The non-orthogonal multiple access (NOMA) has been designed to significantly enhance the spectral efficiency for the massive connections of the mobile devices. Moreover, to guarantee the short latency requirements for 5G multimedia wireless services, researchers have designed the statistical delaybounded quality-of-service (QoS) provisioning for supporting the video transmissions over the statistically varying wireless channels. Accordingly, researchers have proposed the finite blocklength coding (FBC) techniques to model the relationship between data transmission rate and channel capacity in the non- asymptotic regime while supporting short-packet communications under the QoS constraints. Due to its potential to significantly improve spectral efficiency and reduce the transmission latency, a NOMA system can be exploited while being integrated with FBC to guarantee the QoS requirements for both the latency and reliability over mobile wireless ad-hoc networks. However, due to the complexity of the effective-capacity maximization problem in the non-asymptotic regime, the FBC based NOMA scheme have imposed new challenges for determining the convexity of the optimization problem and deriving optimal resource allocation policies for NOMA subject to the statistical delay-bounded and error-rate bounded QoS constraints. In order to solve the abovementioned problems, we define a new concept of /spl epsilon/-effective capacity and propose a corresponding system architecture model for NOMA system under FBC. In particular, we characterize the FBC based NOMA system models in wireless ad hoc networks. Considering the statistical delay- bounded and error-rate bounded QoS constraints, we formulate and solve the max- min fairness problem under FBC. Simulation results are included, which evaluate and validate our proposed FBC based NOMA scheme subject to the statistical delay-bounded and error-rate bounded QoS constraints.