Introduction

In this chapter, we supplement the problem of minimizing the flow time with multiple fixedspeed servers considered in Chapter 13 by allowing the flexibility for each server to have a tuneable speed. This setup is typically referred to as speed scaling in the literature. Increasing server speed decreases the flow time but comes at a cost of increased energy consumption, which is typically accounted for by defining a speed based power cost function P(s) that is an increasing function of speed s.

A variety of problems can now be formulated, such as minimizing flow time subject to a constraint on the total energy cost, or minimizing a linear combination of flow time and total energy cost, etc. The most popular problem studied in literature is the linear combination of flow time and total energy cost, generally referred to as the flow time plus energy problem.

In this chapter, we consider the flow time plus energy problem and begin with the single server case. With a single server, SRPT remains an optimal scheduling algorithm, and the only non-trivial decision is the speed choice. We will show that an algorithm that chooses speed such that the power consumption at each time instant is equal to the number of incomplete jobs in the system plus one is 3-competitive, independent of the power cost function P.

Next, we consider the more challenging speed scaling problem when there are multiple identical servers, i.e., with the same power cost function P. We consider the multi-server version of the SRPT algorithm for scheduling, and the following speed choice: the total power consumption across all servers at any time instant is equal to the number of incomplete jobs in the system, and all servers process their jobs at the same speed. We show that this algorithm is constant competitive, where the constant depends only on the cost function P. Recall from Chapter 13 that the competitive ratio of the SRPT algorithm with fixed speed servers to minimize the flow time is, where n is the number of jobs and m is the number of servers, and wmax and wmin are the maximum and minimum job sizes.