Abstract:
This paper considers a simple repairable system with a warning device and a repairman who can have delayed-multiple vacations. By Markov renewal process theory and the probability analysis method, the system is first transformed into a group of integrodifferential equations. Then, the existence and uniqueness as well as regularity of the system dynamic solution are discussed with the functional analysis method. Further, the asymptotic stability, especially the exponential stability of the system dynamic solution, is studied by using the strongly continuous semigroup theory or semigroup theory. The reliability indices and some applications (such as the comparisons of indices and profit of systems with and without warning device), as well as numerical examples, are presented at the end of the paper.

Abstract:
This paper studies the Gaver's parallel repairable system attended by a cold standby unit and a repairman with multiple vacations.It is assumed that the operating time of units has an exponential distribution,while the repair-time and the vacation time have general continuous distributions.Using vector Markov process theory and Laplace transform method,we obtain the Laplace transform of the reliability,the mean time to first failure,the steady-state availability and the steady-state failure frequency and so on.In addition,we also investigate the parameters' effect on the steady-state availability by numerical comparison and analyze the benefit of the system.

Abstract:
We study a series-parallel repairable system consisting of three units with multiple vacations of a repairman. We first show that all points on the imaginary axis except zero belong to the resolvent set of the operator and zero is an eigenvalue of the operator, and then we prove that the semigroup generated by the operator is irreducible. By combining these results with our previous result we deduce that the dynamic solution of the system converges strongly to its steady-state solution. Thus we obtain asymptotic stability of the dynamic solution of the system.

Abstract:
A model of N-unit series repairable systems with a repairman doing other work is studied and the impact on the system reliability because of a replaceable facility is also considered. It is assumed that the life of each unit, the life of the facility are exponentially distributed, the repair time of the unit, the replace time of the facility and the work time of the repairman outside the system are all generally distributed. By using approach of supplementary variables and method of generalized Markov progress, some important reliability indicates of the system and the facility are obtained.

Abstract:
In this paper, we discussed the reliability of an N-unit series repairable system with the repairman doing other work and priority in repair. In this system, it is assumed that the working time distributions of the n components and the arrival interval time distributions of the customers which are out of the system are both exponential and the components in system is given priority in repair. It is also assumed that the repair time distributions of the n components and the service time distributions of the customers are both general continuous distributions. After repair, components are “as good as new”. Under these assumptions, using a supplementary variable technique and Laplace transform technique, some important reliability indices such as the system availability, the idle probability of the repairman and the rate of service for customers are derived. Our problem is to determine whether or not given the priority to components in repairing such that the benefit of the system is maximized.

We investigate Gaver’s parallel system
attended by a cold standby unit and a repairman with multiple vacations. By
using C0-semigroup theory of linear operators in the functional analysis, we
prove well-posedness and the existence of the unique positive dynamic solution
of the system.

We investigate Gaver’s parallel system
attended by a cold standby unit and a repairman with multiple vacations. By
analysing the spectral distribution of the system operator and taking into account
the irreducibility of the semigroup generated by the system operator we prove
that the dynamic solution converges strongly to the steady state solution. Thus
we obtain asymptotic stability of the dynamic solution of the system.

We investigate a series-parallel repairable system consisting of three-unit with multiple vacations of a repairman. By using C0-semigroup theory of linear operators in the functional analysis, we prove that the system is well-posed and has a unique positive dynamic solution.

Abstract:
To study one unit repairable system with single repairman vacation,this paper proposes a kind of new maintenance and replacement model.Supposing that the system is repairable,under that the consecutive repair time of the system constitute an increasing geometric process,while the successive survival time of the system and the repairman every vacation time constitute a decreasing geometric process stochastically,we consider two kinds of replacement policies based on the working age T and the failure number N for the system and choose the long-run expected profit per unit time as objective function,and establish mathematic models by using renewal process and geometric process theory,the explicit expressions of the objective function are respectively derived.Under the some conditions,we derive that the policy N is better than the policy T and validate the effectiveness of the method through a numerical example also.Finally,we discuss the results.

Abstract:
This paper considers a one unit repairable systems with single repairman va-cation. If the system fails the repair starts immediately unless the repairman is in vaca-tion time, in which case a waiting repairable time is needed. So that the system may be in one of three states: the working state, the waiting repair state and the being repaired state. By using the total probability decomposition and the tool of the Laplace trans-form, the following reliability problems are discussed: 1) the reliability, pointwise a-vailability and the stead-state availability ; 2) the expected failure number during a gi-ven (0,t] and the stead-state failure frequency. Some important reliability results such as the pointwise availability and the stead-state availability are obtained.