We consider the problem of scheduling to minimize mean response time in M/G/1 queues where only estimated job sizes (processing times) are known to the scheduler, where a job of true size $s$ has estimated size in the interval $[βs, αs]$ for some $α\geq β> 0$. We evaluate each scheduling policy by its approximation ratio, which we define to be the ratio between its mean response time and that of Shortest Remaining Processing Time (SRPT), the optimal policy when true sizes are known. Our question: is there a scheduling policy that (a) has approximation ratio near 1 when $α$ and $β$ are near 1, (b) has approximation ratio bounded by some function of $α$ and $β$ even when they are far from 1, and (c) can be implemented without knowledge of $α$ and $β$? We first show that naively running SRPT using estimated sizes in place of true sizes is not such a policy: its approximation ratio can be arbitrarily large for any fixed $β< 1$. We then provide a simple variant of SRPT for estimated sizes that satisfies criteria (a), (b), and (c). In particular, we prove its approximation ratio approaches 1 uniformly as $α$ and $β$ approach 1. This is the first result showing this type of convergence for M/G/1 scheduling. We also study the Preemptive Shortest Job First (PSJF) policy, a cousin of SRPT. We show that, unlike SRPT, naively running PSJF using estimated sizes in place of true sizes satisfies criteria (b) and (c), as well as a weaker version of (a).

This talk is about scheduling and load balancing in a multi-server system, with the goal of minimizing mean response time in a general stochastic setting. We will specifically concentrate on the common case of a load balancing system, where a front-end load balancer (a.k.a. dispatcher) dispatches requests to multiple back-end servers, each with their own queue. Much is known about load balancing in the case where the scheduling at the servers is First-Come-First-Served (FCFS). However, to minimize mean response time, we need to use Shortest-Remaining-Processing-Time (SRPT) scheduling at the servers. Unfortunately, there is almost nothing known about optimal dispatching when SRPT scheduling is used at the servers. To make things worse, it turns out that the traditional dispatching policies that are used in practice with FCFS servers often have poor performance in systems with SRPT servers. In this talk, we devise a simple fix that can be applied to any dispatching policy. This fix, called "guardrails" ensures that the dispatching policy yields optimal mean response time under heavy traffic, when used in a system with SRPT servers. Any dispatching policy, when augmented with guardrails becomes heavy-traffic optimal. Our results also yield the first analytical bounds on mean response time for load balancing systems with SRPT scheduling at the servers. Load balancing and scheduling are highly studied both in the stochastic and the worst-case scheduling communities. One aim of this talk is to contrast some differences in the approaches of the two communities when tackling multi-server scheduling problems.

We consider scheduling to minimize mean response time of the M/G/k queue with unknown job sizes. In the single-server k=1 case, the optimal policy is the Gittins policy, but it is not known whether Gittins or any other policy is optimal in the multiserver case. Exactly analyzing the M/G/k under any scheduling policy is intractable, and Gittins is a particularly complicated policy that is hard to analyze even in the single-server case. In this work we introduce monotonic Gittins (M-Gittins), a new variation of the Gittins policy, and show that it minimizes mean response time in the heavy-traffic M/G/k for a wide class of finite-variance job size distributions. We also show that the monotonic shortest expected remaining processing time (M-SERPT) policy, which is simpler than M-Gittins, is a 2-approximation for mean response time in the heavy traffic M/G/k under similar conditions. These results constitute the most general optimality results to date for the M/G/k with unknown job sizes. Our techniques build upon work by Grosof et al. (2018), who study simple policies, such as SRPT, in the M/G/k; Bansal et al. (2018), Kamphorst and Zwart (2020), and Lin et al. (2010), who analyze mean response time scaling of simple policies in the heavy-traffic M/G/1; and Aalto et al. (2009,2011) and Scully et al. (2018,2020), who characterize and analyze the Gittins policy in the M/G/1.