Abstract
This paper considers an online hierarchical scheduling problem on two parallel identical machines. The objective is to minimize the makspan. It is assumed that all jobs have bounded processing times in between \(p\) and \(rp\), where \(p>0\) and \(r\ge 1\). We first improve a previous result by giving an optimal online algorithm for the non-preemptive version. For the preemptive version, we present an optimal preemptive algorithm without introducing idle time for all \(r\ge 1\). If the algorithm is allowed to use idle time, we show that the semi-online information that jobs are tightly-grouped cannot help improve the bound of the pure online problem.
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References
Bar-Noy A, Freund A, Naor J (2001) On-line load balancing in a hierarchical server topology. SIAM Journal on Computing. 31:527–549
Chassid O, Epstein L (2008) The hierarchical model for load balancing on two machines. Journal of Combinatorial Optimization. 15(4):305–314
Crescenzi P, Gambosi G, Penna P (2004) On-line algorithms for the channel assignment problem in cellular networks. Discrete Applied Mathematics. 137:237–266
Dósa G, Epstein L (2008) Preemptive scheduling on a small number of hierarchical machines. Information and Computation. 206(5):602–619
He Y, Zhang G (1999) Semi on-line scheduling on two identical machines. Computing. 62:179–187
Jiang Y (2008) Online scheduling on parallel machines with two GoS levels. Journal of Combinatorial Optimization. 16:28–38
Jiang Y, He Y, Tang C (2006) Optimal online algorithms for scheduling on two identical machines under a grade of service. Journal of Zhejiang University Science. 7A:309–314
Liu M, Chu C, Xu Y, Zheng F (2011) Semi-online scheduling on 2 machines under a grade of service provision with bounded processing times. Journal of Combinatorial Optimization. 21:138–149
Park J, Chang S, Lee K (2006) Online and semi-online scheduling of two machines under a grade of service provision. Operations Research Letters. 34:692–696
Tan Z, Zhang A (2010) A note on hierarchical scheduling on two uniform machines. Journal of Combinatorial Optimization. 20:85–95
Tan Z, Zhang A (2011) Online hierarchical scheduling: An approach using mathematical programming. Theoretical Computer Science. 412:246–256
Wu Y, Ji M, Yang Q (2012) Optimal semi-online scheduling algorithms on two parallel identical machines under a grade of service provision. International Journal of Production Economics. 135:367–371
Zhang A, Jiang Y, Tan Z (2009) Online parallel machines scheduling with two hierarchies. Theoretical Computer Science. 410:3597–3605
Acknowledgments
An Zhang, Yiwei Jiang, and Jueliang Hu were supported by the National Natural Science Foundation of China through Grant Nos. 11201105, 11001242, and 11071220 respectively. Yiwei Jiang is also supported by Zhejiang Province Natural Science Foundation of China (Y6090175).
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Zhang, A., Jiang, Y., Fan, L. et al. Optimal online algorithms on two hierarchical machines with tightly-grouped processing times. J Comb Optim 29, 781–795 (2015). https://doi.org/10.1007/s10878-013-9627-7
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DOI: https://doi.org/10.1007/s10878-013-9627-7