  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.
