In this paper, we study the performance of wireless information and power transfer in the presence of statistical queuing constraints. We consider harvest-then-transmit protocol in which users first harvest energy from a dedicated source and then transmit information through an uplink multiple access channel (MAC). Each user is subject to limitations on the buffer overflow probability, specified by the quality of service (QoS) exponent θ, and the optimal time allocation for energy harvesting and information decoding operations depends on these constraints in addition to the channel characteristics. We formulate optimization problems to maximize the throughput with and without QoS constraints. In both cases, the problems are convex, and hence Karush-Kuhn-Tucker (KKT) conditions are necessary and sufficient for global optimality. However, it is difficult to obtain closed-form expressions for optimal time interval since we assume that operating intervals are independent of each fading state realization. Hence, we develop an algorithm to obtain optimal solutions numerically. Simulation results justify that QoS constraints primarily affect achievable rate distribution among the users, and override the channel conditions.