failed component has been repaired it is as good as new. Formulae for the availability of the system using Markov renewal and semi-regenerative processes are derived. We assume that either the repair times of the components are generally distributed and the components' lifetimes are phase-type distributed or vice versa. Some duality results between the two systems are obtained. Numerical examples are given for several distributions of lifetimes and of repair times.

Analysis of R-out-of-N repairable systems: the case of phase-type distributions

2004 ◽  
Vol 36 (01) ◽  
pp. 116-138
Author(s):  
Yonit Barron ◽  
Esther Frostig ◽  
Benny Levikson
Keyword(s):  
Repairable System ◽  
Repairable Systems ◽  
Regenerative Processes ◽  
Numerical Examples ◽  
Independent Components ◽  
Duality Results ◽  
Phase Type ◽  
Phase Type Distributions ◽  
Two Systems ◽  
Repair Facility

An R-out-of-N repairable system, consisting of N independent components, is operating if at least R components are functioning. The system fails whenever the number of good components decreases from R to R-1. A failed component is sent to a repair facility. After a failed component has been repaired it is as good as new. Formulae for the availability of the system using Markov renewal and semi-regenerative processes are derived. We assume that either the repair times of the components are generally distributed and the components' lifetimes are phase-type distributed or vice versa. Some duality results between the two systems are obtained. Numerical examples are given for several distributions of lifetimes and of repair times.

Discrete Repairable Systems With External and Internal Failures Under Phase-Type Distributions

2009 ◽  
Vol 58 (1) ◽  
pp. 41-52 ◽  
Author(s):  
J.E. Ruiz-Castro ◽  
G. Fernandez-Villodre ◽  
R. Perez-Ocon
Keyword(s):  
Repairable Systems ◽  
Phase Type ◽  
Phase Type Distributions

Random Fuzzy Repairable Coherent Systems with Independent Components

2016 ◽  
Vol 24 (06) ◽  
pp. 859-872 ◽  
Author(s):  
Ying Liu ◽  
Xiaozhong Li ◽  
Yiying Zhang
Keyword(s):  
Fuzzy Theory ◽  
Probability Distributions ◽  
State Failure ◽  
Coherent System ◽  
Sufficient Data ◽  
Repairable Systems ◽  
Coherent Systems ◽  
Numerical Examples ◽  
Failure Frequency ◽  
Independent Components

Nondeterministic variables of certain distributions are employed to represent uncertain lifetimes and repair times in repairable systems, which are usually treated as stochastic variables and the probability distributions of the stochastic variables have crisp parameters. However, in practice, the parameters may not be precisely represented due to a lack of sufficient data. The main purpose of this paper is to apply the random fuzzy theory to the establishment of repairable coherent system models. Then the expressions of limiting availability and steady state failure frequency of the random fuzzy coherent systems are proposed, respectively. Finally, some numerical examples are given for illustration.

Research on the model of a multistate aggregated Markov repairable system

2019 ◽  
pp. 1748006X1988765
Author(s):  
Quan Zhang ◽  
Shihang Yu ◽  
Yang Han ◽  
Yanjun Li
Keyword(s):  
System Performance ◽  
The State ◽  
Theory And Practice ◽  
Process Theory ◽  
Repairable System ◽  
Repairable Systems ◽  
Numerical Examples ◽  
Markov Repairable System ◽  
Special Case ◽  
Theoretical Results

In theory and practice, system performance is one of the most important issues. Therefore, a series of indexes has been proposed for evaluating the system performance, such as availability. However, these indexes still cannot meet the variant requirements in the reliability and other fields. The purpose of the article is to develop some theoretical results that may be used in modeling the evolution of system performance. So, based on the aggregated stochastic process theory, some new indexes are introduced and established in Markov repairable systems. In this model, the state space is partitioned into working subset W and failure subset F. The system is regarded as stable if the state of system enters one subset, either W or F, at any instance and sojourns within the subset exceeding a given non-negative threshold [Formula: see text]. Otherwise, the system is regarded as unstable. Under these assumptions, the concepts of point-wise and interval-wise are proposed, and the computation formulae of two types of indexes are derived in the theory. Finally, a special case and a few of numerical examples are given to illustrate the results obtained in the paper.

scholarly journals A k-out-of-n reliability system with an unreliable server and phase type repairs and services: the (N,T) policy

2001 ◽  
Vol 14 (4) ◽  
pp. 361-380 ◽  
Author(s):  
Srinivas R. Chakravarthy ◽  
A. Krishnamoorthy ◽  
P. V. Ushakumari
Keyword(s):  
Performance Measures ◽  
Matrix Analytic Methods ◽  
Steady State Analysis ◽  
Numerical Examples ◽  
Unreliable Server ◽  
Failure Times ◽  
Vacation Time ◽  
Phase Type ◽  
Repair Facility ◽  
Random Amount

In this paper we study a k-out-of-n reliability system in which a single unreliable server maintains n identical components. The reliability system is studied under the (N,T) policy. An idle server takes a vacation for a random amount of time T and then attends to any failed component waiting in line upon completion of the vacation. The vacationing server is recalled instantaneously upon the failure of the Nth component. The failure times of the components are assumed to follow an exponential distribution. The server is subject to failure with failure times exponentially distributed. Repair times of the component, fixing times of the server, and vacationing times of the server are assumed to be of phase type. Using matrix-analytic methods we perform steady state analysis of this model. Time spent by a failed component in service, total time in the repair facility, vacation time of the server, non-vacation time of the server, and time until failure of the system are all shown to be of phase type. Several performance measures are evaluated. Illustrative numerical examples are presented.

Reliability evaluation of consecutive k-out-of-m: F balanced systems with a symmetry line

2021 ◽  
pp. 1748006X2110381
Author(s):  
Chen Fang ◽  
Lirong Cui
Keyword(s):  
System Reliability ◽  
System Structure ◽  
Finite Markov Chain ◽  
Finite Markov Chain Imbedding ◽  
Numerical Examples ◽  
Symmetry Line ◽  
Markov Chain Imbedding ◽  
Binary State ◽  
Phase Type ◽  
Phase Type Distributions

Based on some real backgrounds, a new balanced system structure, a consecutive k-out-of- m: F system with a symmetry line, is proposed in this paper. Considering different state numbers of a subsector, the new balanced system is analyzed under two situations respectively: the subsector with binary-state and the subsector with multi-state, while the multi-state balanced systems have not been studied in the previous research. Besides, two models are developed in terms of assumptions for the two situations, respectively. For this system, several methods, such as the finite Markov chain imbedding approach, the order statistics technique and the phase-type distributions, are used on the models. In addition to system reliability formulas, the means and variances of the system lifetimes under two models for different situations are given. Finally, numerical examples are presented to illustrate the results obtained in this paper.

Phase Type Distributions with Finite Support

2014 ◽  
Vol 30 (4) ◽  
pp. 576-597 ◽  
Author(s):  
V. Ramaswami ◽  
N. C. Viswanath
Keyword(s):  
Finite Support ◽  
Phase Type ◽  
Phase Type Distributions

scholarly journals The Algebraic Degree of Phase-Type Distributions

2010 ◽  
Vol 47 (03) ◽  
pp. 611-629
Author(s):  
Mark Fackrell ◽  
Qi-Ming He ◽  
Peter Taylor ◽  
Hanqin Zhang
Keyword(s):  
Lower Bound ◽  
Upper Bound ◽  
Polynomial Algorithm ◽  
Algebraic Degree ◽  
Nonnegative Matrices ◽  
Stieltjes Transform ◽  
Special Cases ◽  
Phase Type ◽  
Phase Type Distributions ◽  
The Matrix

This paper is concerned with properties of the algebraic degree of the Laplace-Stieltjes transform of phase-type (PH) distributions. The main problem of interest is: given a PH generator, how do we find the maximum and the minimum algebraic degrees of all irreducible PH representations with that PH generator? Based on the matrix exponential (ME) order of ME distributions and the spectral polynomial algorithm, a method for computing the algebraic degree of a PH distribution is developed. The maximum algebraic degree is identified explicitly. Using Perron-Frobenius theory of nonnegative matrices, a lower bound and an upper bound on the minimum algebraic degree are found, subject to some conditions. Explicit results are obtained for special cases.

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