Heavy-traffic limits for server idle times with customary server-assignment rules

Sun and Whitt (2017) developed a class of server-assignment rules that can help create effective breaks for servers from naturally available idleness. They found that the standard longest-idle-server-first rule and the alternative random routing generate breaks too infrequently. This paper provides theoretical justifications for these empirical findings by establishing the many-server heavy-traffic limits for server idle times with customary assignment rules. We gain insights into the server idle-busy activity patterns resulting from the server-assignment schemes in the system.