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On the optimality of LEPT and rules for machines in parallel

  • Cheng-Shang Chang (a1), Xiuli Chao (a2), Michael Pinedo (a3) and Richard Weber (a4)


We consider scheduling problems with m machines in parallel and n jobs. The machines are subject to breakdown and repair. Jobs have exponentially distributed processing times and possibly random release dates. For cost functions that only depend on the set of uncompleted jobs at time t we provide necessary and sufficient conditions for the LEPT rule to minimize the expected cost at all t within the class of preemptive policies. This encompasses results that are known for makespan, and provides new results for the work remaining at time t. An application is that if the rule has the same priority assignment as the LEPT rule then it minimizes the expected weighted number of jobs in the system for all t. Given appropriate conditions, we also show that the rule minimizes the expected value of other objective functions, such as weighted sum of job completion times, weighted number of late jobs, or weighted sum of job tardinesses, when jobs have a common random due date.


Corresponding author

Postal address: IBM Research Division, T.J. Watson Research Center, P.O. Box 704, Yorktown Heights, NY 10598, USA.
∗∗ Postal address: Division of Industrial and Management Engineering, New Jersey Institute of Technology, Newark, NJ 07102, USA.
∗∗∗ Postal address: Department of Industrial Engineering and Operations Research, Columbia University, New York, NY 10027, USA.
∗∗∗∗ Postal address: Cambridge University Engineering Department, Mill Lane, Cambridge, CB2 1RX, UK.


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The research of this author was partially supported by the NSF under grant ECS 86–14689.



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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, 186209.
Buyukkoc, C., Varaiya, P. and Walrand, J. (1985) The cµ rule revisited. Adv. Appl. Prob. 17, 237238.
Chang, C. S. (1992) Ordering for stochastic majorization: theory and applications. Adv. Appl. Prob. 24(3).
Chang, C. S. and Yao, D. D. (1990) Rearrangement, majorization and stochastic scheduling.
Frederickson, G. N., Bruno, J. and Downey, P. (1981) Sequencing tasks with exponential service times to minimize the expected flow time or makespan. J. Assoc. Comput. Mach. 28, 100113.
Kämpke, T. (1987) On the optimality of static priority policies in stochastic scheduling on parallel machines. J. Appl. Prob. 24, 430448.
Kämpke, T. (1989) Optimal scheduling of jobs with exponential service times on identical parallel processors. Operat. Res. 37, 126133.
Keilson, T. (1979) Markov Chain Models - Rarity and Exponentiality. Springer-Verlag, New York.
Marshall, A. W. and Olkin, I. (1979) Inequalities: Theory of Majorization and its Applications. Academic Press, New York.
Muirhead, R. F. (1903) Some methods applicable to identities and inequalities of symmetric algebraic functions with n letters. Proc. Edinburgh Math. Soc. 21, 144157.
Pinedo, M. (1983) Stochastic scheduling with release dates and due dates. Operat. Res. 31, 559572.
Pinedo, M. and Rammouz, E. (1988) A note on stochastic scheduling on a single machine subject to breakdown and repair. Prob. Eng. Inf. Sci. 2, 4149.
Pinedo, M. and Weiss, G. (1979) Scheduling stochastic tasks on two parallel processors. Naval Res. Log. Quart. 26, 527535.
Ross, S. M. (1983) Introduction to Stochastic Dynamic Programming. Academic Press, New York.
Shanthikumar, J. G. and Yao, D. D. W. (1991) Multiclass queueing systems: polymatroidal structure and optimal scheduling control. Operat. Res.
Van Der Heyden, L. (1981) Scheduling jobs with exponential processing and arrival times on identical processors so as to minimize expected makespan. Math. Operat. Res. 6, 305312.
Walrand, J. (1988) An Introduction to Queueing Networks. Prentice Hall, Englewood Cliffs, NJ.
Weber, R. R. (1982) Scheduling jobs with stochastic processing requirements on parallel machines to minimize makespan or flowtime. J. Appl. Prob. 19, 167182.
Weber, R. R. (1983) Scheduling stochastic jobs on parallel machines to minimize makespan or flowtime. In Applied Probability - Computer Science: The Interface , ed. Disney, R. and Ott, T., pp. 327338. Birkhauser, Boston, MA.
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., pp. 601609. Springer-Verlag, New York.
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., pp. 157179. Reidel, Dordrecht.
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, 187202.


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