Optimal Control of a Parallel-Server Queueing system under Heavy Traffic Conditions

Abstract: This paper investigates the optimal control of parallel-server queueing systems operating under heavy traffic conditions. The study formulates the system as a Quasi-Birth-Death (QBD) process and applies the Matrix Geometric Method (MGM) to obtain steady-state probabilities and performance measures. Heavy traffic analysis is incorporated to approximate system behavior when utilization approaches unity, where congestion effects dominate and performance deteriorates rapidly. The research embeds control mechanisms including service rate adjustment, admission control, rejection penalties, and server breakdown considerations into the queueing framework. Both scalar and matrix formulations are examined, including breakdown repair dynamics that require solving nonlinear matrix equations numerically. Cost functions incorporating holding, service, and rejection penalties are developed, and numerical results demonstrate significant cost reductions through optimal service rate selection and controlled admission policies. The study highlights that heavy traffic approximations often push optimal solutions toward boundary controls unless nonlinear cost structures are introduced. Overall, the results reveal the economic trade-off between congestion, service capacity, and rejection penalties, providing valuable managerial insights for designing efficient service systems near capacity.

Keywords: Parallel-server, heavy traffic, matrix geometric method, quasi-birth-death process, optimal control, server breakdown, cost optimization