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We prove a heavy traffic scaling limit for a shortest remaining processingtime queue. We are interested in the case where the processing timedistribution has a tail that decays rapidly, i.e., has light tails. Inparticular, we revisit the work in Puha (2015), which shows that the diffusionscaled queue length process multiplied by a processing time distributiondependent factor that tends to infinity converges to a nontrivial reflectingBrownian motion, under the condition that this distribution dependent factor isslowly varying and obeys a certain rate of convergence condition. Here, weprove that the rate of convergence condition is not needed and the result holdsmore generally. We further show convergence of a sequence of nonstandardlyscaled measure valued state descriptors to a point mass at one such that thetotal mass fluctuates randomly in accordance with the diffusion limit for theworkload process. This is a sharp concentration result which shows that, underthis nonstandard scaling, there are a very small number of tasks in the systemand the remaining work for each such task is large and of the same order ofmagnitude as that of other tasks. This is due to the prioritization of the taskwith the least remaining work, and is in contrast to the case of heavy tailedprocessing times studied in Banerjee, Budhiraja, and Puha (2022). There it isshown that, while there is some concentration, the remaining times of the verysmall number of tasks in the system spread out over the nonnegative real lineaccording to a random profile under this nonstandard scaling. Thus, this workcompletes the description of the two fundamentally different behaviors of SPRTby characterizing it in the case of light tailed processing time distributions.

Heavy Traffic Scaling Limits for Shortest Remaining Processing Time Queues with Light Tailed Processing Time Distributions