For stochastic models, ergodicity and the decay rate of stationary distribution in tail are two fundamental issues. They are usually studied separately since their concepts are obviously different. In this paper, we study the waiting time process of GI/G/1 queuing system in the two aspects. We shall give that geometric ergodicity, the geometric decay of stationary distribution in tail, and the geometric decay of the service distribution in tail are equivalent. Then we shall prove that l_-ergodicity, (l-1)_-th declay of stationary distribution in tail, and the l_-th decay of the service distribution in tail are equivalent. Finally, we prove that it is not strong ergodicity.