The Annals of Applied Probability

Justifying diffusion approximations for multiclass queueing networks under a moment condition

Heng-Qing Ye and David D. Yao

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Multiclass queueing networks (MQN) are, in general, difficult objects to study analytically. The diffusion approximation refers to using the stationary distribution of the diffusion limit as an approximation of the diffusion-scaled process (say, the workload) in the original MQN. To validate such an approximation amounts to justifying the interchange of two limits, $t\to\infty$ and $k\to\infty$, with $t$ being the time index and $k$, the scaling parameter. Here, we show this interchange of limits is justified under a $p^{*}$th moment condition on the primitive data, the interarrival and service times; and we provide an explicit characterization of the required order ($p^{*}$), which depends naturally on the desired order of moment of the workload process.

Article information

Ann. Appl. Probab., Volume 28, Number 6 (2018), 3652-3697.

Received: August 2015
Revised: January 2018
First available in Project Euclid: 8 October 2018

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60K25: Queueing theory [See also 68M20, 90B22] 90B15: Network models, stochastic
Secondary: 60F17: Functional limit theorems; invariance principles 60J20: Applications of Markov chains and discrete-time Markov processes on general state spaces (social mobility, learning theory, industrial processes, etc.) [See also 90B30, 91D10, 91D35, 91E40]

Multiclass queueing network diffusion limit interchange of limits uniform stability


Ye, Heng-Qing; Yao, David D. Justifying diffusion approximations for multiclass queueing networks under a moment condition. Ann. Appl. Probab. 28 (2018), no. 6, 3652--3697. doi:10.1214/18-AAP1401.

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  • [1] Billingsley, P. (1999). Convergence of Probability Measures, 2nd ed. Wiley, New York.
  • [2] Bramson, M. (1996). Convergence to equilibria for fluid models of FIFO queueing networks. Queueing Syst. Theory Appl. 22 5–45.
  • [3] Bramson, M. (1996). Convergence to equilibria for fluid models of head-of-the-line proportional processor sharing queueing networks. Queueing Syst. Theory Appl. 23 1–26.
  • [4] Bramson, M. (1998). State space collapse with application to heavy traffic limits for multiclass queueing networks. Queueing Syst. Theory Appl. 30 89–148.
  • [5] Bramson, M. (1998). Stability of two families of queueing networks and a discussion of fluid limits. Queueing Syst. Theory Appl. 28 7–31.
  • [6] Bramson, M. and Dai, J. G. (2001). Heavy traffic limits for some queueing networks. Ann. Appl. Probab. 11 49–90.
  • [7] Braverman, A. and Dai, J. G. (2017). Stein’s method for steady-state diffusion approximations of $M/Ph/n+M$ systems. Ann. Appl. Probab. 27 550–581.
  • [8] Braverman, A., Dai, J. G. and Feng, J. (2016). Stein’s method for steady-state diffusion approximations: An introduction through the Erlang-A and Erlang-C models. Stoch. Syst. 6 301–366.
  • [9] Budhiraja, A. and Lee, C. (2009). Stationary distribution convergence for generalized Jackson networks in heavy traffic. Math. Oper. Res. 34 45–56.
  • [10] Chen, H. (1995). Fluid approximations and stability of multiclass queueing networks: Work-conserving disciplines. Ann. Appl. Probab. 5 637–665.
  • [11] Chen, H. and Ye, H. Q. (2012). Asymptotic optimality of balanced routing. Oper. Res. 60 163–179.
  • [12] Chen, H. and Ye, H. Q. (2001). Existence condition for the diffusion approximations of multiclass priority queueing networks. Queueing Syst. 38 435–470.
  • [13] Chen, H. and Zhang, H. (2000). Diffusion approximations for some multiclass queueing networks with FIFO service disciplines. Math. Oper. Res. 25 679–707.
  • [14] Chen, H. and Zhang, H. (2000). A sufficient condition and a necessary condition for the diffusion approximations of multiclass queueing networks under priority service disciplines. Queueing Syst. Theory Appl. 34 237–268.
  • [15] Dai, J. G. (1995). On positive Harris recurrence of multiclass queueing networks: A unified approach via fluid limit models. Ann. Appl. Probab. 5 49–77.
  • [16] Dai, J. G., Dieker, A. B. and Gao, X. (2014). Validity of heavy-traffic steady-state approximations in many-server queues with abandonment. Queueing Syst. 78 1–29.
  • [17] Dai, J. G. and Harrison, J. M. (1992). Reflected Brownian motion in an orthant: Numerical methods for steady-state analysis. Ann. Appl. Probab. 2 65–86.
  • [18] Dai, J. G. and Lin, W. (2008). Asymptotic optimality of maximum pressure policies in stochastic processing networks. Ann. Appl. Probab. 18 2239–2299.
  • [19] Dai, J. G. and Meyn, S. P. (1995). Stability and convergence of moments for multiclass queueing networks via fluid limit models. IEEE Trans. Automat. Control 40 1889–1904.
  • [20] Davis, M. H. A. (1984). Piecewise-deterministic Markov processes: A general class of nondiffusion stochastic models. J. Roy. Statist. Soc. Ser. B 46 353–388.
  • [21] Dupuis, P. and Williams, R. J. (1994). Lyapunov functions for semimartingale reflecting Brownian motions. Ann. Probab. 22 680–702.
  • [22] Durrett, R. (2010). Probability: Theory and Examples, 4th ed. Cambridge Series in Statistical and Probabilistic Mathematics 31. Cambridge Univ. Press, Cambridge.
  • [23] Gamarnik, D. and Goldberg, D. A. (2013). On the rate of convergence to stationarity of the $\mathrm{M}/\mathrm{M}/\mathrm{N}$ queue in the Halfin–Whitt regime. Ann. Appl. Probab. 23 1879–1912.
  • [24] Gamarnik, D. and Stolyar, A. L. (2012). Multiclass multiserver queueing system in the Halfin–Whitt heavy traffic regime: Asymptotics of the stationary distribution. Queueing Syst. 71 25–51.
  • [25] Gamarnik, D. and Zeevi, A. (2006). Validity of heavy traffic steady-state approximation in generalized Jackson networks. Ann. Appl. Probab. 16 56–90.
  • [26] Gurvich, I. (2014). Validity of heavy-traffic steady-state approximations in multiclass queueing networks: The case of queue-ratio disciplines. Math. Oper. Res. 39 121–162.
  • [27] Gut, A. (1988). Stopped Random Walks: Limit Theorems and Applications. Applied Probability. A Series of the Applied Probability Trust 5. Springer, New York.
  • [28] Kang, W. N., Kelly, F. P., Lee, N. H. and Williams, R. J. (2009). State space collapse and diffusion approximation for a network operating under a fair bandwidth sharing policy. Ann. Appl. Probab. 19 1719–1780.
  • [29] Katsuda, T. (2010). State-space collapse in stationarity and its application to a multiclass single-server queue in heavy traffic. Queueing Syst. 65 237–273.
  • [30] Katsuda, T. (2012). Stationary distribution convergence for a multiclass single-server queue in heavy traffic. Sci. Math. Jpn. 75 317–334.
  • [31] Krichagina, E. V. and Taksar, M. I. (1992). Diffusion approximation for $\mathrm{GI}/\mathrm{G}/1$ controlled queues. Queueing Syst. Theory Appl. 12 333–367.
  • [32] Mandelbaum, A. and Stolyar, A. L. (2004). Scheduling flexible servers with convex delay costs: Heavy-traffic optimality of the generalized $c\mu$-rule. Oper. Res. 52 836–855.
  • [33] Rybko, A. N. and Stolyar, A. L. (1992). On the ergodicity of random processes that describe the functioning of open queueing networks. Problemy Peredachi Informatsii 28 3–26.
  • [34] Shah, D., Tsitsiklis, J. N. and Zhong, Y. (2014). Qualitative properties of $\alpha$-fair policies in bandwidth-sharing networks. Ann. Appl. Probab. 24 76–113.
  • [35] Stolyar, A. L. (2004). Maxweight scheduling in a generalized switch: State space collapse and workload minimization in heavy traffic. Ann. Appl. Probab. 14 1–53.
  • [36] Stolyar, A. L. (2015). Diffusion-scale tightness of invariant distributions of a large-scale flexible service system. Adv. in Appl. Probab. 47 251–269.
  • [37] Stolyar, A. L. (2015). Tightness of stationary distributions of a flexible-server system in the Halfin–Whitt asymptotic regime. Stoch. Syst. 5 239–267.
  • [38] Stolyar, A. L. and Yudovina, E. (2012). Tightness of invariant distributions of a large-scale flexible service system under a priority discipline. Stoch. Syst. 2 381–408.
  • [39] Williams, R. J. (1998). Diffusion approximations for open multiclass queueing networks: Sufficient conditions involving state space collapse. Queueing Syst. Theory Appl. 30 27–88.
  • [40] Ye, H. Q. and Yao, D. D. (2008). Heavy-traffic optimality of a stochastic network under utility-maximizing resource allocation. Oper. Res. 56 453–470.
  • [41] Ye, H. Q. and Yao, D. D. (2010). Utility-maximizing resource control: Diffusion limit and asymptotic optimality for a two-bottleneck model. Oper. Res. 58 613–623.
  • [42] Ye, H. Q. and Yao, D. D. (2012). A stochastic network under proportional fair resource control-diffusion limit with multiple bottlenecks. Oper. Res. 60 716–738.
  • [43] Ye, H. Q. and Yao, D. D. (2016). Diffusion limit of fair resource control—Stationarity and interchange of limits. Math. Oper. Res. 41 1161–1207.