## The Annals of Applied Probability

### Stein’s method for steady-state diffusion approximations of $M/\mathit{Ph}/n+M$ systems

#### Abstract

We consider $M/\mathit{Ph}/n+M$ queueing systems in steady state. We prove that the Wasserstein distance between the stationary distribution of the normalized system size process and that of a piecewise Ornstein–Uhlenbeck (OU) process is bounded by $C/\sqrt{\lambda}$, where the constant $C$ is independent of the arrival rate $\lambda$ and the number of servers $n$ as long as they are in the Halfin-Whitt parameter regime. For each integer $m>0$, we also establish a similar bound for the difference of the $m$th steady-state moments. For the proofs, we develop a modular framework that is based on Stein’s method. The framework has three components: Poisson equation, generator coupling, and state space collapse. The framework, with further refinement, is likely applicable to steady-state diffusion approximations for other stochastic systems.

#### Article information

Source
Ann. Appl. Probab., Volume 27, Number 1 (2017), 550-581.

Dates
Revised: November 2015
First available in Project Euclid: 6 March 2017

https://projecteuclid.org/euclid.aoap/1488790835

Digital Object Identifier
doi:10.1214/16-AAP1211

Mathematical Reviews number (MathSciNet)
MR3619795

Zentralblatt MATH identifier
1362.60077

#### Citation

Braverman, Anton; Dai, J. G. Stein’s method for steady-state diffusion approximations of $M/\mathit{Ph}/n+M$ systems. Ann. Appl. Probab. 27 (2017), no. 1, 550--581. doi:10.1214/16-AAP1211. https://projecteuclid.org/euclid.aoap/1488790835

#### References

• [1] Aksin, Z., Armony, M. and Mehrotra, V. (2007). The modern call center: A multi-disciplinary perspective on operations management research. Prod. Oper. Manag. 16 665–688.
• [2] Armony, M., Israelit, S., Mandelbaum, A., Marmor, Y. N., Tseytlin, Y. and Yom-Tov, G. B. (2015). On patient flow in hospitals: A data-based queueing-science perspective. Stoch. Syst. 5 146–194.
• [3] Asmussen, S. (2003). Applied Probability and Queues. Stochastic Modelling and Applied Probability, 2nd ed. Applications of Mathematics (New York) 51. Springer, New York.
• [4] Barbour, A. D. (1988). Stein’s method and Poisson process convergence. J. Appl. Probab. 25 175–184.
• [5] Barbour, A. D. (1990). Stein’s method for diffusion approximations. Probab. Theory Related Fields 84 297–322.
• [6] Bell, S. L. and Williams, R. J. (2005). Dynamic scheduling of a parallel server system in heavy traffic with complete resource pooling: Asymptotic optimality of a threshold policy. Electron. J. Probab. 10 1044–1115.
• [7] Borovkov, A. (1964). Some limit theorems in the theory of mass service, I. Theory Probab. Appl. 9 550–565.
• [8] Borovkov, A. (1965). Some limit theorems in the theory of mass service, II. Theory Probab. Appl. 10 375–400.
• [9] Bramson, M. (1998). State space collapse with application to heavy traffic limits for multiclass queueing networks. Queueing Syst. Theory Appl. 30 89–148.
• [10] Budhiraja, A. and Lee, C. (2009). Stationary distribution convergence for generalized Jackson networks in heavy traffic. Math. Oper. Res. 34 45–56.
• [11] Chatterjee, S. (2014). A short survey of Stein’s method. Proceedings of ICM 2014. To appear.
• [12] Chen, L. H. Y., Goldstein, L. and Shao, Q.-M. (2011). Normal Approximation by Stein’s Method. Springer, Heidelberg.
• [13] 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.
• [14] Dai, J. G. and He, S. (2013). Many-server queues with customer abandonment: Numerical analysis of their diffusion model. Stoch. Syst. 3 96–146.
• [15] Dai, J. G., He, S. and Tezcan, T. (2010). Many-server diffusion limits for $G/\mathit{Ph}/n+\mathit{GI}$ queues. Ann. Appl. Probab. 20 1854–1890.
• [16] Dai, J. G. and Tezcan, T. (2011). State space collapse in many-server diffusion limits of parallel server systems. Math. Oper. Res. 36 271–320.
• [17] Dieker, A. B. and Gao, X. (2013). Positive recurrence of piecewise Ornstein–Uhlenbeck processes and common quadratic Lyapunov functions. Ann. Appl. Probab. 23 1291–1317.
• [18] Eryilmaz, A. and Srikant, R. (2012). Asymptotically tight steady-state queue length bounds implied by drift conditions. Queueing Syst. 72 311–359.
• [19] Ethier, S. N. and Kurtz, T. G. (1986). Markov Processes. Characterization and Convergence. Wiley, New York.
• [20] Foschini, G. J. and Salz, J. (1978). A basic dynamic routing problem and diffusion. IEEE Trans. Commun. 26 320–327.
• [21] 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.
• [22] Gamarnik, D. and Zeevi, A. (2006). Validity of heavy traffic steady-state approximation in generalized Jackson networks. Ann. Appl. Probab. 16 56–90.
• [23] Gans, N., Koole, G. and Mandelbaum, A. (2003). Telephone call centers: Tutorial, review, and research prospects. Manuf. Serv. Oper. Manag. 5 79–141.
• [24] Gibbs, A. L. and Su, F. E. (2002). On choosing and bounding probability metrics. Int. Stat. Rev. 70 419–435.
• [25] Gilbarg, D. and Trudinger, N. S. (1983). Elliptic Partial Differential Equations of Second Order, 2nd ed. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 224. Springer, Berlin.
• [26] Glynn, P. W. and Zeevi, A. (2008). Bounding stationary expectations of Markov processes. In Markov Processes and Related Topics: A Festschrift for Thomas G. Kurtz. Inst. Math. Stat. Collect. 4 195–214. IMS, Beachwood, OH.
• [27] Gurvich, I. (2014). Diffusion models and steady-state approximations for exponentially ergodic Markovian queues. Ann. Appl. Probab. 24 2527–2559.
• [28] 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.
• [29] Gurvich, I., Huang, J. and Mandelbaum, A. (2014). Excursion-based universal approximations for the Erlang—A queue in steady-state. Math. Oper. Res. 39 325–373.
• [30] Halfin, S. and Whitt, W. (1981). Heavy-traffic limits for queues with many exponential servers. Oper. Res. 29 567–588.
• [31] Harrison, J. M. (1978). The diffusion approximation for tandem queues in heavy traffic. Adv. in Appl. Probab. 10 886–905.
• [32] Harrison, J. M. (1998). Heavy traffic analysis of a system with parallel servers: Asymptotic optimality of discrete-review policies. Ann. Appl. Probab. 8 822–848.
• [33] Harrison, J. M. and López, M. J. (1999). Heavy traffic resource pooling in parallel-server systems. Queueing Syst. Theory Appl. 33 339–368.
• [34] Harrison, J. M. and Reiman, M. I. (1981). Reflected Brownian motion on an orthant. Ann. Probab. 9 302–308.
• [35] Henderson, S. G. (1997). Variance reduction via an approximating Markov process. Ph.D. thesis, Dept. Operations Research, Stanford Univ. Available at http://people.orie.cornell.edu/shane/pubs/thesis.pdf.
• [36] Iglehart, D. L. and Whitt, W. (1970). Multiple channel queues in heavy traffic. I. Adv. in Appl. Probab. 2 150–177.
• [37] Iglehart, D. L. and Whitt, W. (1970). Multiple channel queues in heavy traffic. II. Sequences, networks, and batches. Adv. in Appl. Probab. 2 355–369.
• [38] 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.
• [39] 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.
• [40] Knoblauch, A. (2008). Closed-form expressions for the moments of the binomial probability distribution. SIAM J. Appl. Math. 69 197–204.
• [41] Luk, H. M. (1994). Stein’s method for the gamma distribution and related statistical applications. Ph.D. thesis, Univ. Southern California, Los Angeles, CA.
• [42] Meyn, S. P. and Tweedie, R. L. (1993). Stability of Markovian processes. III. Foster–Lyapunov criteria for continuous-time processes. Adv. in Appl. Probab. 25 518–548.
• [43] Peterson, W. P. (1991). A heavy traffic limit theorem for networks of queues with multiple customer types. Math. Oper. Res. 16 90–118.
• [44] Reed, J. (2009). The $G/\mathit{GI}/N$ queue in the Halfin–Whitt regime. Ann. Appl. Probab. 19 2211–2269.
• [45] Reiman, M. I. (1984). Some diffusion approximations with state space collapse. In Modelling and Performance Evaluation Methodology (Paris, 1983) (F. Baccelli and G. Fayolle, eds.). Lecture Notes in Control and Inform. Sci. 60 209–240. Springer, Berlin.
• [46] Reiman, M. I. (1984). Open queueing networks in heavy traffic. Math. Oper. Res. 9 441–458.
• [47] Ross, N. (2011). Fundamentals of Stein’s method. Probab. Surv. 8 210–293.
• [48] Shi, P., Chou, M., Dai, J. G., Ding, D. and Sim, J. (2016). Models and insights for hospital inpatient operations: Time-dependent ED boarding time. Manage. Sci. 62 1–28. DOI:10.1287/mnsc.2014.2112.
• [49] Stein, C. (1972). A bound for the error in the normal approximation to the distribution of a sum of dependent random variables. In Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability (Univ. California, Berkeley, Calif., 1970/1971). Probability Theory II 583–602. Univ. California Press, Berkeley, CA.
• [50] Stein, C. (1986). Approximate Computation of Expectations. Institute of Mathematical Statistics Lecture Notes—Monograph Series 7. IMS, Hayward, CA.
• [51] Tezcan, T. (2008). Optimal control of distributed parallel server systems under the Halfin and Whitt regime. Math. Oper. Res. 33 51–90.
• [52] Whitt, W. (1971). Weak convergence theorems for priority queues: Preemptive-resume discipline. J. Appl. Probab. 8 74–94.
• [53] Whitt, W. (2002). Stochastic-Process Limits. An Introduction to Stochastic-Process Limits and Their Application to Queues. Springer, New York.
• [54] Williams, R. J. (1998). Diffusion approximations for open multiclass queueing networks: Sufficient conditions involving state space collapse. Queueing Syst. Theory Appl. 30 27–88.
• [55] 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.
• [56] Zhang, J. and Zwart, B. (2008). Steady state approximations of limited processor sharing queues in heavy traffic. Queueing Syst. 60 227–246.