Journal of Scheduling

, Volume 6, Issue 2, pp 183–211 | Cite as

Patience is a Virtue: The Effect of Slack on Competitiveness for Admission Control

  • Michael H. Goldwasser


We consider the on-line competitiveness for scheduling a single resource non-preemptively in order to maximize its utilization. Our work examines this model when parameterizing an instance by a new value which we term the patience. This parameter measures each job's willingness to endure a delay before starting, relative to this same job's processing time. Specifically, the slack of a job is defined as the gap between its release time and the last possible time at which it may be started while still meeting its deadline. We say that a problem instance has patience κ, if each job with length ‖J‖ has a slack of at least κ·‖J‖.

Without any restrictions placed on the job characteristics, previous lower bounds show that no algorithm, deterministic or randomized, can guarantee a constant bound on the competitiveness of a resulting schedule. Previous researchers have analyzed a problem instance by parameterizing based on the ratio between the longest job's processing time and the shortest job's processing time. Our main contribution is to provide a fine-grained analysis of the problem when simultaneously parameterized by patience and the range of job lengths. We are able to give tight or almost tight bounds on the deterministic competitiveness for all parameter combinations.

If viewing the analysis of each parameter individually, our evidence suggests that parameterizing solely on patience provides a richer analysis than parameterizing solely on the ratio of the job lengths. For example, in the special case where all jobs have the same length, we generalize a previous bound of 2 for the deterministic competitiveness with arbitrary slacks, showing that the competitiveness for any κ ≥ 0 is exactly 1+1/(⌊κ⌋+1). Without any bound on the job lengths, a simple greedy algorithm is (2+(1/κ))-competitive for any κ<0. More generally we will find that for any fixed ratio of job lengths, the competitiveness of the problem tends towards 1 as the patience is increased. The converse is not true, as for any fixed κ<0 we find that the competitiveness is bounded away from 1, no matter what further restrictions are placed on the ratio of job lengths.

patience slack admission control scheduling online analysis 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Awerbuch, B., Y. Azar, and S. Plotkin, “Throughput-competitive on-line routing,” in Proc. 34th Symp. on Foundations of Computer Science, Palo Alto, California, November, 1993, pp. 32-40.Google Scholar
  2. Awerbuch, B., Y. Bartal, A. Fiat, and A. Rosén, “Competitive non-preemptive call control,” in Proc. Fifth Annu. ACM-SIAM Symp. on Discrete Algorithms, Arlington, Virginia, January, 1994, pp. 312-320.Google Scholar
  3. Bar-Noy, A., R. Canetti, S. Kutten, Y. Mansour, and B. Schiber, “Bandwidth allocation with preemption,” SIAM J. Comput., 28(5), 1806-1828 (1999).Google Scholar
  4. Bar-Noy, A., J.A. Garay, A. Herzberg, and S. Aggarwal, “Sharing video on demand,” Manuscript Presented at the Workshop on Algorithmic Aspects of Communication, Bologna, Italy, July, 1997.Google Scholar
  5. Baruah, S., G. Koren, D. Mao, B. Mishra, A. Ragunathan, L. Rosier, D. Shasta, and F. Wang, “On the competitiveness of on-line real-time task scheduling,” Real-Time Syst., 4, 125-144 (1992).Google Scholar
  6. Baruah, S.K. and J.R. Haritsa, “Scheduling for overload in real-time systems,” IEEE Trans. Comput., 46(9), 1034-1039 (1997).Google Scholar
  7. Ben-David, S., A. Borodin, R. Karp, G. Tardos, and A. Widgerson, “On the power of randomization in on-line algorithms,” Algorithmica, 11(1), 2-14 (1994).Google Scholar
  8. Feldmann, A., B. Maggs, J. Sgall, D. Sleator, and A. Tomkins, “Competitive analysis of call admission algorithms that allow delay,” Technical Report CMU-CS-95-102. School of Computer Science Carnegie Mellon University, Pittsburgh, PA, 1995.Google Scholar
  9. Garey, J., I. Gopal, and S. Kutten, “Efficient on-line call control algorithms,” J. Algorithms, 23(1), 180-194 (1997).Google Scholar
  10. Goldman, S., J. Parwatikar, and S. Suri, “On-line scheduling with hard deadlines,” J. Algorithms, 34(2), 370-389 (2000).Google Scholar
  11. Graham, R., E.L. Lawler, J.K. Lenstra, and A. Rinnooy Kan, “Optimization and approximation in deterministic sequencing and scheduling: A survey,” in Discrete Optimization II, Vol. 5 of Annals of Discrete Mathematics, North-Holland, Amsterdam, 1979, pp. 287-326.Google Scholar
  12. Kalyanasundaram, B. and K. Pruhs, “Speed is as powerful as clairvoyance,” J. ACM, 47(4), 617-743 (2000).Google Scholar
  13. Karlin, A., M. Manasse, L. Rudolph, and D. Sleator, “Competitive snoopy paging,” Algorithmica, 3(1), 70-119 (1988).Google Scholar
  14. Kopetz, H., Real-time Systems: Design Principles for Distributed Embedded Applications, Kluwer Academic Publishers, Boston, 1997.Google Scholar
  15. Lawler, E. L., J. K. Lenstra, A. Rinnooy Kan, and D. B. Shmoys, “Sequencing and scheduling: Algorithms and complexity” in S. Graves, A. Rinnooy Kan, and P. Zipken (eds.), Logistics of Production and Inventory, Vol. 4 of Handbooks in Operations Research and Management Science, North-Holland, Amsterdam 1993, pp. 445-522.Google Scholar
  16. Lipton, R. and A. Tomkins, “Online interval scheduling,” in Proc. 5th Annu. ACM-SIAM Symp. on Discrete Algorithms, Arlington, Virginia, January, 1994, pp. 302-311.Google Scholar
  17. Plotkin, S.A., “Competitive routing of virtual circuits in ATM networks,” IEEE J. Selected Areas Commun., 13(6), 1128-1136 (1995).Google Scholar
  18. Raghavan, P. and M. Snir, “Memory versus randomization in on-line algorithms,” IBM J. Res. Dev., 38, 683-707 (1994).Google Scholar
  19. Sleator, D. and R. Tarjan, “Amortized efficiency of list update and paging rules,” Commun. ACM, 28, 202-208 (1985).Google Scholar

Copyright information

© Kluwer Academic Publishers 2003

Authors and Affiliations

  • Michael H. Goldwasser
    • 1
  1. 1.Department of Computer ScienceLoyola University ChicagoChicago

Personalised recommendations