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Diffusion Approximation for Fair Resource Control—Interchange of Limits Under a Moment Condition

Author

Listed:
  • Heng-Qing Ye

    (Department of Logistics and Maritime Studies, Hong Kong Polytechnic University, Hong Kong)

  • David D. Yao

    (Department of Industrial Engineering and Operations Research, Columbia University, New York 10027)

Abstract

In a prior study [Ye HQ, Yao DD (2016) Diffusion limit of fair resource control–Stationary and interchange of limits. Math. Oper. Res . 41(4):1161–1207.] focusing on a class of stochastic processing network with fair resource control, we justified the diffusion approximation (in the context of the interchange of limits) provided that the p th moment of the workloads are bounded. To this end, we introduced the so-called bounded workload condition, which requires the workload process be bounded by a free process plus the initial workload. This condition is for a derived process, the workload, as opposed to primitives such as arrival processes and service requirements; as such, it could be difficult to verify. In this paper, we establish the interchange of limits under a moment condition of suitable order on the primitives directly: the required order is p ∗ > 2 ( p + 2 ) on the moments of the primitive processes so as to bound the p th moment of the workload. This moment condition is trivial to verify, and indeed automatically holds in networks where the primitives have moments of all orders, for instance, renewal arrivals with phase-type interarrival times and independent and identically distributed phase-type service times.

Suggested Citation

  • Heng-Qing Ye & David D. Yao, 2021. "Diffusion Approximation for Fair Resource Control—Interchange of Limits Under a Moment Condition," Mathematics of Operations Research, INFORMS, vol. 46(3), pages 869-894, August.
    • Handle: RePEc:inm:ormoor:v:46:y:2021:i:3:p:869-894
    DOI: 10.1287/moor.2020.1065
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    References listed on IDEAS

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    1. Heng-Qing Ye & David D. Yao, 2016. "Diffusion Limit of Fair Resource Control—Stationarity and Interchange of Limits," Mathematics of Operations Research, INFORMS, vol. 41(4), pages 1161-1207, November.
    2. Heng-Qing Ye & David D. Yao, 2012. "A Stochastic Network Under Proportional Fair Resource Control---Diffusion Limit with Multiple Bottlenecks," Operations Research, INFORMS, vol. 60(3), pages 716-738, June.
    3. Itai Gurvich, 2014. "Validity of Heavy-Traffic Steady-State Approximations in Multiclass Queueing Networks: The Case of Queue-Ratio Disciplines," Mathematics of Operations Research, INFORMS, vol. 39(1), pages 121-162, February.
    4. Amarjit Budhiraja & Chihoon Lee, 2009. "Stationary Distribution Convergence for Generalized Jackson Networks in Heavy Traffic," Mathematics of Operations Research, INFORMS, vol. 34(1), pages 45-56, February.
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