On the steady-state probability of delay and large negative deviations for the GI/GI/n queue in the Halfin-Whitt regime

We consider the FCFS GI/GI/n queue in the Halfin-Whitt heavy traffic regime, and prove bounds for the steady-state probability of delay (s.s.p.d.) for generally distributed processing times. We prove that there exists ε > 0, depending on the inter-arrival and processing time distributions, such that the s.s.p.d. is bounded from above by exp ( − εB ) as the associated excess parameter B →∞; and by 1− εB as B → 0. We also prove that the tail of the steady-state number of idle servers has a Gaussian decay, and use known results to show that our bounds are tight (in an appropriate sense). Our main proof technique is the derivation of new stochastic comparison bounds for the FCFS GI/GI/n queue, which are of a structural nature, hold for all n and times t, and build on the recent work of Gamarnik and Goldberg [21].