Novel heavy-traffic regimes for large-scale service systems

We introduce a family of heavy-traffic regimes for large scale service systems, presenting a range of scalings that include both moderate and extreme heavy traffic, as compared to classical heavy traffic. The heavy-traffic regimes can be translated into capacity sizing rules that lead to Economies-of-Scales, so that the system utilization approaches 100% while congestion remains limited. We obtain heavy-traffic approximations for stationary performance measures in terms of asymptotic expansions, using a non-standard saddle point method, tailored to the specific form of integral expressions for the performance measures, in combination with the heavy-traffic regimes. Keywords: heavy-traffic approximations, heavy-traffic regimes, large service systems, queueing theory, asymptotic analysis, saddle point method, Riemann zeta function