Abstract
We consider an online interval scheduling problem on two related machines. If one machine is at least as twice as fast as the other machine, we say the machines are distinct; otherwise the machines are said to be similar. Each job j in J is characterized by a length p_j, and an arrival time t_j; the question is to determine whether there exists a feasible schedule such that each job starts processing at its arrival time. For the case of unit-length jobs, we prove that when the two machines are distinct, there is an amount of lookahead allowing an online algorithm to solve the problem. When the two machines are similar, we show that no finite amount of lookahead is sufficient to solve the problem in an online fashion. We extend these results to jobs having arbitrary lengths, and consider an extension focused on minimizing total waiting time.
Highlights
(i) For the case of unit-length jobs, we show that there exists an online algorithm with lookahead 2T1 if and only if the ratio between the two standard processing times is at least 2 (Sect. 3); in addition, we show that there cannot exist an online algorithm with lookahead less than 2T1
We consider jobs of arbitrary length. It becomes relevant whether we are given an upperbound, called P, on the length of the longest job in the instance. In case such a bound is given, and if the two machines are distinct, there exists an online algorithm with lookahead; and otherwise, there does not exist an online algorithm with any amount of lookahead (Sect. 4.2)
We have focussed on the potential that lookahead offers for online algorithms to solve an interval scheduling problem with two related machines
Summary
Each job j ∈ J must be assigned to either machine M1 or M2; the resulting schedule is feasible if and only if there is no overlap between any pair of jobs assigned to the same machine; in other words, for each pair of distinct jobs j1, j2 ∈ J , with j1 ≤ j2, assigned to a same machine i , t j2 ≥ t j1 + p j1 Ti (i = 1, 2). An instance of this problem is called feasible if a feasible schedule exists, otherwise the instance is called infeasible. Subsection 1.2 describes the practical application motivating this work, and Sect. 1.3 summarizes our results
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