In this paper, we study the performance of non-orthogonal multiple access (NOMA) schemes in wireless powered communication networks (WPCN) focusing on the system energy efficiency (EE). We consider multiple energy harvesting user equipments (UEs) that operate based on harvest-then-transmit protocol. The uplink information transfer is carried out by using power-domain multiplexing, and the receiver decodes each UE's data in such a way that the UE with the best channel gain is decoded without interference. In order to determine optimal resource allocation strategies, we formulate optimization problems considering two models, namely half-duplex and asynchronous transmission, based on how downlink and uplink operations are coordinated. In both cases, we have concave-linear fractional problems, and hence Dinkelbach's method can be applied to obtain the globally optimal solutions. Thus, we first derive analytical expressions for the harvesting interval, and then we provide an algorithm to describe the complete procedure. Furthermore, we incorporate delay-limited sources and investigate the impact of statistical queuing constraints on the energy-efficient allocation of operating intervals. We formulate an optimization problem that maximizes the system effective-EE while UEs are applying NOMA scheme for uplink information transfer. Since the problem satisfies pseudo-concavity, we provide an iterative algorithm using bisection method to determine the unique solution. In the numerical results, we observe that broadcasting at higher power level is more energy efficient for WPCN with uplink NOMA. Additionally, exponential decay quality of service parameter has considerable impact on the optimal solution, and in the presence of strict constraints, more time is allocated for downlink interval under half-duplex operation with uplink time-division multiple access mode.