In this paper, we propose a buffer state based relay selection scheme for a finite buffer-aided cooperative relaying system. Our proposed relay selection scheme selects a relay node based on both the channel quality and the buffer state of the relay nodes. The Markov chain model is adopted to analyze the buffer state transition properties and the outage probability of the proposed relay selection scheme is obtained in a closed-form expression. Our analytical results show that if the buffer size of each relay node is greater than 2, the proposed relay selection scheme can achieve full diversity. Furthermore, it is shown that our proposed relay selection scheme has lower average packet delay compared with the max-link relay selection scheme. In addition, the average packet delay of the proposed relay selection scheme does not increase as the buffer size increases.