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Transmission over wireless fading channels under quality of service (QoS) constraints is studied when only the receiver has channel side information. Being unaware of the channel conditions, transmitter is assumed to send the information at a fixed rate. Under these assumptions, a two-state (ON-OFF) transmission model is adopted, where information is transmitted reliably at a fixed rate in the ON state while no reliable transmission occurs in the OFF state. QoS limitations are imposed as constraints on buffer violation probabilities, and effective capacity formulation is used to identify the maximum throughput that a wireless channel can sustain while satisfying statistical QoS constraints. Energy efficiency is investigated by obtaining the bit energy required at zero spectral efficiency and the wideband slope in both wideband and low-power regimes assuming that the receiver has perfect channel side information (CSI). Initially, the wideband regime with multipath sparsity is investigated, and the minimum bit energy and wideband slope expressions are found. It is shown that the minimum bit energy requirements increase as the QoS constraints become more stringent. Subsequently, the low-power regime, which is also equivalent to the wideband regime with rich multipath fading, is analyzed. In this case, bit energy requirements are quantified through the expressions of bit energy required at zero spectral efficiency and wideband slope. It is shown for a certain class of fading distributions that the bit energy required at zero spectral efficiency is indeed the minimum bit energy for reliable communications. Moreover, it is proven that this minimum bit energy is attained in all cases regardless of the strictness of the QoS limitations. The impact upon the energy efficiency of multipath sparsity and richness is quantified, and comparisons with variable-rate/fixed-power and variable-rate/variable-power cases are provided.