Delay asymptotics of the SRPT scheduler

In this paper we look at the shortest-remaining-processing-time (SRPT) scheduling algorithm. It has been long thought that although it exhibits optimal mean packet delay, the larger sized packets will "suffer" a very large delay. In this paper, we consider this scheduling rule for a discrete-time queueing system that is accessed by a large number of flows (a many flows regime). In such an asymptotic regime where the number of flows are large, we derive expressions for the packet delay distributions for batch arrival processes, and with bounded packet sizes. Using these results, we compare the delay asymptote (i.e., for any finite delay, and asymptotic in the number of flows) of the SRPT scheduler with that of a FIFO (first in first out) scheduler, when there is a mix of packet sizes. In particular, we apply the asymptotic delay result to a system accessed by packets which are one of two sizes: \´1\´ or \´k\´, and the arrival process is i.i.d. in time. We show that the difference in rate function of the delay asymptote between SRPT and FIFO for the size \´k\´ packet decays as 1/k. Thus, for large packets, the delay distributions under FIFO and SRPT look similar. On the other hand, for the size \´1\´ packet, the delay rate function under SRPT is invariant with k. However for FIFO, the delay rate function for the size \´1\´ packet decays as 1/k. This indicates that for size \´1\´ packets, SRPT performs increasingly better as the range in packet size increases. Thus, these results indicate that SRPT is a good policy to implement for Web-servers, where empirical evidence suggests a large variability in packet sizes.