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We consider an M/M/Infinity service system in which an arriving customer isserved by the first idle server in an infinite sequence S_1, S_2, ... ofservers. We determine the first two terms in the asymptotic expansions of themoments of L as lambda tends to infinity, where L is the index of the serverS_L serving a newly arriving customer in equilibrium, and lambda is the ratioof the arrival rate to the service rate. The leading terms of the moments showthat L/lambda tends to a uniform distribution on [0,1].

The M/M/Infinity Service System with Ranked Servers in Heavy Traffic