In this paper, we propose a cross-layer design framework for transmitting Markov modulated Poisson process (MMPP) traffic over Nakagami-m fading channel with delay demands. The adaptive modulation and coding (AMC) technique is used at the physical layer. The energy efficiency is described as the average throughput over the average transmission power, where both of throughput and transmit power have full consideration of the queuing system. We first derive the closed-form expressions of the delay and the energy efficiency with the stationary distribution of the system. We then derive the energy efficient thresholds to choose the AMC transmission modes. At last, we derive the transmission policy to maximize the energy efficiency with delay constraints. Numerical results are provided to support the theoretical development.Index Terms-Cross-layer design, energy efficiency, adaptive modulation and coding, queuing theory, finite state Markov channel, Markov modulated Poisson process, delay.