In this paper we investigate some structure properties of the tail o-field and the invariant o-field of both homogeneous and nonhomogeneous Markov chains as representations for asymptotic events, descriptions of completely nonatomic and atomic sets and global characterizations of asymptotic o-flelds. It is shown that the Martin boundary theor.y can provide a unified approach to the asymptotic o...

The most commonly used models for the failure process of a repairable system are nonhomogeneous Poisson processes, corresponding to minimal repairs, and renewal processes, corresponding to perfect repairs. This paper studies a more general model for recurrent events, called the Trend-Renewal Process (TRP). This is a time transformed renewal process, having both the ordinary renewal process and ...

in this paper at rst, a history of mathematical models is given.next, some basic information about random variables, stochastic processesand markov chains is introduced. as follows, the entropy for a discrete timemarkov process is mentioned. after that, the entropy for sis stochastic modelsis computed, and it is proved that an epidemic will be disappeared after a longtime.

In a networked control system scenario, the packet dropout is usually modeled by time-invariant (homogeneous) Markov chain (MC) process. However, from practical point of view, probabilities loss can vary in time and/or probability parameter dependency. Therefore, to design fault detection filter (FDF) implemented semi-reliable communication network, it important consider variation network param...