Queues with Many Servers: The Virtual Waiting-Time Process in the QED Regime

We consider a first-come first-served multiserver queue in the Qualityand Efficiency-Driven (QED) regime. In this regime, which was first formalized by Halfin and Whitt, the number of servers N is not small, servers’ utilization is 1−O 1/√N (Efficiency-Driven) while waiting time is O 1/ √ N (Quality-Driven). This is equivalent to having the number of servers N being approximately equal to R+ √R, where R is the offered load and is a positive constant. For the G/DK /N queue in the QED regime, we analyze the virtual waiting time VN t , as N increases indefinitely. Assuming that the service-time distribution has a finite support (hence the DK in G/DK /N), it is shown that, in the limit, the scaled virtual waiting time VN t = √ NVN t /ƐS is representable as a supremum over a random weighted tree (S denotes a service time). Informally, it is then argued that, for large N ,