In this paper, we consider a wireless powered communication system, where an energy harvesting (EH) node harvests energy from a radio frequency (RF) signal broadcasted by an access point (AP) in the downlink (DL). The node stores the harvested energy in an energy buffer and uses the stored energy to transmit data to the AP in the uplink (UL). We investigate two simple online transmission policies for the EH node, namely a best-effort policy and an on-off policy, which do not require knowledge of the EH profile nor channel knowledge. In particular, for both policies, the EH node transmits in each time slot with a constant desired power if sufficient energy is available in its energy buffer. Otherwise, the node transmits with the maximum possible power in the best-effort policy and remains silent in the on-off policy. For both policies, we use the theory of discrete-time continuous-state Markov chains to analyze the limiting distribution of the stored energy for finite- and infinite-size energy buffers. We provide this limiting distribution in closed form for a Nakagami-m fading DL channel and analyze the outage probability for a Nakagami-m fading UL channel. All derived analytical results are not limited to EH via RF WPT but are applicable for any independent and identically distributed EH process, originating from e.g. solar and wind energy. Our results reveal that, for low outage probabilities, the best-effort policy is superior to the on-off policy and the optimal constant UL transmit power of the EH node that minimizes the outage probability is always less than the average harvested power but increases with the capacity of the energy buffer. The opposite behaviour is observed for high outage probabilities. Furthermore, we show that the minimum outage probability of the two proposed policies closely approaches the outage probability of the optimal offline power allocation policy.