Advances in Applied Probability

Reducing response time in fork-join systems under heavy traffic via imbalance control

Saul C. Leite and Marcelo D. Fragoso

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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.

Article information

Adv. in Appl. Probab., Volume 45, Number 4 (2013), 1137-1156.

First available in Project Euclid: 12 December 2013

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60K25: Queueing theory [See also 68M20, 90B22]
Secondary: 93E20: Optimal stochastic control

Queueing theory parallel system heavy traffic analysis


Leite, Saul C.; Fragoso, Marcelo D. Reducing response time in fork-join systems under heavy traffic via imbalance control. Adv. in Appl. Probab. 45 (2013), no. 4, 1137--1156. doi:10.1239/aap/1386857861.

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  • Barroso, L., Dean, J. and Holzle, U. (2003). Web search for a planet: the google cluster architecture. IEEE Micro 23, 22–28.
  • 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.
  • Billingsley, P. (1999). Convergence of Probability Measures, 2nd edn. John Wiley, New York.
  • Borovkov, A. (1964). Some limit theorems in the theory of mass service. Theory Prob. Appl. 9, 550–565.
  • Borovkov, A. (1965). Some limit theorems in the theory of mass service. II. Multiple channels systems. Theory Prob. Appl. 10, 375–400.
  • 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 O. J. Boxma and G. M. Koole, CWI, Amsterdam, pp. 1–24.
  • 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.
  • Flatto, L. and Hahn, S. (1984). Two parallel queues created by arrivals with two demands. I. SIAM J. Appl. Math. 44, 1041–1053.
  • Gonçalves, C. B. \et (2007). A capacity planning model for web search engines. Unpublished manuscript.
  • 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.
  • Harrison, J. M and López, M. J. (1999). Heavy traffic resource pooling in parallel-server systems. Queueing Systems 33, 339–368.
  • Ikeda, N. and Watanabe, S. (1989). Stochastic Differential Equations and Diffusion Processes. North-Holland, Amsterdam.
  • Kemper, B. and Mandjes, M. (2012). Mean sojourn times in two-queue fork-join systems: bounds and approximations. OR Spectrum 34, 723–742.
  • Kingman, J. (1961). The single server queue in heavy traffic. Proc. Camb. Phil. Soc. 57, 902–904.
  • Ko, S.-S. and Serfozo, R. F. (2008). Sojourn times in G/M/1 fork-join networks. Naval Res. Logistics 55, 432–443.
  • Kushner, H. J. (2001). Heavy Traffic Analysis of Controlled Queueing and Communication Networks (Appl. Math. (New York) 47). Springer, New York.
  • 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.
  • 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.
  • 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.
  • 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.
  • Prohorov, Y. (1963). Transition phenomena in queueing processes, I. Litovsk. Mat. Sb. 3, 199–205 (in Russian).
  • Taylor, L. and Williams, R. (1993). Existence and uniqueness of semimartingale reflecting Brownian motions in an orthant. Prob. Theory Relat. Fields 96, 283–317.
  • Whitt, W. (2002). Stochastic-Process Limits. Springer, New York.
  • 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.