Optimality of the Second Order Metrics of the M/M/2 System with Heterogeneous Service Rates

In a previous paper [4], we described an approach to improve the performance of the M/M/2 system by replacing its homogeneous servers with heterogeneous servers. Denoting by beta the ratio of the slow service rate to the fast service rate, we showed that there exists an optimal ratio at which the first order metrics reach their optimums. In the present work, we extend the study to the second order metrics. We first derive the distributions of the queuing time and system sojourn time, then prove the optimality of the second order metrics.