Advertisement

Online Scheduling on Two Uniform Machines to Minimize the Makespan with a Periodic Availability Constraint

  • Ming Liu
  • Chengbin Chu
  • Yinfeng Xu
  • Lu Wang
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6124)

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 T u  = αT a , where T u and T a 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 \(1+\frac{1}{s}\) and prove that LS algorithm is optimal. We also show that for the problem P2, M1PU|online, T u  = αT a |C max , 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 < 1. At last, we study a special case P2, M1PU|online,T u  = αT a , non − increasing sequence|C max , 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 \(\frac{3}{2}\).

Keywords

Online scheduling Makespan Competitive analysis Uniform machines Periodic availability constraint 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Borodin, A., El-Yaniv, R.: Online Computation and Competitive Analysis. Cambridge University Press, Cambridge (1998)zbMATHGoogle Scholar
  2. 2.
    Pruhs, K., Sgall, J., Torng, E.: Online scheduling. In: Leung, J.Y.-T. (ed.) Handbook of Scheduling: Algorithms, Models, and Performance Analysis (2004)Google Scholar
  3. 3.
    Lee, C.Y., Lei, L., Piendo, M.: Current trends in deterministic scheduling. Ann. Oper. Res. 70, 1–41 (1997)zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Liao, C.J., Shyur, D.L., Lin, C.H.: Makespan minimization for two parallel machines with an availability constraint. European Journal of Operational Research 160, 445–456 (2005)zbMATHCrossRefGoogle Scholar
  5. 5.
    Lee, C.Y.: Machine scheduling with an availability constraint. Jounral of Global Optimization 9, 395–416 (1996)zbMATHCrossRefGoogle Scholar
  6. 6.
    Tan, Z., He, Y.: Optimal online algorithm for scheduling on two identical machines with machine availability constraints. Information Processing Letters 83, 323–329 (2002)zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Xu, D., Cheng, Z., Yin, Y., Li, H.: Makespan minimization for two parallel machines scheduling with a periodic availability constraint. Computers and Operations Research (2008), doi:10.1016/j.cor.2008.05.001Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • Ming Liu
    • 1
  • Chengbin Chu
    • 1
    • 2
  • Yinfeng Xu
    • 1
  • Lu Wang
    • 3
  1. 1.School of ManagementXi’an Jiaotong UniversityXi’anP.R. China
  2. 2.Laboratoire Génie IndustrielEcole Centrale Paris, Grande Voie des VignesChâtenay-Malabry CedexFrance
  3. 3.Shanghai Vocational School of CAACShanghaiP.R. China

Personalised recommendations