Heavy-tailed (HT) traffic (e.g., the Internet and multimedia traffic) fundamentally challenges the validity of classic scheduling algorithms, designed under conventional light-tailed (LT) assumptions. To address such a challenge, this paper investigates the impact of HT traffic on delay-based maximum weight scheduling (DMWS) algorithms, which have been proven to be throughput-optimal with enhanced delay performance under the LT traffic assumption. First, it is proven that the DMWS policy is not throughput-optimal anymore in the presence of hybrid LT and HT traffic by inducing unbounded queuing delay for LT traffic. Then, to solve the unbounded delay problem, a delay-based maximum power-weight scheduling (DMPWS) policy is proposed that makes scheduling decisions based on queuing delay raised to a certain power. It is shown by the fluid model analysis that DMPWS is throughput-optimal with respect to moment stability by admitting the largest set of traffic rates supportable by the network, while guaranteeing bounded queuing delay for LT traffic. Moreover, a variant of the DMPWS algorithm, namely the IU-DMPWS policy, is proposed, which operates with infrequent queue state updates. It is also shown that compared with DMPWS, the IU-DMPWS policy preserves the throughput optimality with much less signaling overhead, thus expediting its practical implementation.