Improved Bounds on the Rate of Convergence to Stationarity of the M/M/N Queue as Both the Number of Servers and the Arrival Rate Grow Large

Consideration is given to the stable M/M/N/∞ queueing system as both the arrival rate and the number of servers tend to infinity, including the Halflin–Whitt regime. One is interested in the asymptotic behavior of the rate of convergence to stationarity of the total number of customers in the system. Bounds for the rate of convergence are provided, which improve upon those available in the literature. The derivations, simpler than those available in the literature, are based on the well-known logarithmic norm method. It is also shown that the obtained results can be adopted to M/M/N/N+R queues as well.