可修系统中半马氏过程的稳态分布

The Steady-distribution of Semi-Markov Process in the Repairable System

众所周知,可修系统是可靠性理论中讨论的一类非常重要的系统,也是可靠性数学主要研究对象之一,研究可修系统的主要数学工具是马氏理论.当构成系统各部件的寿命分布和故障后的修理时间分布,及其出现的有关分布均为指数分布时,只要适当的定义系统的状态,这样的系统总可以用马氏过程来描述.大部分学者为了方便,均是在马氏框架下研究问题的.但是在实践中经常遇到部件的寿命或修理时间分布不是指数分布的情形,这时可修系统所构成的随机过程是半马氏过程,用现有的马氏理论无法解决相关问题.目前,关于半马氏的理论研究的研究又很少,基于此,针对半马氏的随机模型给出了与马氏理论相平行的稳态分布的求解方法.

It is well known, a repairable system is a kind of important system in the theory of reliability and one of the main research object in the mathematical theory of reliability. The theory of Markov is major mathematical tool. When all distributions of involved components of the system are exponential distribution, as long as we properly define the state space of the system, the system can be described by a Markov process. But, in fact, the distributions of the operating and repair times are not exponential, the stochastic process is a semi-Markov process. The theory of Markov can＇ t solve the relevant problems. Everyone should further discuss Semi-Markov. At present, the theories of semi-Markov process are rarely studied. Based on this, the paper gives the method of the steady-distribution of the parallel with Markov theory in the Semi-Markov model.