### Abstract

We consider semi on-line scheduling on two uniform machines. The speed of the slow machine is normalized to 1 while the speed of the fast machine is assumed to be s ≥ 1. Jobs of size J_{1},J_{2}, . . . arrive one at a time, and each J_{i} (i ≥ 1) has to be assigned to one of the machines before J_{i}+1 arrives. The assignment cannot be changed later. The processing time of the ith job is J_{i} on the slow machine and J_{i}/s on the fast one. The objective is to minimize the makes pan. We study both the case where the only information known in advance is the total sizeΣi≥1 Ji of the jobs and the case where the only information known in advance is the optimum makes pan. For each of these two cases, we almost completely determine the best possible competitive ratio of semi on-line algorithms compared to the off-line optimum, as a function of s in the range 1 ≤s < 1+ √ 17 4 ≈ 1.2808, except for a very short subinterval around s = 1.08. We also prove that the best competitive ratio achievable for known optimum is at least as good as the one for known sum, even for any number of uniform machines of any speeds.

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

Pages (from-to) | 458-480 |

Number of pages | 23 |

Journal | Journal of Combinatorial Optimization |

Volume | 21 |

Issue number | 4 |

DOIs | |

Publication status | Published - May 1 2011 |

### Keywords

- Competitive analysis
- Semi on-line scheduling
- Uniform machines

### ASJC Scopus subject areas

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