When transmitting stochastic traffic flows over wireless networks, there exists an inherent tradeoff between average transmit power and corresponding queuing-delay bound. In this paper, we investigate such a tradeoff and show how average power increases as delay-bound requirement for wireless network traffics becomes stringent. Specifically, we propose the resource allocation schemes to minimize the power consumption subject to a delay quality-of-service (QoS) constraint, where the delay constraint is in terms of queue-length decay rate when an arrival traffic is transmitted through the wireless networks. We focus on orthogonal-frequency-division-multiplexing (OFDM) communications under three different network infrastructures, namely, point-to-point link, multihop amplify-and-forward (AF) network, and multiuser cellular network. We derive the optimal resource allocation policies for each scenario, and compare their performances with other existing resource-allocation policies. The obtained simulation and numerical results show that using our proposed optimal resource-allocation policies, significant power saving can be achieved. Furthermore, our OFDM-based communications systems can significantly reduce the power consumption, especially under stringent delay constraint.