# How small are shifts required in optimal preemptive schedules?

- 177 Downloads
- 1 Citations

## Abstract

An event in a schedule is a job start, interruption, resumption or completion time. A shift in a schedule is an non-idling interval between two events that does not contain other events. If a scheduling problem with a regular criterion has only integer data (and we consider here only such problems), then the length of smallest shifts required in optimal nonpreemptive schedules is obviously 1. The length of smallest shifts required in optimal preemptive schedules, however, can be infinitely small. As Sauer and Stone (Order 4:195–206, 1987) showed more than 25 years ago, shifts of length less than \(m^{-n}\) are not required in shortest preemptive schedules of \(n\) unit-length jobs with precedence constraints on \(m\) identical parallel machines. They showed, on the other hand, that there are instances for infinitely many \(n\) such that shifts of length less than \((m-1)^{-n/(3m)}(m-2)/m\) can be required if \(m\ge 3\). In this paper, we continue research in the same direction and strengthen their results, finding the respective tighter bounds, \(m^{-(n+1)/2}\) and \((m-1)^{-n/(m+3)}\). We also obtain similar results for other preemptive scheduling problems on identical parallel machines. A useful consequence of certain of these results is that preemptive scheduling problems with unequal release dates and/or unequal due dates can require even smaller shifts for optimality. We also identify problems whose optimal preemptive schedules do not require shifts of length less than \(1/m\).

## Keywords

Preemptive scheduling Identical parallel machines Release dates Due dates## Notes

### Acknowledgments

We are thankful to the anonymous reviewer who found a serious omission in the original version of this paper and also observed that Corollary 5 can be extended to the criteria of a minimum number of late jobs and a minimum weighted number of late jobs.

## References

- Baptiste, Ph. (2002).
*On preemption redundancy*. Technical report. Yorktown Heights, NY: IBM Watson Research Center.Google Scholar - Baptiste, Ph, Carlier, J., Kononov, A., Sevastyanov, S., & Sviridenko, M. (2009a). Integrality property in preemptive parallel machine scheduling.
*Lectue Notes in Computer Science*,*5675*, 38–46.Google Scholar - Baptiste, Ph, Carlier, J., Kononov, A., Queyranne, M., Sevastyanov, S., & Sviridenko, M. (2011). Properties of optimal schedules in preemptive shop scheduling.
*Discrete Applied Mathematics*,*159*(5), 272–280.CrossRefGoogle Scholar - Baptiste, Ph, Carlier, J., Kononov, A., Queyranne, M., Sevastianov, S., & Sviridenko, M. (2009b). Structural properties of preemptive schedules.
*Discrete Analysis and Operations Research*,*16*(1), 3–36.Google Scholar - Baptiste, Ph, & Timkovsky, V. G. (2001). On preemption redundancy in scheduling unit processing time jobs on two parallel machines.
*Operations Research Letters*,*28*(5), 205–212.CrossRefGoogle Scholar - Baptiste, Ph, & Timkovsky, V. G. (2004). Shortest path to nonpreemptive schedules of unit-time jobs on two identical parallel machines with minimum total completion time.
*Mathematical Methods of Operations Research*,*60*(1), 145–153.CrossRefGoogle Scholar - Bertogna, M., Buttazzo, G. C., Marinoni, M., Yao, G., Esposito, F. & Caccamo. M. (2010). Preemption points placement for sporadic task sets. In
*ECRTS ’10 proceedings of the 2010 22nd Euromicro conference on real-time systems*(pp. 251–260). Washington, DC: IEEE Computer Society.Google Scholar - Brucker, P., Heitman, S., & Hurink, J. (2003). How useful are preemptive schedules.
*Operations Research Letters*,*31*, 129–136.Google Scholar - Brucker, P., Hurink, J., & Knust, S. (2002). A polynomial algorithm for \({P}\vert p_j=1,r_j,\text{ outtree }\vert \sum {C}j\).
*Mathematical Methods of Operations Research*,*56*(3), 407–412. Google Scholar - Buttazzo, G. C., Bertogna, M., & Yao, G. (2013). Limited preemptive scheduling for real-time systems: A survey.
*IEEE Transactions on Industrial Informatics*,*9*(3), 15.Google Scholar - Coffman, E. G, Jr, Sethuraman, J., & Timkovsky, V. G. (2003). Ideal preemptive schedules on two processors.
*Acta Informatica*,*39*(8), 597–612.CrossRefGoogle Scholar - Du, J., Leung, J. Y.-T., & Young, G. H. (1991). Scheduling chain-structured tasks to minimize makespan and mean flow time.
*Information and Computation*,*92*(2), 219–236.CrossRefGoogle Scholar - Graham, R. L., Lawler, E. L., Lenstra, J. K., & Rinnooy Kan, A. H. G. (1979). Optimization and approximation in deterministic sequencing and scheduling: A survey.
*Annals of Discrete Mathematics*,*5*, 287–326.CrossRefGoogle Scholar - Lawler, E. L. (1982). Preemptive scheduling of precedence-constrained jobs on parallel machines. In M. A. H. Dempster, J. K. Lenstra, & A. H. G. Rinnooy Kan (Eds.),
*Deterministic and stochastic scheduling*(pp. 101–123). Proceedings of the NATO advanced study and research institute on theoretical approaches to scheduling problems held in Durham, July 6–17, 1981, Vol. 84 of NATO advanced study institute series C: Mathematical and Physical Sciences Dordrecht: D. Reidel Publishing Co.Google Scholar - Lawler, E. L., Lenstra, J. K., Rinnooy Kan, A. H. G., & Shmoys, D. B. (1993). Sequencing and scheduling: Algorithms and complexity. In S. C. Graves, A. H. G. Rinnooy Kan, & P. Zipkin (Eds.),
*Handbook on operations research and management science, Vol. 4 of logistics of production and inventory*(pp. 445–552). Amsterdam: Elsevier.Google Scholar - McNaughton, R. (1959). Scheduling with deadlines and loss functions.
*Management Science*,*6*, 1–12.CrossRefGoogle Scholar - Meumeu Yomsi, P., & Sorel, Y. (2007). Extending rate monotonic analysis with exact cost of preemptions for hard real-time systems. In
*ECRTS ’07 Proceedings of the 19th Euromicro conference on real-time systems*(pp. 280–290). Washington, DC: IEEE Computer Society.Google Scholar - Muntz, R. R., & Coffman, E. G, Jr. (1969). Optimal preemptive scheduling on two-processor systems.
*IEEE Transactions of Computers*,*C18*, 1014–1020.CrossRefGoogle Scholar - Muntz, R. R., & Coffman, E. G, Jr. (1970). Preemptive scheduling of real-time tasks on multiprocessor system.
*Journal of the ACM*,*17*(2), 324–338.CrossRefGoogle Scholar - Ndoye, F. & Sorel, Y. (2011). Preemptive multiprocessor real-time scheduling with exact preemption cost. In
*Proceedings of 5th Junior researcher workshop on real-time computing*(pp. 25–28). Nantes, 29–30 September.Google Scholar - Ryser, H. J. (1956). Maximal determinants in combinatorial investigations.
*Canadian Journal of Mathematics*,*30*, 756–762.Google Scholar - Sauer, N. W., & Stone, M. G. (1987). Rational preemptive scheduling.
*Order*,*4*, 195–206.CrossRefGoogle Scholar - Shachnai, H., Tamir, T., & Woeginger, G. J. (2005). Minimizing makespan and preemption costs on a system of uniform machines.
*Algorithmica*,*42*, 309–334.CrossRefGoogle Scholar - Timkovsky, V. G. (2004). Reducibility among scheduling classes. In J. Y. -T. Leung (Ed.),
*Handbook of scheduling: Algorithms, models, and performance analysis*, Chapter 8 (pp. 8.1–8.42). London: CRC Press.Google Scholar