Large-scale Join-Idle-Queue system with general service times

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A parallel server system with n identical servers is considered. The service time distribution has a finite mean 1/μ, but otherwise is arbitrary. Arriving customers are be routed to one of the servers immediately upon arrival. Join-Idle-Queue routing algorithm is studied, under which an arriving customer is sent to an idle server, if such is available, and to a randomly uniformly chosen server, otherwise. We consider the asymptotic regime where n→∞ and the customer input flow rate is λn. Under the condition λ/μ<1/2, we prove that, as n→∞, the sequence of (appropriately scaled) stationary distributions concentrates at the natural equilibrium point, with the fraction of occupied servers being constant equal λ/μ. In particular, this implies that the steady-state probability of an arriving customer waiting for service vanishes.