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Large loss networks

Author

Listed:
  • Hunt, P. J.
  • Kurtz, T. G.

Abstract

A general loss network is considered in the limit as the arrival rates and link capacities become large with their ratio held fixed. We show that the network obeys a functional law of large numbers (along a subsequence) and that the free circuit process acts as a control for the network. The network exhibits a separation of time-scales, with the free circuit process operating on the fast time-scale as a random walk on +J, and this leads to an interesting conjecture for transient random walks. The techniques used to prove the results are of independent interest and can be applied to a wide range of models in which a similar separation of time-scale occurs, or in which the transition rates of the process undergo a discontinuity at or near a boundary. Finally, we give examples and show that commonly employed fixed point approximations are not valid in this limit.

Suggested Citation

  • Hunt, P. J. & Kurtz, T. G., 1994. "Large loss networks," Stochastic Processes and their Applications, Elsevier, vol. 53(2), pages 363-378, October.
    • Handle: RePEc:eee:spapps:v:53:y:1994:i:2:p:363-378
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    Cited by:

    1. Yiran Liu & Harsha Honnappa & Samy Tindel & Nung Kwan Yip, 2021. "Infinite server queues in a random fast oscillatory environment," Queueing Systems: Theory and Applications, Springer, vol. 98(1), pages 145-179, June.
    2. Castiel, Eyal & Borst, Sem & Miclo, Laurent & Simatos, Florian & Whiting, Phil, 2020. "Induced idleness leads to deterministic heavy traffic limits for queue-based random-access algorithms," TSE Working Papers 20-1129, Toulouse School of Economics (TSE).
    3. Golshid Baharian & Tolga Tezcan, 2011. "Stability analysis of parallel server systems under longest queue first," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 74(2), pages 257-279, October.
    4. Mor Harchol-Balter, 2022. "The multiserver job queueing model," Queueing Systems: Theory and Applications, Springer, vol. 100(3), pages 201-203, April.
    5. Andjel, Enrique & López, F. Javier & Sanz, Gerardo, 2002. "Ergodicity of one-dimensional resource sharing systems," Stochastic Processes and their Applications, Elsevier, vol. 98(1), pages 1-22, March.
    6. Varun Gupta & Neil Walton, 2019. "Load Balancing in the Nondegenerate Slowdown Regime," Operations Research, INFORMS, vol. 67(1), pages 281-294, January.
    7. Ellen Cardinaels & Sem C. Borst & Johan S. H. Leeuwaarden, 2019. "Job assignment in large-scale service systems with affinity relations," Queueing Systems: Theory and Applications, Springer, vol. 93(3), pages 227-268, December.
    8. Ohad Perry & Ward Whitt, 2013. "A Fluid Limit for an Overloaded X Model via a Stochastic Averaging Principle," Mathematics of Operations Research, INFORMS, vol. 38(2), pages 294-349, May.
    9. Itai Gurvich & Ohad Perry, 2012. "Overflow Networks: Approximations and Implications to Call Center Outsourcing," Operations Research, INFORMS, vol. 60(4), pages 996-1009, August.
    10. Jun Luo & Jiheng Zhang, 2013. "Staffing and Control of Instant Messaging Contact Centers," Operations Research, INFORMS, vol. 61(2), pages 328-343, April.
    11. Debankur Mukherjee & Sem C. Borst & Johan S. H. van Leeuwaarden & Philip A. Whiting, 2020. "Asymptotic Optimality of Power-of- d Load Balancing in Large-Scale Systems," Mathematics of Operations Research, INFORMS, vol. 45(4), pages 1535-1571, November.
    12. Edieal Pinker & Tolga Tezcan, 2013. "Determining the Optimal Configuration of Hospital Inpatient Rooms in the Presence of Isolation Patients," Operations Research, INFORMS, vol. 61(6), pages 1259-1276, December.
    13. Ohad Perry & Ward Whitt, 2011. "A Fluid Approximation for Service Systems Responding to Unexpected Overloads," Operations Research, INFORMS, vol. 59(5), pages 1159-1170, October.
    14. Mor Harchol-Balter, 2021. "Open problems in queueing theory inspired by datacenter computing," Queueing Systems: Theory and Applications, Springer, vol. 97(1), pages 3-37, February.

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