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Diffusion approximations for multiclass queueing networks under preemptive priority service discipline

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Abstract

We prove a heavy traffic limit theorem to justify diffusion approximations for multiclass queueing networks under preemptive priority service discipline and provide effective stochastic dynamical models for the systems. Such queueing networks appear typically in high-speed integrated services packet networks about telecommunication system. In the network, there is a number of packet traffic types. Each type needs a number of job classes (stages) of processing and each type of jobs is assigned the same priority rank at every station where it possibly receives service. Moreover, there is no inter-routing among different traffic types throughout the entire network.

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References

  1. Dai W. A heavy traffic limit theorem for queueing networks with finite capacity[C]. In: INFORMS Applied Probability Conference, Atlanta, U.S.A., June 14–16, 1995.

  2. Dai W. Brownian approximations for queueing networks with finite buffers: modeling, heavy traffic analysis and numerical implementations[D]. Ph D Dissertation, School of Mathematics, Georgia Institute of Technology, 1996. Aslo published in UMI Dissertation Services, A Bell & Howell Company, 300 N.Zeeb Road, Ann Arbor, Michican 48106, U.S.A. 1997.

  3. Dai J G, Dai W. A heavy traffic limit theorem for a class of open queueing networks with finite buffers[J]. Queueing Systems, 1999, 32(1/3):5–40.

    Article  MATH  Google Scholar 

  4. Reiman M I. Open queueing networks in heavy traffic[J]. Mathematics of Operations Research, 1984, 9(3):441–458.

    Article  MATH  MathSciNet  Google Scholar 

  5. Bramson M. State space collapse with application to heavy traffic limits for multiclass queueing networks[J]. Queueing Systems, 1998, 30(1/2):89–148.

    Article  MATH  MathSciNet  Google Scholar 

  6. Bramson M. State space collapse for queueing networks[C]. In: Proceedings of the International Congress of Mathematicians, 1998, Vol III, 213–222.

    MathSciNet  Google Scholar 

  7. Williams R J. Diffusion approximations for open multiclass queueing networks: sufficient conditions involving state space collapse[J]. Queueing Systems: Theory and Applications, 1998, 30(1/2):27–88.

    Article  MathSciNet  Google Scholar 

  8. Williams R J. Reflecting diffusions and queueing networks[C]. In: Proceedings of the International Congress of Mathematicians, 1998, Vol III, 321–330.

    Google Scholar 

  9. Bramson M, Dai J G. Heavy traffic limits for some queueing networks[J]. Annals of Applied Probability, 2001, 11(1):49–90.

    MATH  MathSciNet  Google Scholar 

  10. Chen H, Zhang H. A sufficient condition and a necessary condition for the diffusion approximations of multiclass queueing networks under priority service displines[J]. Queueing Systems, 2000, 34(1/4):237–268.

    Article  MATH  Google Scholar 

  11. Chen H, Zhang H. Diffusion approximations for some multiclass queueing networks with FIFO service disciplines[J]. Mathematics of Operations Research, 2000, 25(4):679–707.

    Article  MATH  MathSciNet  Google Scholar 

  12. Harrison J M, Williams R J. Multidimensional reflected Brownian motions having exponential stationary distributions[J]. Annals of Probability, 1987, 15(1):115–137.

    MATH  MathSciNet  Google Scholar 

  13. Dai J G, Harrison J M. Reflected Brownian motion in an orthant: numerical methods for steadystate analysis[J]. Annals of Applied Probability, 1992, 2(1):65–86.

    MATH  MathSciNet  Google Scholar 

  14. Shen X, Chen H, Dai J G, Dai W. The finite element method for computing the stationary distribution of an SRBM in a hypercube with applications to finite buffer queueing networks[J]. Queueing Systems, 2002, 42(1):33–62.

    Article  MATH  MathSciNet  Google Scholar 

  15. Dai J G, Wang Y. Nonexistence of Brownian models of certain multicalss queueing networks[J]. Queueing Systems, 1993, 13(1/3):41–46.

    Article  MATH  MathSciNet  Google Scholar 

  16. Williams R J. An invariance principle for semimartingale reflecting Brownian motions in an orthant[J]. Queueing Systems, 1998, 30(1/2):5–25.

    Article  MATH  Google Scholar 

  17. Ethier S N, Kurtz T G. Markov processes: charaterization and convergence[M]. New York: Wiley, 1986.

    Google Scholar 

  18. Bernard A, Kharroubi A El. Régulation déterministes et stochastiques dans le premier “orthant” de R n[J]. Stochastics Stochastics Rep, 1991, 34(3/4):149–167.

    MathSciNet  MATH  Google Scholar 

  19. Harrison J M, Reiman M I. Reflected Brownian motion on an orthant[J]. Annals of Probability, 1981, 9(2):302–308.

    MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, Nanjing University, Nanjing, 210093, P. R. China

    Dai Wan-yang  (戴万阳)

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  1. Dai Wan-yang  (戴万阳)
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Correspondence to Dai Wan-yang  (戴万阳).

Additional information

Communicated by GUO Mao-zheng

Project supported by the National Natural Science Foundation of China (No. 10371053)

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Dai, Wy. Diffusion approximations for multiclass queueing networks under preemptive priority service discipline. Appl Math Mech 28, 1331–1342 (2007). https://doi.org/10.1007/s10483-007-1006-x

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  • DOI: https://doi.org/10.1007/s10483-007-1006-x

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2000 Mathematics Subject Classification

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