Online Job Scheduling on a Single Machine with General Cost Functions

Abstract

We consider the problem of online job scheduling on a single machine with general job-dependent cost functions. In this model, each job j has a processing requirement (length) v <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">j</inf> and arrives with a nonnegative nondecreasing cost function g <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">j</inf> (t), and this information is revealed to the system upon arrival of job j at time r <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">j</inf> . The goal is to schedule the jobs preemptively on the machine in an online fashion so as to minimize the generalized completion time $\sum\limits_{} {_j{g_j}} \left({{C_j}}\right)$, where C <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">j</inf> is the completion time of job j on the machine. It is assumed that the machine has a unit processing speed that can work on a single job at any time instance. In particular, we are interested in finding an online scheduling policy whose objective cost is competitive with respect to a slower optimal offline benchmark, i.e., the one that knows all the job specifications a priori and is slower than the online algorithm. Under some mild assumptions, we provide a speed-augmented competitive algorithm for general nondecreasing cost functions g <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">j</inf> (t) by utilizing a novel optimal control framework.