Steady-state analysis of load-balancing algorithms in the sub-Halfin-Whitt regime

We study a class of load-balancing algorithms for many-server systems (N servers). Each server has a buffer of size with, i.e. a server can have at most one job in service and jobs queued. We focus on the steady-state performance of load-balancing algorithms in the heavy traffic regime such that the load of the system is for [CDATA[0 which we call the sub-Halfin-Whitt regime (and 0,] is the so-called Halfin-Whitt regime). We establish a sufficient condition under which the probability that an incoming job is routed to an idle server is 1 asymptotically (as) at steady state. The class of load-balancing algorithms that satisfy the condition includes join-the-shortest-queue, idle-one-first, join-the-idle-queue, and power-of-d-choices with (r a positive integer). The proof of the main result is based on the framework of Stein's method. A key contribution is to use a simple generator approximation based on state space collapse.