The IEEE 802.16e standard specifies the sleep mode and the idle mode of a mobile station (MS) for power saving. In this paper, to reduce the energy consumption of the MS, we employ the sleep mode while the MS is on-session, and the idle mode while it is off-session. Under the assumption that the time duration from the end of a session to the arrival of a new downlink session request follows an exponential distribution of the mean 1/ν and that arrivals of messages during an on-session follow a Poisson process with rate λ, we analyze the awake mode period and the sleep mode period by using the busy period analysis of the M/G/1 queue, and then we derive the total mean length of an on-session which consists of a geometric number of awake mode periods and sleep mode periods. Since the sum of an on-session and an off-session constitutes a cycle, we can express the average power consumption in terms of the mean lengths of an awake mode period, a sleep mode period and an idle mode period. The average power consumption indicates how much the MS can save energy by employing the sleep mode and the idle mode. We also derive the Laplace Stieltjes transform (and the mean) of the queueing delay of messages to examine a tradeoff between the power consumption and the delay of messages. Analytical results, which are shown to be well-matched by simulations, address that our employment of the sleep mode and the idle mode provides a considerable reduction in the energy consumption of the MS.