On Large Delays in Multi-Server Queues with Heavy Tails

We present upper and lower bounds for the tail distribution of the stationary waiting time $D$ in the stable $GI/GI/s$ FCFS queue. These bounds depend on the value of the traffic load $ρ$ which is the ratio of mean service and mean interarrival times. For service times with intermediate regularly varying tail distribution the bounds are exact up to a constant, and we are able to establish a `principle of $s-k$ big jumps' in this case (here $k$ is the integer part of $ρ$), which gives the most probable way for the stationary waiting time to be large. Another corollary of the bounds obtained is to provide a new proof of necessity and sufficiency of conditions for the existence of moments of the stationary waiting time.

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