A sufficient condition of the recurrence and a sufficient and ne- cessary condition of the transience for an irreducible and aperiodic Markvo chain are given in this paper. The theorem 4 and- theorem 5 proved by F, G. Foster [1] are just the particular cases of them. Finally, these results are applied to the embedded Markov chain associated with queueing process so as to resolve the problem of the state classification for M/G/1 queueing process with bulk service studied by Bailey [2].