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On the optimality of LEPT and μc rules for parallel processors and dependent arrival processes

Part of: Operations research and management science

Published online by Cambridge University Press:  01 July 2016

Arie Hordijk  and
Ger Koole
Arie Hordijk*
Affiliation:
University of Leiden
Ger Koole
Affiliation:
University of Leiden
*
* Postal address: Dept. of Mathematics and Computer Science, University of Leiden P.O. Box 9512, 2300 RA Leiden, The Netherlands.
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Abstract

In this paper we study scheduling problems of multiclass customers on identical parallel processors. A new type of arrival process, called a Markov decision arrival process, is introduced. This arrival process can be controlled and allows for an indirect dependence on the numbers of customers in the queues. As a special case we show the optimality of LEPT and the µc-rule in the last node of a controlled tandem network for various cost structures. A unifying proof using dynamic programming is given.

Keywords

STOCHASTIC SCHEDULINGDYNAMIC PROGRAMMINGDEPENDENT ARRIVALS

MSC classification

Primary: 90B22: Queues and service
Secondary: 90C40: Markov and semi-Markov decision processes
Type
Research Article
Information
Advances in Applied Probability , Volume 25 , Issue 4 , December 1993 , pp. 979 - 996
Copyright
Copyright © Applied Probability Trust 1993 

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Footnotes

**

Present address: CWI, Kruislaan 413, 1098 SJ Amsterdam, The Netherlands.

References

[1] Asmussen, S. and Koole, G. (1993) Marked point processes as limits of Markovian arrival streams. J. Appl. Prob. 30, 365–372.Google Scholar
[2] Baras, J. S., Dorsey, A. J. and Makowski, A. M. (1985) Two competing queues with linear costs and geometric service requirements: the µc-rule is often optimal. Adv. Appl. Prob. 17, 186–209.CrossRefGoogle Scholar
[3] Baras, J. S., Ma, D.-J. and Makowski, A. M. (1985) K competing queues with geometric service requirements and linear costs: the µc-rule is always optimal. Syst. Control Lett. 6, 173–180.Google Scholar
[4] Bruno, J., Downey, P. and Frederickson, G. N. (1981) Sequencing tasks with exponential service times to minimize the expected flow time or makespan. J. Assoc. Comput. Mach. 28, 100–113.Google Scholar
[5] Buyukkoc, C., Varaiya, P. and Walrand, J. (1985) The cp rule revisited. Adv. Appl. Prob. 17, 237–238.Google Scholar
[6] Chang, C. S., Chao, X., Pinedo, M. and Weber, R. R. (1992) On the optimality of LEPT and cµ-rules for machines in parallel. J. Appl. Prob. 29, 667–681.Google Scholar
[7] Hordijk, A. and Koole, G. (1992) The µc-rule is not optimal in the second node of the tandem queue: a counterexample. Adv. Appl. Prob. 24, 234–237.Google Scholar
[8] Hordijk, A. and Koole, G. (1992) On the assignment of customers to parallel queues. Prob. Eng. Inf. Sci. 6, 495–511.Google Scholar
[9] Kämpke, T. (1989) Optimal scheduling of jobs with exponential service times on parallel processors. Operat. Res. 37, 126–133.Google Scholar
[10] Koole, G. (1992) Stochastic scheduling and dynamic programming. Ph.D. thesis, University of Leiden. (Available from the author on request.) Google Scholar
[11] Neuts, M. F. (1981) Matrix-Geometric Solutions in Stochastic Models: An Algorithmic Approach. Johns Hopkins University Press, Baltimore.Google Scholar
[12] Pinedo, M. (1983) Stochastic scheduling with stochastic release dates and due dates. Operat. Res. 31, 559–572.Google Scholar
[13] Pinedo, M. and Weiss, G. (1971) Scheduling of stochastic tasks on two parallel processors. Naval Res. Logist. Quart. 26, 527–535.Google Scholar
[14] Righter, R. and Shanthikumar, J. G. (1989) Scheduling multiclass single server queueing systems to stochastically maximize the number of successful departures. Prob. Eng. Inf. Sci., 3, 323–333.Google Scholar
[15] Righter, R. and Shanthikumar, J. G. (1992) Extension of the bivariate characterization for stochastic orders. Adv. Appl. Prob. 24, 506–508.CrossRefGoogle Scholar
[16] Ross, S. M. (1983) Introduction to Stochastic Dynamic Programming. Academic Press, New York.Google Scholar
[17] Shanthikumar, J. G. and Yao, D. D. W. (1992) Multiclass queueing systems: polymatroidal structure and optimal scheduling control. Operat. Res. 40, S293–S299.CrossRefGoogle Scholar
[18] Van Der Heyden, L. (1981) Scheduling jobs with exponential service times on non-identical processors so as to minimize expected makespan. Math. Operat. Res. 6, 305–312.Google Scholar
[19] Walrand, J. (1988) An Introduction to Queueing Networks. Prentice-Hall, Englewood Cliffs, NJ.Google Scholar
[20] Weber, R. R. (1982) Scheduling jobs with stochastic processing requirements on parallel machines to minimize makespan or flowtime. J. App. Prob. 19, 167–182.Google Scholar
[21] Weber, R. R. (1983) Scheduling stochatic jobs on parallel machines to minimize makespan or flowtime. In Applied Probability-Computer Science: The Interface, ed. Disney, R. and Ott, T. Birkhauser, Boston.Google Scholar
[22] Weber, R. R. (1988) Stochastic scheduling on parallel processors and minimization of concave functions of completion times. In Stochastic Differential Systems, Stochastic Control Theory and Applications 10, ed. Fleming, W. and Lions, P.-L. Springer-Verlag, New York.Google Scholar
[23] Weiss, G. (1982) Multiserver stochastic scheduling. In Deterministic and Stochastic Scheduling, ed. Dempster, M. A. H., Lenstra, J. K. and Rinnooy Kan, A. H. G. Reidel, Dordrecht.Google Scholar
[24] Weiss, G. and Pinedo, M. (1980) Scheduling tasks with exponential service times on non-identical processors to minimize various cost functions. J. Appl. Prob. 17, 187–202.Google Scholar