# Multiple Server SRPT with speed scaling is competitive

## TL;DR

This paper demonstrates that multi-server SRPT with speed scaling can achieve a constant competitive ratio for flow time plus energy, depending only on power functions, not on job or server count.

## Contribution

It proves multi-server SRPT with speed scaling is constant competitive, unlike in the no-speed-scaling case, and identifies optimal routing strategies under various conditions.

## Key findings

- Multi-server SRPT with speed scaling is constant competitive.
- Round-robin routing is optimal for unit job sizes.
- Greedy dispatch policies lack constant competitiveness.

## Abstract

Can the popular shortest remaining processing time (SRPT) algorithm achieve a constant competitive ratio on multiple servers when server speeds are adjustable (speed scaling) with respect to the flow time plus energy consumption metric? This question has remained open for a while, where a negative result in the absence of speed scaling is well known. The main result of this paper is to show that multi-server SRPT can be constant competitive, with a competitive ratio that only depends on the power-usage function of the servers, but not on the number of jobs/servers or the job sizes (unlike when speed scaling is not allowed). When all job sizes are unity, we show that round-robin routing is optimal and can achieve the same competitive ratio as the best known algorithm for the single server problem. Finally, we show that a class of greedy dispatch policies, including policies that route to the least loaded or the shortest queue, do not admit a constant competitive ratio. When job arrivals are stochastic, with Poisson arrivals and i.i.d. job sizes, we show that random routing and a simple gated-static speed scaling algorithm achieves a constant competitive ratio.