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Biomedical sensors (BMSs) monitor the heterogeneous vital signs of patients. They have diverse Quality of Service (QoS) requirements including reduced collision, delay, loss, and energy consumption in the transmission of data, which are non-constrained, delay-constrained, reliability-constrained, and critical. In this context, this paper proposes a traffic priority-aware adaptive slot allocation-based medium access control (TraySL-MAC) protocol. Firstly, a reduced contention adaptive slot allocation algorithm is presented to minimize contention rounds. Secondly, a low threshold vital signs criticality-based adaptive slot allocation algorithm is developed for high priority data. Thirdly, a high threshold vital signs criticality-based adaptive slot allocation algorithm is designed for low priority data. Simulations are performed to comparatively evaluate the performance of the proposed protocol with state-of-the-art MAC protocols. From the analysis of the results, it is evident that the proposed protocol is beneficial in terms of lower packet delivery delay and energy consumption, and higher throughput in realistic biomedical environments.