Asymptotic Evaluation of Delay in the SRPT Scheduler

In this note, the shortest remaining processing time (SRPT) scheduler is considered. We focus on the SRPT scheduling policy for a discrete-time queueing system that is accessed by a large number of flows (a many flows regime). In such an asymptotic regime (large capacity and large number of flows), we derive the expression for the delay rate function, which describes the complete delay distribution, for batch arrival processes, and with bounded job sizes. Based on these results, we compare the delay rate function (i.e., for any finite delay, and asymptotic in the number of flows) of the SRPT scheduler with that of a first-in-first-out (FIFO) scheduler, when there is a mix of job sizes. Our analysis holds for any finite mix of job sizes and for truncated heavy-tailed arrival processes. We apply the result to a system accessed by jobs which are one of two sizes: 1 or M. We investigate the unfairness of SRPT by comparing the delay rate function of size M jobs for SRPT and FIFO. We show that the difference in the delay rate function between SRPT and FIFO for size M jobs decay as O(1/M gamma) for some gamma that satisfies 0<gamma<1 and for M sufficiently large. In other words, the delay distributions under FIFO and SRPT becomes increasingly similar for increasingly larger jobs. On the other hand, for size 1 jobs, the delay rate function under SRPT is invariant with M. However, for FIFO, the delay rate function decays as O(1/Mgamma) for some 0<gamma<1 and M large. Thus, for size 1 jobs, SRPT performs increasingly better compared to FIFO as the range in job size increases. These results indicate that SRPT is a good policy to implement for web-servers, where empirical evidence suggests a large variability in job sizes