Quantum communication networks require transmission of high-fidelity, uncoded qubits for applications such as entanglement distribution and quantum key distribution. However, current implementations are constrained by limited buffer capacity and qubit decoherence, which degrades qubit quality while waiting in the buffer. A key challenge arises from the stochastic nature of qubit generation, there exists a random delay (D) between the initiation of a generation request and the availability of the qubit. This induces a fundamental trade off early initiation increases buffer waiting time and hence decoherence, whereas delayed initiation leads to server idling and reduced throughput. We model this system as an admission control problem in a finite buffer queue, where the reward associated with each job is a decreasing function of its sojourn time. We derive analytical conditions under which a simple "no lag" policy where a new qubit is generated immediately upon the availability of buffer space is optimal. To address scenarios with unknown system parameters, we further develop a Bayesian learning framework that adaptively optimizes the admission policy. In addition to quantum communication systems, the proposed model is applicable to delay sensitive IoT sensing and service systems.