### 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 (Kellerer et al., Oper. Res. Lett. 21, 235-242, 1997; Zhang, Inf. Process. Lett. 61, 145-148, 1997; Englert et al., Proc. 48th Symp. Foundations of Computer Science (FOCS), 2008) 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 (Englert et al., Proc. 48th Symp. Foundations of Computer Science (FOCS), 2008). 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.

Original language | English |
---|---|

Pages (from-to) | 161-179 |

Number of pages | 19 |

Journal | Journal of Combinatorial Optimization |

Volume | 20 |

Issue number | 2 |

DOIs | |

Publication status | Published - Aug 2010 |

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### Keywords

- Scheduling
- Semi-online algorithms
- Uniformly related machines

### ASJC Scopus subject areas

- Discrete Mathematics and Combinatorics
- Applied Mathematics
- Computational Theory and Mathematics
- Computer Science Applications
- Control and Optimization

### Cite this

*Journal of Combinatorial Optimization*,

*20*(2), 161-179. https://doi.org/10.1007/s10878-008-9200-y

**Online scheduling with a buffer on related machines.** / Dósa, G.; Epstein, Leah.

Research output: Contribution to journal › Article

*Journal of Combinatorial Optimization*, vol. 20, no. 2, pp. 161-179. https://doi.org/10.1007/s10878-008-9200-y

}

TY - JOUR

T1 - Online scheduling with a buffer on related machines

AU - Dósa, G.

AU - Epstein, Leah

PY - 2010/8

Y1 - 2010/8

N2 - 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 (Kellerer et al., Oper. Res. Lett. 21, 235-242, 1997; Zhang, Inf. Process. Lett. 61, 145-148, 1997; Englert et al., Proc. 48th Symp. Foundations of Computer Science (FOCS), 2008) 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 (Englert et al., Proc. 48th Symp. Foundations of Computer Science (FOCS), 2008). 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.

AB - 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 (Kellerer et al., Oper. Res. Lett. 21, 235-242, 1997; Zhang, Inf. Process. Lett. 61, 145-148, 1997; Englert et al., Proc. 48th Symp. Foundations of Computer Science (FOCS), 2008) 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 (Englert et al., Proc. 48th Symp. Foundations of Computer Science (FOCS), 2008). 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.

KW - Scheduling

KW - Semi-online algorithms

KW - Uniformly related machines

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U2 - 10.1007/s10878-008-9200-y

DO - 10.1007/s10878-008-9200-y

M3 - Article

VL - 20

SP - 161

EP - 179

JO - Journal of Combinatorial Optimization

JF - Journal of Combinatorial Optimization

SN - 1382-6905

IS - 2

ER -