Online scheduling on two uniform machines to minimize the makespan with a periodic availability constraint

  • Authors:

  • Ming Liu;Chengbin Chu;Yinfeng Xu;Lu Wang

  • Affiliations:

  • School of Management, Xi'an Jiaotong University, Xi'an, Shaanxi Province, P.R. China;School of Management, Xi'an Jiaotong University, Xi'an, Shaanxi Province, P.R. China and Laboratoire Génie Industriel, Ecole Centrale Paris, Châtenay-Malabry Cedex, France;School of Management, Xi'an Jiaotong University, Xi'an, Shaanxi Province, P.R. China;Shanghai Vocational School of CAAC, Shanghai, P.R. China

  • Venue:
  • AAIM'10 Proceedings of the 6th international conference on Algorithmic aspects in information and management
  • Year:
  • 2010

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Abstract

We consider the problem of online scheduling on 2 uniform machines where one machine is periodically unavailable. The problem is online in the sense that when a job presents, we have to assign it to one of the 2 uniform machines before the next one is seen. Preemption is not allowed. The objective is to minimize makespan. Assume that the speed of the periodically unavailable machine is normalized to 1, while the speed of the other one is s. Given a constant number α 0, we also suppose that Tu = αTa, where Tu and Ta are the length of each unavailable time period and the length of the time interval between two consecutive unavailable time periods, respectively. In the case where s ≥ 1, we show a lower bound of the competitive ratio 1 + 1/s and prove that LS algorithm is optimal. We also show that for the problem P2, M1PU|online, Tu = αTa|Cmax, LS algorithm proposed in [7] is optimal with a competitive ratio 2. After that, we give some lower bounds of competitive ratio in the case 0 s P2, M1PU|online, Tu = αTa, non - increasing sequence|Cmax, where non-increasing sequence means that jobs arrive in a non-increasing order of their processing times. We show that LS algorithm is optimal with a competitive ratio 3/2.