Large Deviations of Square Root Insensitive Random Sums

We provide a large deviation result for a random sum ∑Nx n=0Xn, where Nx is a renewal counting process and Xn n≥0 are i.i.d. random variables, independent of Nx , with a common distribution that belongs to a class of square root insensitive distributions. Asymptotically, the tails of these distributions are heavier than e− √ x and have zero relative decrease in intervals of length √ x, hence square root insensitive. Using this result we derive the asymptotic characterization of the busy period distribution in the stable GI/G/1 queue with square root insensitive service times; this characterization further implies that the tail behavior of the busy period exhibits a functional change for distributions that are lighter than e− √ x .