The Annals of Applied Probability

On the rate of convergence to stationarity of the M/M/N queue in the Halfin–Whitt regime

David Gamarnik and David A. Goldberg

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We prove several results about the rate of convergence to stationarity, that is, the spectral gap, for the $M/M/n$ queue in the Halfin–Whitt regime. We identify the limiting rate of convergence to steady-state, and discover an asymptotic phase transition that occurs w.r.t. this rate. In particular, we demonstrate the existence of a constant $B^{\ast}\approx1.85772\mbox{ s.t.}$ when a certain excess parameter $B\in(0,B^{\ast}]$, the error in the steady-state approximation converges exponentially fast to zero at rate $\frac{B^{2}}{4}$. For $B>B^{\ast}$, the error in the steady-state approximation converges exponentially fast to zero at a different rate, which is the solution to an explicit equation given in terms of special functions. This result may be interpreted as an asymptotic version of a phase transition proven to occur for any fixed $n$ by van Doorn [Stochastic Monotonicity and Queueing Applications of Birth-death Processes (1981) Springer].

We also prove explicit bounds on the distance to stationarity for the $M/M/n$ queue in the Halfin–Whitt regime, when $B<B^{\ast}$. Our bounds scale independently of $n$ in the Halfin–Whitt regime, and do not follow from the weak-convergence theory.

Article information

Ann. Appl. Probab., Volume 23, Number 5 (2013), 1879-1912.

First available in Project Euclid: 28 August 2013

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Zentralblatt MATH identifier

Primary: 60K25: Queueing theory [See also 68M20, 90B22]

Many-server queues rate of convergence spectral gap weak convergence orthogonal polynomials parabolic cylinder functions


Gamarnik, David; Goldberg, David A. On the rate of convergence to stationarity of the M/M/N queue in the Halfin–Whitt regime. Ann. Appl. Probab. 23 (2013), no. 5, 1879--1912. doi:10.1214/12-AAP889.

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  • [1] Atkinson, J. D. and Caughey, T. K. (1968). Spectral density of piecewise linear first order systems excited by white noise. Internat. J. Non-Linear Mech. 3 137–156.
  • [2] Bo, R. and Wong, R. (1994). Uniform asymptotic expansion of Charlier polynomials. Methods Appl. Anal. 1 294–313.
  • [3] Buchholz, H. (1969). The Confluent Hypergeometric Function. Springer, New York.
  • [4] Chen, M. (1998). Estimate of exponential convergence rate in total variation by spectral gap. Acta Math. Sinica (N.S.) 14 9–16.
  • [5] Chen, M. F. (1991). Exponential $L^{2}$-convergence and $L^{2}$-spectral gap for Markov processes. Acta Math. Sinica (N.S.) 7 19–37.
  • [6] Chruściński, D. (2004). Quantum mechanics of damped systems. II. Damping and parabolic potential barrier. J. Math. Phys. 45 841–854.
  • [7] Clarke, A. B. (1956). A waiting line process of Markov type. Ann. Math. Statist. 27 452–459.
  • [8] Diaconis, P. (1996). The cutoff phenomenon in finite Markov chains. Proc. Natl. Acad. Sci. USA 93 1659–1664.
  • [9] Dominici, D. (2007). Asymptotic analysis of the Askey-scheme. I. From Krawtchouk to Charlier. Cent. Eur. J. Math. 5 280–304 (electronic).
  • [10] Dominici, D. (2008). Asymptotic analysis of the Krawtchouk polynomials by the WKB method. Ramanujan J. 15 303–338.
  • [11] Dunster, T. M. (2001). Uniform asymptotic expansions for Charlier polynomials. J. Approx. Theory 112 93–133.
  • [12] Erdelyi, A., Magnus, W., Oberhettinger, F., Tricomi, F. and Bateman, H. (1953). Higher Transcendental Functions, Volume II. McGraw-Hill, New York.
  • [13] Erlang, A. K. (1948). On the Rational Determination of the Number of Circuits. The Copenhagen Telephone Company, Copenhagen.
  • [14] Ethier, S. N. and Kurtz, T. G. (2005). Markov Processes: Characterization and Convergence. Wiley, New York.
  • [15] Fricker, C., Robert, P. and Tibi, D. (1999). On the rates of convergence of Erlang’s model. J. Appl. Probab. 36 1167–1184.
  • [16] Gans, N., Koole, G. and Mandelbaum, A. (2003). Telephone call centers: Tutorial, review, and research prospects. Manufacturing and Service Operations Management 5 79–141.
  • [17] Gradshteyn, I. S. and Ryzhik, I. M. (1980). Table of Integrals, Series, and Products. Academic Press, New York.
  • [18] Halfin, S. and Whitt, W. (1981). Heavy-traffic limits for queues with many exponential servers. Oper. Res. 29 567–588.
  • [19] Hille, E. (1959). Analytic Function Theory. Ginn and Company, Boston.
  • [20] Iglehart, D. L. (1965). Limiting diffusion approximations for the many server queue and the repairman problem. J. Appl. Probab. 2 429–441.
  • [21] Jagerman, D. L. (1974). Some properties of the Erlang loss function. Bell System Tech. J. 53 525–551.
  • [22] Janssen, A. J. E. M. and van Leeuwaarden, J. S. H. (2008). Back to the roots of the $M/D/s$ queue and the works of Erlang, Crommelin and Pollaczek. Stat. Neerl. 62 299–313.
  • [23] Kang, W. and Ramanan, K. (2012). Asymptotic approximations for stationary distributions of many-server queues with abandonment. Ann. Appl. Probab. 22 477–521.
  • [24] Karlin, S. and McGregor, J. (1958). Many server queueing processes with Poisson input and exponential service times. Pacific J. Math. 8 87–118.
  • [25] Karlin, S. and McGregor, J. L. (1957). The differential equations of birth-and-death processes, and the Stieltjes moment problem. Trans. Amer. Math. Soc. 85 489–546.
  • [26] Kijima, M. (1992). Evaluation of the decay parameter for some specialized birth–death processes. J. Appl. Probab. 29 781–791.
  • [27] Ledermann, W. and Reuter, G. E. H. (1954). Spectral theory for the differential equations of simple birth and death processes. Philos. Trans. Roy. Soc. Lond. Ser. A 246 321–369.
  • [28] Morse, P. M. (1955). Stochastic properties of waiting lines. J. Operations Res. Soc. Amer. 3 255–261.
  • [29] Pollaczek, F. (1946). Sur l’application de la théorie des fonctions au calcul de certaines probabilités continues utilisées dans la théorie des réseaux téléphoniques. Ann. Inst. H. Poincaré 10 1–55.
  • [30] Saaty, T. L. (1960). Time-dependent solution of the many-server Poisson queue. Operations Res. 8 755–772.
  • [31] van Doorn, E. A. (1981). Stochastic Monotonicity and Queueing Applications of Birth-death Processes. Lecture Notes in Statistics 4. Springer, New York.
  • [32] van Doorn, E. A. (1984). On oscillation properties and the interval of orthogonality of orthogonal polynomials. SIAM J. Math. Anal. 15 1031–1042.
  • [33] van Doorn, E. A. (1985). Conditions for exponential ergodicity and bounds for the decay parameter of a birth–death process. Adv. in Appl. Probab. 17 514–530.
  • [34] van Doorn, E. A. (1987). Representations and bounds for zeros of orthogonal polynomials and eigenvalues of sign-symmetric tri-diagonal matrices. J. Approx. Theory 51 254–266.
  • [35] van Doorn, E. A. (2002). Representations for the rate of convergence of birth–death processes. Theory Probab. Math. Statist. 65 37–43.
  • [36] van Doorn, E. A. and Zeifman, A. I. (2009). On the speed of convergence to stationarity of the Erlang loss system. Queueing Syst. 63 241–252.
  • [37] van Doorn, E. A., Zeifman, A. I. and Panfilova, T. L. (2009). Bounds and asymptotics for the rate of convergence of birth–death processes. Teor. Veroyatn. Primen. 54 18–38.
  • [38] van Leeuwaarden, J. S. H. and Knessl, C. (2011). Transient behavior of the Halfin–Whitt diffusion. Stochastic Process. Appl. 121 1524–1545.
  • [39] van Leeuwaarden, J. S. H. and Knessl, C. (2012). Spectral gap of the Erlang A model in the Halfin–Whitt regime. Unpublished manuscript.
  • [40] Xie, S. and Knessl, C. (1993). On the transient behavior of the Erlang loss model: Heavy usage asymptotics. SIAM J. Appl. Math. 53 555–599.
  • [41] Young, R. (1991). Euler’s Constant. The Mathematical Gazette 75 187–190.
  • [42] Zeĭfman, A. I. (1991). Some estimates of the rate of convergence for birth and death processes. J. Appl. Probab. 28 268–277.