Generally, it is not easy to find the exact tail-estimation for the distribution of the minimum value in partial sum sequence of a stationary ergodic Markov chain.

In this paper, the completely ergodic Markov chain with rewards is considered. A definition of reliability in probability form is given, based on the frequency form as used in engineering.

This paper examines port queuing system with single berth. For its imbedded Markov chain, the author gives a necessary and sufficient condition under which the system is ergodic.