Recent problems in wireless cellular networks, such as network capacity and indoor coverage, have been addressed by the orthogonal frequency-division multiple-access (OFDMA) small-cell deployments of next-generation Long-Term Evolution Advanced (LTE-A) cellular systems. In this new paradigm, the deployment of hybrid-access small cells can be seen as an optimal solution since they serve both the registered indoor users and unregistered neighboring users in the small-cell coverage area. However, effective quality-of-service (QoS) provisioning and fair admission control pop up as two crucial challenges in these hybrid accesses. Motivated by these challenges, in this paper, we propose a traffic-aware OFDMA hybrid small-cell deployment for QoS provisioning and an optimal admission control strategy for next-generation cellular systems. The traffic awareness in the proposed framework is provided by deriving a novel traffic-aware utility function, which differentiates the user QoS levels with the user's priority indexes, channel conditions, and traffic characteristics. An optimization procedure is formulated, and a novel heuristic is also developed to solve the traffic-aware scheduling problem under transmitted power constraints. To further enhance the proposed scheme, an admission control algorithm based on the utility function is also proposed. The proposed QoS awareness and admission control mechanism are evaluated by thorough simulations, and we show that our proposed framework achieves an optimum QoS performance in terms of total throughput and traffic delay.