Queues with many servers: The virtual waiting-time process in the QED regime

We consider a first-come first-served multiserver queue in the Quality- and 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/root N) (Efficiency-Driven) while waiting time is O(1/root N) ( Quality- Driven). This is equivalent to having the number of servers N being approximately equal to R+beta root R, where R is the offered load and beta is a positive constant. For the G/D-K/N queue in the QED regime, we analyze the virtual waiting time V-N(t), as N increases indefinitely. Assuming that the service-time distribution has a finite support (hence the D-K in G/D-K/N), it is shown that, in the limit, the scaled virtual waiting time (V) over cap (N)(t) = root NVN (t)/ES is representable as a supremum over a random weighted tree ( S denotes a service time). Informally, it is then argued that, for large N, (V) over cap (N)(t) approximate to (E < s > [(V) over cap (N)(t-S)]+(X) over cap (t) -beta)(+), t is an element of R; here.(E < s > [(V) over cap (N)(t-S)] is the averaging of (V) over cap (N)(t-S) over S, and the process (X) over cap (t) is zero-mean Gaussian that summarizes all relevant information about arrivals and service times (X) over cap (t) arises as a limit of an infinite-server (G/D-K/infinity) process, appropriately scaled). The results are obtained by using both combinatorial and probabilistic arguments. Possible applications of our approximations include fast simulation of queues and estimation/ prediction of customer waiting times in the QED regime.