Bounds and moments for stationary delay in GI/GI/s queue

Only a little is known about the tail properties of the distribution of the stationary waiting time, or delay, in multi-server queues with the FCFS service discipline, in the contrast with the one-dimensional case. In the GI/G/1 queue, the γ-th moment of the stationary delay is finite if and only if the (γ+1)-st moment of the service time distribution is finite - this goes back to Kiefer and Wolfowitz. Under subexponential-type conditions, the tail asymptotics for the stationary delay are also known; they follow the tail of the distribution of the residual service time. Recent results (W. Whitt, A. Scheller-Wolf and K. Sigman, A. Scheller-Wolf and R. Vesilo, etc.) suggest that, in the multi-server queue, the tail distribution of the stationary delay is not always as heavy as that of the residual service time, and the conditions for the finiteness of the γ-th moment differ from those in the single server queue. In our QUESTA paper (2006), we studied the two-server queue GI/GI/2 and obtained sharp asymptotics for the tail distribution of the stationary delay assuming the subexponentiality of the distribution of the residual service time. In particular, these asymptotics differ for different regions of the traffic load &rho. Now we present upper and lower bounds for the distribution tail of the stationary delay in the GI/GI/s queue, for any s ≥2. The derivation of these bounds requires new ideas and techniques. In particular, in the case of (intermediate) regularly varying service times these bounds are exact up to a constant. These bounds again depend on the traffic load. They also allow obtaining conditions for the existence of power moments, which are both necessary and sufficient.