In this paper we study the asymptotic behavior of the tail of the stationary backlog distribution in a single server queue with constant service capacity c, fed by the so-called "M=G=1 input process " or "Cox input process". Asymptotic lower bounds are obtained for any distribution G and asymptotic upper bounds are derived when G is a subexponential distribution. We find the bounds to be tight in some instances, e.g., G corresponding to either the Pareto or lognormal distribution and c \Gamma ae! 1, where ae is the arrival rate to the buffer. Keywords: Asymptotic self-similar process; Long-range dependence; Subexponential
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