Online Algorithms for Scheduling Unit Length Jobs on Unbounded Parallel-Batch Machines with Linearly Lookahead

Abstract

In this paper, we propose a new online scheduling model with linear lookahead intervals, which has the character that at any time [Formula: see text], one can foresee the jobs that will coming in the time interval [Formula: see text] in which [Formula: see text]. In this new lookahead model, the length of the lookahead intervals are variable as the time going on and the number of jobs increasing, and has the tend of steady growth. In this paper, we consider online scheduling of unit length jobs on [Formula: see text] identical parallel-batch machines under this new lookahead model to minimize makespan. The batch capacity is unbounded, that is [Formula: see text]. We present an optimal online algorithm for [Formula: see text], and provide a best possible online algorithm of competitive ratio [Formula: see text] for [Formula: see text], where [Formula: see text] is the positive root of [Formula: see text].

Research on the parallel–batch scheduling with linearly lookahead model

“…In [2], Jiao and Yuan proposed a new scheduling model with linearly lookahead intervals. This model has the character that the length of the lookahead intervals is variable as the time going on and the number of jobs increasing, and has the tend of steady growth.…”

“…For problem P |online, p-batch, b=∞, p j = 1, LK (λ,β) |F max , suppose that λ > 1, 0 ≤ β < λ−1 λ m +λ−2 , then there is no online algorithm with competitive ratio strictly less than 1 + γ m , in which γ m be the positive root of λγ2 m+ (λ + β + λ(λ m −1) λ−1 )γ m + λ m +λ−2 λ−1 β = 1.Proof. Let γ = γ m and H be an arbitrary online algorithm and let be a small positive number.…”