Logarithmic Asymptotics for SteadyState Tail Probabilities in a SingleServer QueueP. W. Glynn and W. Whitt Advances in Applied Probability, 131156 (1994) We consider the standard singleserver queue with unlimited waiting space and the firstin firstout service discipline, but without any explicit independence conditions on the interarrival and service times. We find conditions for the steadystate waitingtime distribution to have smalltail asymptotics of the form x^{1}log P(W>x)>θ^{*} as x>∞ for θ>0. We require only stationarity of the basic sequence of service times minus interarrival times and a Gartner Ellis condition for the cumulant generating function of the associated partial sums, i.e., n^{1}log Eexp(θS_{n})>ψ(θ), plus regularity conditions on the decay rate function ψ. The asymptotic decay rate θ^{*} is the root of the equation ψ(θ)=0. This result in turn implies a corresponding asymptotic result for the steadystate workload in a queue with general nondecreasing input. This asymptotic result covers the case of multiple independent sources, so that it provides additional theoretical support for a concept of effective bandwidths for admission control in multiclass queues based on asymptotic decay rates.
