Online scheduling with a buffer on related machines

Abstract

Online scheduling with a buffer is a semi-online problem which is strongly related to the basic online scheduling problem. Jobs arrive one by one and are to be assigned to parallel machines. A buffer of a fixed capacity K is available for storing at most K input jobs. An arriving job must be either assigned to a machine immediately upon arrival, or it can be stored in the buffer for unlimited time. A stored job which is removed from the buffer (possibly, in order to allocate a space in the buffer for a new job) must be assigned immediately as well. We study the case of two uniformly related machines of speed ratio s ≥ 1, with the goal of makespan minimization.Two natural questions can be asked. The first question is whether this model is different from standard online scheduling, that is, is any size of buffer K > 0 already helpful to the algorithm, compared to the case K = 0. The second question is whether there exists a constant K, so that a larger buffer is no longer beneficial to an algorithm, that is, increasing the size of the buffer above this threshold would not change the best competitive ratio further. Previous work [15,19,5] shows that in the case s = 1, already K = 1 allows to design a 4 3 -competitive algorithm, which is best possible for any K ≥ 1, whereas the best possible ratio for K = 0 is 3 2 . Similar results have been show for multiple identical machines [5]. We answer both questions affirmatively, and show that a buffer of size K = 2 is sufficient to achieve the a competitive ratio which matches the lower bound for K → ∞ for any s > 1. In fact, we show that a buffer of size K = 1 can evidently be exploited by the algorithm for any s > 1, but for a range of values of s, it is still weaker than a buffer of size 2. On the other hand, in the case s ≥ 2, a buffer of size K = 1 already allows to achieve optimal bounds.

Preemptive Online Scheduling with Reordering

“…Intuitively, this seems to be the correct approach; the algorithm is aware of the exact sizes of the largest jobs and takes them into account in the other scheduling decisions, but it postpones their assignment until the later. Nevertheless, in [3] one of the algorithms of optimal competitive ratio, which uses K = 1, has two cases, where in one of the cases the larger available job is assigned while the smaller job is stored in the buffer. We note an interesting difference with the algorithms of [6].…”

An overview of some results for reordering buffers

“…About a decade ago, some initial results about reordering buffers for non-preemptive minimum makespan scheduling where obtained [11,21,23,28]. These results are mainly concerned with only two machines.…”

Online Minimum Makespan Scheduling With a Buffer

“…Related models: Some similar models have been investigated in the last years such as (i) online scheduling problems with bounded migration [19]: when a new job comes some already scheduled jobs can be reassigned. Here the bound migration means that the total size of rescheduled jobs is bounded or the total cost of the rescheduled jobs is bounded; (ii) Online scheduling problems with bounded rearrangement : where we are allowed to reschedule a bounded number of jobs in order to allocate a new job and there are several variants of this model [4,6,17,21]; (iii) related machines: on two machines with a buffer [5], Dósa and Epstein gave an optimal online algorithm with a buffer of size 2, and also an optimal online algorithm with a buffer of size one if the speed ratio of the fast and slow machines is larger than 2; on two machines without a buffer, Epstein and Favrholdt gave an optimal algorithm if preemption is allowed [8]; the upper bound 5.828 and lower bound are by Berman and others in 2000 [3]. Our contribution: (i) For m > 51 identical machines, we give a 1.5-competitive online algorithm with a buffer of size ⌈1.5m⌉; although our result is worse than the best result ⌈1.477m⌉ [20] published in 2013, our result [16] was the first result to beat ⌈1.6197m⌉ [7] in 2012; (ii) for three identical machines, we propose an optimal online algorithm with a buffer of size six, which is smaller than a buffer of size nine [7] and a buffer of size eight [20]; (iii) for m uniform machines, using a buffer of size m, we improve the competitive ratio from 2 + ǫ in [7] to 2 − 1/m + ǫ, where ǫ > 0 is arbitrarily small.…”