The Stochastic Comparison And Reliability Study Of A Kind Of Warm Standby System

In reliability design,standby redundancy techniques are often used to increase the reliability of complex systems.Generally,there are three standby systems: cold standby,hot standby,and warm standby.Cold standby means that the component has zero failure rate in standby state.Hot standby means that both the standby component and the work component operate in a normal environment.Warm standby is between cold standby and hot standby.When the component is in standby state,it works in a mild environment.When the normal operating component fails,it immediately switches to work under normal environment.Warm standby system have the effect of balancing energy loss,and have a wide range of applications in many fields,such as power systems,wireless sensor systems,and gas supply systems,etc.This paper mainly considers the reliability and some stochastic properties of a type of warm standby system.The main content has three parts.In the first part,we consider the warm standby repairable system which consists of a single warm standby component,two normal operating components and a maintenance worker.Assuming that components lifetime,standby lifetime and repair time of the system are independently distributed as the exponential distribution,the Markov model is established by analyzing the state of the system applying.We obtain the transition probability function of the system between different states and transfer rate matrix.Thus we calculate the Laplace transform expressions of instantaneous availability and reliability,as well as several reliability indicators such as the steady state availability of the system at the time t ? ?.In the second part,firstly,we discuss the optimal allocation problem of a single warm standby component in a parallel system which consists of two separate components.It is possible to improve the performance of the system by assigning the warm standby component to a working component with larger reversed hazard rate order or likelihood ratio order.Secondly,comparing the stochastic performance of the parallel or series system and the warm standby system,it is concluded that the performance of the warm standby system is superior to that of the parallel or series system in the sense of usual stochastic order.Finally,we study the stochastic properties of a warm standby system which consists of n normal operating components and m warm standby components.When the warm standby components in the system fail at the k time,stochastic relationship between the component lifetime and the number of normal operating components remaining in the system is obtained.And we demonstrate the above conclusions by specific numerical examples.In the third part,we consider the warm standby system which consists of a single warm standby component and a single operating component.Under the condition that the component lifetime of the system are independently distributed as the exponential distribution and Weibull distribution,we estimate the component failure rate and the parameters in the distribution function by using the method of maximum likelihood estimation and Bayes estimation.Then using the conditional probability formula,the reliability function of the conditional residual lifetime of surviving components in a system is deduced.And the reliability function of residual lifetime and inactivity time of the system is derived.The stochastic order relationship between component lifetime and system residual lifetime(inactivity time)is established.