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

Published Online:https://doi.org/10.1287/moor.2020.1065

References

  • [1] Billingsley P (1999) Convergence of Probability Measures, 2nd ed. (John Wiley & Sons, New York).Google Scholar
  • [2] Bramson M (1998) State space collapse with application to heavy traffic limits for multiclass queueing networks. Queueing Systems 30(1–2):89–148.CrossrefGoogle Scholar
  • [3] Budhiraja A , Lee C (2009) Stationary distribution convergence for generalized Jackson networks in heavy traffic. Math. Oper. Res. 34(1):45–56.Google Scholar
  • [4] Dai JG (1995) On positive Harris recurrence of multi-class queueing networks: A unified approach via fluid limit models. Ann. Appl. Probab. 5(1):49–77.CrossrefGoogle Scholar
  • [5] Dai JG , Meyn SP (1995) Stability and convergence of moments for multiclass queueing networks via fluid models. IEEE Trans. Automatic Control . 40(11):1899–1904.CrossrefGoogle Scholar
  • [6] Davis MHA (1984) Piecewise-deterministic Markov processes: A general class of nondiffusion models. J. Royal Statist. Soc. B 46(3):353–388.CrossrefGoogle Scholar
  • [7] Dupuis P , Williams RJ (1994) Lyaponov functions for semimartingale reflected Brownian motions. Ann. Probab. 22(2):680–702.CrossrefGoogle Scholar
  • [8] Durrett R (2010) Probability: Theory and Examples, 4th ed. (Cambridge University Press, Cambridge, UK).Google Scholar
  • [9] Gamarnik D , Zeevi A (2006) Validity of heavy traffic steady-state approximations in generalized Jackson networks. Ann. Appl. Probab. 16(1):56–96.CrossrefGoogle Scholar
  • [10] Gurvich I (2014) Validity of heavy-traffic steady-state approximations in multiclass queueing networks: The case of queue-ratio disciplines. Math. Oper. Res. 39(1):121–162.LinkGoogle Scholar
  • [11] Gut A (1988) Stopped Random Walks: Limit Theorems and Applications, Applied Probability, vol. 5 (Springer-Verlag, New York).Google Scholar
  • [12] Kang WN , Kelly FP , Lee NH , Williams RJ (2009) State space collapse and diffusion approximation for a network operating under a fair bandwidth sharing policy. Ann. Appl. Probab. 19(5):1719–1780.CrossrefGoogle Scholar
  • [13] Katsuda T (2010) State-space collapse in stationarity and its application to a multiclass single-server queue in heavy traffic. Queueing Systems 65(3):237–273.CrossrefGoogle Scholar
  • [14] Katsuda T (2012) Stationary distribution convergence for a multiclass single-server queue in heavy traffic. Sci. Math. Japan 75:317–334.Google Scholar
  • [15] Krichagina EV , Taksar MI (1992) Diffusion approximation for GI/G/1 controlled queues. Queueing Systems 12(3–4):333–368.CrossrefGoogle Scholar
  • [16] Shah D , Tsitsiklis JN , Zhong Y (2014) Qualitative properties of alpha-fair policies in bandwidth-sharing networks. Ann. Appl. Probab . 24(1):76–113.Google Scholar
  • [17] Wang W , Maguluri ST , Srikant R , Ying L (2018) Heavy-traffic insensitive bounds for weighted proportionally fair bandwidth sharing policies. Preprint, submitted August 6, https://arxiv.org/abs/1808.02120.Google Scholar
  • [18] Williams RJ (1998) Diffusion approximations for open multi-class queueing networks: Sufficient conditions involving state space collapse. Queueing Systems 30(1–2):27–88.CrossrefGoogle Scholar
  • [19] Ye HQ , and Yao DD (2012) A stochastic network under fair resource control—Diffusion limit with multiple bottlenecks. Oper. Res . 60(3):716–738.Google Scholar
  • [20] Ye HQ , Yao DD (2016) Diffusion limit of fair resource control—Stationary and interchange of limits. Math. Oper. Res . 41(4):1161–1207.Google Scholar
  • [21] Ye HQ , Yao DD (2018) Justifying diffusion approximations for stochastic processing networks under a moment condition. Ann. Appl. Probab . 28(6):3652–3697.CrossrefGoogle Scholar
INFORMS site uses cookies to store information on your computer. Some are essential to make our site work; Others help us improve the user experience. By using this site, you consent to the placement of these cookies. Please read our Privacy Statement to learn more.