Reliability Modeling And Analysis Of Repairable (k₁,k₂)/n(G) Systems Based On Phase-type Distribution

In some engineering systems,the reliability and availability of the system can be improved by using the way of redundancy in general.As a typical redundant structure system,k-out-of-n:G system is widely used in practical systems.The reliability problems of k-outof-n:G repairable systems with identical components have been investigated extensively by many researchers,but practical systems often have different types of components.Based on the k-out-of-n:G system,this paper establishes the reliability mathematical model of(k1,k2)-out-of-n:G system with two types of components,considers the vacation strategy of repairman and the retrial characteristics of components,and analyzes and calculates some main reliability indexes of the model.Firstly,the system reliability for a(k1,k2)-out-of-n:G repairable system considering multiple vacations is analyzed.It is assumed that the working times of two different types of components are distributed exponentially with different parameters,and the repairman implements multiple vacation strategy and the vacation time follows a phase-type distribution.By using Markov process theory and Cramer's rule,the steady availability and the mean time to the first failure of the system are obtained.The rationality of the model is verified by a numerical example.Under different vacation time,the relative sensitivity and overall sensitivity of the mean time to the first failure to the system parameters are analyzed.When other conditions are the same,the transient availability and reliability of the system with or without vacation policy are compared and analyzed.Secondly,the reliability of the(k1,k2)-out-of-n:G repairable retrial system is analyzed.It is assumed that the working time and retrial time of type 1 and type 2 components follow exponential distributions,and the repairman's repair time follows a phase-type distribution.The system reliability measures are obtained using Markov process theory and matrix analysis method.The rationality of the model is verified by a numerical example,and the influence of parameters on the steady availability and the mean time to first failure are obtained.Under different repair time,the relative sensitivity and overall sensitivity of the mean time to first failure to each parameter are analyzed.The effects of retrial characteristics on the transient availability and reliability of the system are compared.Finally,this paper discusses the reliability of(k1,k2)-out-of-n:G repairable system with retrial and working vacation.It is assumed that the working time and retrial time of the two types of components follow different exponential distributions,and the vacation time and repair time of the repairman follow phase-type distributions.The steady-state availability,the mean time to first failure and steady-state failure frequency of the system are obtained by using Markov process theory,Cramer's rule and the properties of phase-type distribution.A numerical example is used to verify the correctness and rationality of the model.In terms of reliability measures and benefit,the(k1,k2)-out-of-n:G repairable system with retrial and working vacation is compared with(k1,k2)-out-of-n:G repairable retrial system.