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In this paper, we study the performance of an uplink non-orthogonal multiple access (NOMA) network under statistical quality of service (QoS) delay constraints, captured through each user s effective capacity (EC). We first propose novel closed-form expressions for the EC in a two-user NOMA network and show that in the high signal-to-noise ratio (SNR) region, the 'strong' NOMA user, referred to as U2, has a limited EC, assuming the same delay constraint as the 'weak' user, referred to as U1. We demonstrate that for the weak userU1, OMA and NOMA have comparable performance at low transmit SNRs, while NOMA outperforms OMA in terms of EC at high SNRs. On the other hand, for the strong user U2, NOMA achieves higher EC than OMA at small SNRs, while OMA becomes more beneficial at high SNRs. Furthermore, we show that at high transmit SNRs, irrespective of whether the application is delay tolerant, or not, the performance gains of NOMA over OMA for U1, and OMA over NOMA for U2 remain unchanged. When the delay QoS of one user is fixed, the performance gap between NOMA and OMA in terms of total EC increases with decreasing statistical delay QoS constraints for the other user. Next, by introducing pairing, we show that NOMA with user-pairing outperforms OMA, in terms of total uplink EC. The best pairing strategies are given in the cases of four and six users NOMA, raising once again the importance of power allocation in the optimization of NOMA s performance.