In this paper, the throughput of relay networks with multiple source-destination pairs under queueing constraints has been investigated for both variable-rate and fixed-rate schemes. When channel side information (CSI) is available at the transmitter side, transmitters can adapt their transmission rates according to the channel conditions, and achieve the instantaneous channel capacities. In this case, the departure rates at each node have been characterized for different system parameters, which control the power allocation, time allocation and decoding order. In the other case of no CSI at the transmitters, a simple automatic repeat request (ARQ) protocol with fixed rate transmission is used to provide reliable communication. Under this ARQ assumption, the instantaneous departure rates at each node can be modeled as an ON-OFF process, and the probabilities of ON and OFF states are identified. With the characterization of the arrival and departure rates at each buffer, stability conditions are identified and effective capacity analysis is conducted for both cases to determine the system throughput under statistical queueing constraints. In addition, for the variable-rate scheme, the concavity of the sum rate is shown for certain parameters, helping to improve the efficiency of parameter optimization. Finally, via numerical results, the influence of system parameters and the behavior of the system throughput are identified.