Introduction

In Chapter 13, we considered the problem of minimizing flow time for both single and multiple servers, when server speeds were fixed, and the only decision variable was which job to schedule or process at each time. With speed tuneable servers, a natural extension of this problem, called speed scaling, was studied in Chapter 14, where the problem of minimizing flow time plus energy was studied, with two decision variables: which job to schedule or process and its processing speed.

In this chapter, we consider an alternate formulation of the speed scaling problem, where jobs have deadlines, and server speeds are tuneable with corresponding power functions. The problem is to find which job to schedule or process at each time, and its processing speed, so as to minimize the total energy used, such that each job is complete by its deadline. For this formulation, both the common deadline case (all deadlines are identical) and the individual deadline case are of interest.

With a single server, for the commonly used power function P(s) = s,> 1 with speed s, we present an online algorithm for both the common and the individual deadlines case, for which the competitive ratio is upper bounded by , and the upper bound is also tight for the considered algorithm. For P(s) = s,> 1, we also consider the more modern paradigm of machine learning augmented algorithm for going beyond the worst case, where prediction about job arrival times and their sizes is available, but with uncertain accuracy.

Another power function of interest, motivated by information theory, is given by P(s) = 2s − 1, which results in fundamentally different results than P(s) = s. For P(s) = 2s − 1, with a single server, we present an online algorithm whose competitive ratio is at most 3 only for the common deadline case. For the common deadline case, we also show how to extend results from the single server case to the multiple server case without changing the competitive ratio.