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Abstract This paper studies load balancing for many‐server (Nservers) systems. Each server has a buffer of sizeb − 1, and can have at most one job in service andb − 1 jobs in the buffer. The service time of a job follows the Coxian‐2 distribution. We focus on steady‐state performance of load balancing policies in the heavy traffic regime such that the normalized load of system isλ = 1 − N−αfor 0 < α < 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) withd = O(Nα 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.
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