Energy-efficiency of wireless communication systems has been an important research topic in recent years. For such a system, a transmission profile is described by the transmission power and the modulation and coding scheme (MCS) to be used for packet transmission. For a given channel condition, higher order MCSs offer higher throughput at the expense of requiring more transmission power. Average power consumption of the system can be reduced by using lower order MCSs at the expense of increased queuing delays. Using this observation, the goal of this study is the development of transmission profile selection policies so as to minimize the average power consumption while meeting a statistical delay constraint for a wireless link. For the purpose of assessing the proposed policies, the system is modeled as an M/M/1 queue where transmission speeds of packets are dynamically selected based on the queuing delay already experienced by them. This setting is shown to give rise to a multi-regime Markov uid queue model which is used to obtain the waiting time distributions of packets as well as the average power consumption. In the numerical examples, proposed pro- file selection policies are evaluated for different system parameters using realistic transmission profiles obtained from LTE simulations. A proposed energy-aware profile selection policy is shown to consistently outperform all other proposed policies in terms of energy-efficiency whereas a reasonable performance is also obtained with a simpler-to-implement policy.