The Gated Infinite-Server Queue with Unbounded Service Times and Heavy Traffic

A queueing system with infinitely many servers is considered where access of customers to service is controlled by a gate. The gate is open only if all servers are free. Customers are served in stages. We study the asymptotic behavior of the number of customers served in a stage and of the stage duration in the heavy traffic regime in the case where the service time distribution belongs to the attraction domain of a double exponential law.