Recent years have witnessed an active research on the stochastic network utility maximization (NUM) problem, where the optimal network resource allocation policy, such as congestion control, routing, and scheduling, is formulated as a constrained maximization of some utility function under the stochastic dynamics in users' traffic and time-varying wireless channels. So far, the stochastic NUM problem and its associated solutions are investigated under light-tailed (LT) traffic assumption, which is largely departing from the recent large-scale empirical studies, which verify the emergence of heavy-tailed (HT) traffic in a variety of networked systems. The unique stochastic features of HT traffic fundamentally challenges the feasibility of the classic algorithms developed for the stochastic NUM problem. This paper aims to develop effective algorithms to maximize network utility in the presence of heavy tailed traffic. First, it is proven that without considering the inherent bursty nature of HT traffic, the classic stochastic subgradient algorithm, which is utility optimal, fail to achieve queue stability. Our theoretical analysis reveals that such queue instability phenomenon is due to the fact that the stochastic subgradient algorithm inherently exploits queue length to update Lagrange dual variables. To counter this challenge, a time-average stochastic gradient algorithm is proposed, which decouples the queue-length update process and the dual variable update process in such a way that utility-optimality and queue-stability can be simultaneously achieved. Our convergence and stability analysis shows that the proposed algorithm can avoid the LT queues from competing with the HT queues, thus completely shielding those LT queues from the destructive impact of HT traffic.