Abstract
We address the problem of scheduling a multiclass queueing network on M parallel servers to minimize time-average linear holding costs. We analyze a heuristic priority-index rule, based on Klimov's solution to the single-server model: Compute the indices given by Klimov's adaptive greedy algorithm and, when a server becomes free, select a customer with largest index. We present closed-form performance guarantees for this heuristic, with respect to (1) the optimal cost in the original parallel-servers network, and (2) the optimal cost in a “corresponding” single-server network, attended by a server working M times faster. Simpler expressions are derived for the special case that there is no customer feedback, where the heuristic becomes the cµ-rule. Our analysis is based on a primal-dual approach: We compare the cost of the heuristic to the value of (the dual of) a strong linear programming (LP) relaxation, which represents the optimal cost for the “corresponding” single-server network. This relaxation follows from a set of approximate conservation laws (ACLs) satisfied by the network. Our proof of these laws relies on the first set of work decomposition laws known for this model, which we obtain from a classical flow conservation law.
Preview
Unable to display preview. Download preview PDF.
References
Bertsimas, D. and Niño-Mora, J.: Conservation laws, extended polymatroids and multi-armed bandit problems; a polyhedral approach to indexable systems. Math. Oper. Res. 21 (1996a) 257–306
Bertsimas, D. and Niño-Mora, J.: Optimization of multiclass queueing networks with changeover times via the achievable region approach: Part II, the multi-station case. Working paper, Operations Research Center, MIT, (1996b)
Federgruen, A. and Groenevelt, H.: Characterization and optimization of achievable performance in general queueing systems. Oper. Res. 36 (1988) 733–741
Fuhrmann, S.W. and Cooper, R.B.: Stochastic decompositions in the M/C/1 queue with generalized vacations. Oper. Res. 33 (1985) 1117–1129
Glazebrook, K.D. and Garbe, R.: Almost optimal policies for stochastic systems which almost satisfy conservation laws. Working paper, Department of Statistics, Newcastle University, (1996)
Glazebrook, K.D. and Niño-Mora, J.: Scheduling multiclass queueing networks on parallel servers: Approximate and heavy traffic optimality of Klimov's rule. CORE Discussion Paper 9710, Université catholique de Louvain, (1997a)
Glazebrook, K.D. and Niño-Mora, J.: An LP approach to stability, optimization and performance analysis for Markovian multiclass queueing networks. CORE Discussion Paper, Université catholique de Louvain, (1997b)
Klimov, G.P.: Time sharing service systems I. Theory Probab. Appl. 19 (1974) 532–551
Nemhauser, G.L. and Wolsey, L.A.: Integer and Combinatorial Optimization. Wiley, New York, (1988)
Niño-Mora, J. Optimal Resource Allocation in a Dynamic and Stochastic Environment: A Mathematical Programming Approach. PhD Dissertation, Sloan School of Management, MIT, (1995)
Weiss, G.: Approximation results in parallel machines stochastic scheduling. Ann. Oper. Res. Special Volume on Production Planning and Scheduling, M. Queyranne, ed., 26 (1990) 195–242
Weiss, G.: Turnpike optimality of Smith's rule in parallel machines stochastic scheduling. Math. Oper. Res. 17 (1992) 255–270
Weiss, G.: On almost optimal priority rules for preemptive scheduling of stochastic jobs on parallel machines. Adv. in Appl. Probab. 27 (1995) 821–839
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 1997 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Glazebrook, K.D., Niño-Mora, J. (1997). Scheduling multiclass queueing networks on parallel servers: Approximate and heavy-traffic optimality of Klimov's priority rule. In: Burkard, R., Woeginger, G. (eds) Algorithms — ESA '97. ESA 1997. Lecture Notes in Computer Science, vol 1284. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-63397-9_18
Download citation
DOI: https://doi.org/10.1007/3-540-63397-9_18
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-63397-6
Online ISBN: 978-3-540-69536-3
eBook Packages: Springer Book Archive