Mobile Broadband Wireless Access (BWA) networks will offer in the forthcoming years multiple and differentiated services to users with high mobility requirements, connecting via portable or wearable devices which rely on the use of batteries by necessity. Since a relatively large fraction of energy is consumed by such devices for transmitting/receiving data over-the-air, mechanisms are needed to reduce power consumption, in order to increase the lifetime of devices, and hence, improve user's satisfaction. The IEEE 802.16, which supports mobile BWA since its "e" amendment in 2005, defined power saving functions at the Medium Access Control (MAC) layer, which are designed to be operated during open traffic sessions for the greatest energy consumption reduction. However, enabling power saving usually increases the transmission latency, which can negatively affect the Quality of Service (QoS) experienced by users. On the other hand, imposing stringent QoS requirements may limit the amount of energy that can be saved. In this paper, an extensive study of the mutual interaction between power saving mechanisms and QoS support is carried out in the context of the IEEE 802.16e. In particular, two types of delay-constrained applications with different requirements are considered, i.e., Web and Voice over IP (VoIP) for which the IEEE 802.16e standard specifies two different power saving classes. The performance is assessed via detailed packet-level simulation, with respect to several system parameters. To capture the relative contribution of all the factors on the energy- and QoS-related metrics, part of the evaluation is carried out by means of 2k ?? ??! analysis.