Steady‐state analysis of load balancing with Coxian‐2 distributed service times

This paper studies load balancing for many-server ($N$ servers) systems assuming Coxian-$2$ service time and finite buffer with size $b-1$ (i.e. a server can have at most one job in service and $b-1$ jobs in queue). We focus on steady-state performance of load balancing policies in the heavy traffic regime such that the load of system is $\lambda = 1 - N^{-\alpha}$ for $0<\alpha<0.5.$ We identify a set of policies that achieve asymptotic zero waiting. The set of policies include several classical policies such as join-the-shortest-queue (JSQ), join-the-idle-queue (JIQ), idle-one-first (I1F) and power-of-$d$-choices (Po$d$) with $d=O(N^\alpha\log N)$. The proof of the main result is based on Stein's method and state space collapse. A key technical contribution of this paper is the iterative state space collapse approach that leads to a simple generator approximation when applying Stein's method.