We focus on the buffer-aided max link selection (MLS) relaying system in term of outage and delay performance over independent and non-identically distributed Nakagami-m fading channels. By using appropriate mathematical manipulation, the exact closed-form expressions of outage probability and average packet delay are achieved firstly, as well as the diversity and coding gains. The presented numerical results show that the non-identically distributed fading channels impose great loss in diversity gain for the MLS schemes. Moreover, the buffer-aided relaying systems are also unstable due to the fact that the average packet delays at all relays are different under this condition. Therefore, to improve the diversity and delay performance, we propose a novel weighted-based MLS (W-MLS) scheme. By employing weight vectors, the W-MLS scheme on one hand makes the equivalent powers of source-relay and relay-destination links equal. On the other hand, the W-MLS scheme also ensures that the equivalent power of relay-destination links is  greater than the one of source-relay links. As a result, comparing with the traditional MLS, the proposed W-MLS scheme can not only provide enough diversity and coding gains, but also enhance the stability of buffer-aided relaying systems due to that the average packet delays at all relays are entirely the same.