An approximation method for 2-chain flexible queues with preemptive priority

Chaining configurations (e.g. 2-chain) have been widely applied in service systems to improve responsiveness to customer demand. In this paper, we propose an approximation method for the analysis of symmetric parallel queueing systems with 2-chain configuration and preemptive priority. In the queueing system of interest, the arrival processes of different types of customers follow independent Poisson processes with identical arrival rate λ, and service times of different types of customers are exponentially distributed with identical service rate μ. Each server serves its primary customers with preemptive priority. When there are no primary customers in the system, the server serves secondary customers. We first derive the system stability condition. Then, we analyse the system's various performance measures. Our approach relies on the matrix-analytic method coupled with the idle probability of each server, which is proved to be 1 − λ/μ. The average queue length of each queue and the rate that each server serves its secondary customers (i.e. the helping rate) can be estimated using our proposed method. Finally, we conduct numerical studies to demonstrate the accuracy of our proposed approximation method.