Analysis of a Markovian Queueing Model with an Alternating Server and Queue-Length-Based Threshold Control

This paper analyzes a finite-capacity Markovian queueing system with two customer types, each assigned to a separate buffer, and a single alternating server whose service priority is dynamically controlled by a queue-length-based threshold policy. The arrivals of both customer types follow independent Poisson processes, and the service times are generally distributed. The server alternates between the two buffers, granting service priority to buffer 1 when its queue length exceeds a specified threshold immediately after service completion; otherwise, buffer 2 receives priority. Once buffer 1 gains priority, it retains it until it becomes empty, with all priority transitions occurring non-preemptively. We develop an embedded Markov chain model to derive the joint queue length distribution at departure epochs and employ supplementary variable techniques to analyze the system performance at arbitrary times. This study provides explicit expressions for key performance measures, including blocking probabilities and average queue lengths, and demonstrates the effectiveness of threshold-based control in balancing service quality between customer classes. Numerical examples illustrate the impact of buffer capacities and threshold settings on system performance and offer practical insights into the design of adaptive scheduling policies in telecommunications, cloud computing, and healthcare systems.