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Scheduling different types of packets, such as realtime and non-real-time data packets, at sensor nodes with resource constraints in Wireless Sensor Networks (WSN) is of vital importance to reduce sensors' energy consumptions and end-to-end data transmission delays. Most of the existing packet-scheduling mechanisms of WSN use First Come First Served (FCFS), non-preemptive priority and preemptive priority scheduling algorithms. These algorithms incur a high processing overhead and long end-to-end data transmission delay due to the FCFS concept, starvation of high priority real-time data packets due to the transmission of a large data packet in nonpreemptive priority scheduling, starvation of non-real-time data packets due to the probable continuous arrival of real-time data in preemptive priority scheduling, and improper allocation of data packets to queues in multilevel queue scheduling algorithms. Moreover, these algorithms are not dynamic to the changing requirements of WSN applications since their scheduling policies are predetermined. In this paper, we propose a Dynamic Multilevel Priority (DMP) packet scheduling scheme. In the proposed scheme, each node, except those at the last level of the virtual hierarchy in the zone-based topology of WSN, has three levels of priority queues. Real-time packets are placed into the highest-priority queue and can preempt data packets in other queues. Non-real-time packets are placed into two other queues based on a certain threshold of their estimated processing time. Leaf nodes have two queues for real-time and non-real-time data packets since they do not receive data from other nodes and thus, reduce end-to-end delay. We evaluate the performance of the proposed DMP packet scheduling scheme through simulations for real-time and non-real-time data. Simulation results illustrate that the DMP packet scheduling scheme outperforms conventional schemes in terms of average data waiting time and end-to-end delay.