Health-related cloud computing tasks are of utmost urgency and importance, hence requiring prompt attention. In this paper, we therefore propose a queuing model that uses a priority queue for the execution of healthcare tasks independently of other non-priority tasks in a non-priority queue. The proposed model ensures that tasks with stringent deadlines are executed first, while non-priority tasks are processed later. Simulation results demonstrate the effectiveness of the proposed model, in which a large percentage of priority tasks are completed within their deadlines to satisfy the timely processing of critical workloads. The non-priority tasks also exhibit acceptable performance, meaning the model can efficiently manage critical and non-critical tasks in dynamic cloud environments.