Reliability Analysis And Replacement Polices Of A Repairable System Based On Phase-type

Reliability mathematical modeling is a classic model in stochastic operations research,has been widespread concern by the majority of scholars.However,on the reliability of the system in the process of modeling,geometric process of degradation of repairable system is one of our most commonly research object,people often assume that the lifetime,repair time of the system and the vacation time of the repairman follows exponential distribution or other typical distribution.The result was a severe narrowing the scope of the model.Aiming at the problem,on the premise of considering different cause system failure,phase-type distribution was used to modeling.We study reliability analysis of one or two components parallel repairable system based on geometric process and phase-type vacation,optimal replacement policy problem of single component repairable system based on geometric process and phase(PH)impact.The main contents are follows:Firstly,this system considers one unit repairable system with repairman's multiple vacation,where the working time follows geometric process with phase type(PH)distribution,and the repair time follows negative exponential distribution,and the vacation time follows a phase type distribution.When the system is operative,it may be subjected to two types of failures: one is due to Possion shocks,and the other one is due to wear-out.By establishing the quasi-birth-and-death process of the system,we derive the stationary distribution of the system.Using matrix analysis method,we derive some reliability quantities such as steady-state availability and the steady-state failure frequency of the system and present some numerical examples.Secondly,this system investigates a two-different component of parallel repairable system which has the remembering of the failure phase when the component is repaired.In this system,repairman's single vacation,and where the vacation time follows phase-type(PH)distribution is considered.When the component is operative,it may be subjected to two types of failures,and it is assumed that the component is not “as good as new”.Assume that the working time follows geometric process with PH distribution,and the repair time follows negative exponential distribution.By establishing the Markov process of the system and using matrix analysis method,we analyze the reliability of this repairable system and give some numerical examples corresponding reliability index.Finally,this system studies one unit system with repairman's single vacation which is subject to shocks from the outside when it is in the prosess of long-run The system may has two types failures.One is extremely Possion shocks,and the other is due to the age of the system.The every shock follows phase-type(PH)distribution.Once a shock exceeds the threshold value in the running of the system,the system will fail.After continuous failures,both the lifetime and the repairable time of the system follow geometric process with phase type(PH)distribution.However,the system can withstand the impact of the threshold value and the repairable time obeys the decreasing,increasing the negative exponential distribution of geometric process respectively.Under these assumptions,by using renewal process and geometric process theory,the explicit expression of the long-run expected cost of the system time is derived by considering the replacement policies N based on system failure number of the system.Finally,the optimal replacement policy is obtained by numerical examples.