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Reducing Response Time in Fork-Join Systems under Heavy Traffic Via Imbalance Control

Part of: Special processes Stochastic systems and control

Published online by Cambridge University Press:  04 January 2016

Saul C. Leite  and
Marcelo D. Fragoso
Saul C. Leite
Affiliation:
Universidade Federal de Juiz de Fora
Marcelo D. Fragoso*
Affiliation:
Laboratório Nacional de Computação Científica
*
∗ Postal address: Department of Systems and Control, National Laboratory for Scientific Computing (LNCC), Laboratório Nacional de Computação Científica, Av. Getulio Vargas 333, Petrópolis, RJ, CEP:25651-075, Brazil. Email address: frag@lncc.br
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Abstract

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We consider the problem of reducing the response time of fork-join systems by maintaining the workload balanced among the processing stations. The general problem of modeling and finding an optimal policy that reduces imbalance is quite difficult. In order to circumvent this difficulty, the heavy traffic approach is taken, and the system dynamics are approximated by a reflected diffusion process. This way, the problem of finding an optimal balancing policy that reduces workload imbalance is set as a stochastic optimal control problem, for which numerical methods are available. Some numerical experiments are presented, where the control problem is solved numerically and applied to a simulation. The results indicate that the response time of the controlled system is reduced significantly using the devised control.

Keywords

Queueing theoryparallel systemheavy traffic analysis

MSC classification

Primary: 60K25: Queueing theory
Secondary: 93E20: Optimal stochastic control

Information

Type
General Applied Probability
Information
Advances in Applied Probability , Volume 45 , Issue 4 , December 2013 , pp. 1137 - 1156
Copyright
© Applied Probability Trust 

Footnotes

A preliminary version of this paper was presented at the IFAC World Conference 2011, Milan, Italy.

References

Barroso, L., Dean, J. and Holzle, U. (2003). Web search for a planet: the google cluster architecture. IEEE Micro 23, 22–28.Google Scholar
Bell, S. and Williams, R. (2001). Dynamic scheduling of a system with two parallel servers in heavy traffic with resource pooling: asymptotic optimality of a threshold policy. Ann. Appl. Prob. 11, 608–649.Google Scholar
Billingsley, P. (1999). Convergence of Probability Measures, 2nd edn. John Wiley, New York.Google Scholar
Borovkov, A. (1964). Some limit theorems in the theory of mass service. Theory Prob. Appl. 9, 550–565.CrossRefGoogle Scholar
Borovkov, A. (1965). Some limit theorems in the theory of mass service. II. Multiple channels systems. Theory Prob. Appl. 10, 375–400.Google Scholar
Boxma, O., Koole, G. and Liu, Z. (1994). Queueing-theoretic solution methods for models of parallel and distributed systems. In Performance Evaluation of Parallel and Distributed Systems-Solution Methods, eds Boxma, O. J. and Koole, G. M., CWI, Amsterdam, pp. 1–24.Google Scholar
Dai, J. G. and Williams, R. J. (1995). Existence and uniqueness of semimartingale reflecting Brownian motions in convex polyhedrons. Theory Prob. Appl. 40, 1–40.Google Scholar
Flatto, L. and Hahn, S. (1984). Two parallel queues created by arrivals with two demands. I. SIAM J. Appl. Math. 44, 1041–1053.Google Scholar
Gonçalves, C. B. et al. (2007). A capacity planning model for web search engines. Unpublished manuscript.Google Scholar
Harrison, J. M. (1998). Heavy traffic analysis of a system with parallel servers: asymptotic optimality of discrete-review policies. Ann. Appl. Prob. 8, 822–848.Google Scholar
Harrison, J. M. and López, M. J. (1999). Heavy traffic resource pooling in parallel-server systems. Queueing Systems 33, 339–368.Google Scholar
Ikeda, N. and Watanabe, S. (1989). Stochastic Differential Equations and Diffusion Processes. North-Holland, Amsterdam.Google Scholar
Kemper, B. and Mandjes, M. (2012). Mean sojourn times in two-queue fork-join systems: bounds and approximations. OR Spectrum 34, 723–742.Google Scholar
Kingman, J. (1961). The single server queue in heavy traffic. Proc. Camb. Phil. Soc. 57, 902–904.CrossRefGoogle Scholar
Ko, S.-S. and Serfozo, R. F. (2008). Sojourn times in G/M/1 fork-join networks. Naval Res. Logistics 55, 432–443.Google Scholar
Kushner, H. J. (2001). Heavy Traffic Analysis of Controlled Queueing and Communication Networks (Appl. Math. (New York) 47). Springer, New York.CrossRefGoogle Scholar
Kushner, H. J. and Chen, Y. N. (2000). Optimal control of assignment of Jobs to processors under heavy traffic. Stoch. Stoch. Reports 68, 177–228.CrossRefGoogle Scholar
Kushner, H. J. and Dupuis, P. G. (1992). Numerical Methods for Stochastic Control Problems in Continuous Time (Appl. Math. (New York) 24). Springer, New York.Google Scholar
Lebrecht, A. S. and Knottenbelt, W. J. (2007). Response time approximations in fork-join queues. In Proc. 23rd Annual UK Performance Engineering Workshop (UKPEW, 2007), Ormskirk.Google Scholar
Leite, S. and Fragoso, M. (2010). Heavy traffic analysis of state-dependent parallel queues with triggers and an application to web search systems. Performance Evaluation 67, 913–928.CrossRefGoogle Scholar
Prohorov, Y. (1963). Transition phenomena in queueing processes, I. Litovsk. Mat. Sb. 3, 199–205 (in Russian).Google Scholar
Taylor, L. and Williams, R. (1993). Existence and uniqueness of semimartingale reflecting Brownian motions in an orthant. Prob. Theory Relat. Fields 96, 283–317.Google Scholar
Whitt, W. (2002). Stochastic-Process Limits. Springer, New York.CrossRefGoogle Scholar
Williams, R. J. (2000). On dynamic scheduling of a parallel server system with complete resource pooling. In Analysis of Communication Networks: Call Centres, Traffic and Performance (Fields Inst. Commun. 28), American Mathematical Society, Providence, RI, pp. 49–71.Google Scholar