We provide a large deviation result for a random sum ∑n=0Nx Xn, 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.
