Energy efficiency of wireless transmissions is analyzed in the presence of Markov sources and queueing constraints. Not knowing the channel conditions, the transmitter is assumed to send the data at a fixed rate over a Rayleigh fading channel. This fixed-rate transmission is modeled as a two-state (ON/OFF) continuous-time Markov chain. Using the effective bandwidth of Markov sources and the effective capacity of the Markov fluid transmission model, a characterization of the maximum average arrival rate that can be supported in the Rayleigh fading channel under buffer constraints is given, and energy efficiency is analyzed by determining the minimum energy per bit and wideband slope expressions. The impact of queueing constraints, source and channel parameters, and fixed-rate transmissions on the energy efficiency is identified.