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BRAVO for Many-Server QED Systems with Finite Buffers

  • D. J. Daley (a1), Johan S. H. Van Leeuwaarden (a2) and Yoni Nazarathy (a3)

This paper demonstrates the occurrence of the feature called BRAVO (balancing reduces asymptotic variance of output) for the departure process of a finite-buffer Markovian many-server system in the QED (quality and efficiency-driven) heavy-traffic regime. The results are based on evaluating the limit of an equation for the asymptotic variance of death counts in finite birth-death processes.

Corresponding author
Postal address: Department of Mathematics and Statistics, The University of Melbourne, Parkville, VIC 3010, Australia. Email address:
∗∗ Postal address: Department of Mathematics and Computer Science, Eindhoven University of Technology, PO Box 513, 5600 MB Eindhoven, The Netherlands. Email address:
∗∗∗ Postal address: School of Mathematics and Physics, The University of Queensland, St Lucia, Brisbane, 4072, Australia. Email address:
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[1] Al Hanbali, A., Mandjes, M., Nazarathy, Y. and Whitt, W. (2011). “The asymptotic variance of departures in critically loaded queues.” Adv. Appl. Prob. 43, 243263.
[2] Armony, M. and Maglaras, C. (2004). “On customer contact centers with a call-back option: customer decisions, routing rules, and system design.” Operat. Res. 52, 271292.
[3] Borst, S., Mandelbaum, A. and Reiman, M. I. (2004). “Dimensioning large call centers.” Operat. Res. 52, 1734.
[4] Browne, S. and Whitt, W. (1995). “Piecewise-linear diffusion processes.“In Advances in Queueing, CRC, Boca Raton, FL, pp. 463480.
[5] Daley, D. J. (1976). “Queueing output processes.” Adv. Appl. Prob. 8, 395415.
[6] Daley, D. J. (2011). “Revisiting queueing output processes: a point process viewpoint.” Queueing Systems 68, 395405.
[7] Disney, R. L. and König, D. (1985). “Queueing networks: a survey of their random processes.” SIAM Rev. 27, 335403.
[8] Feller, W. (1968). “An Introduction to Probability Theory and Its Applications, Vol. I, 3rd edn. John Wiley, New York.
[9] Gans, N., Koole, G. and Mandelbaum, A. (2003). “Telephone call centers: tutorial, review, and research prospects.” Manufacturing Service Operat. Manag. 5, 79141.
[10] Garnett, O., Mandelbaum, A. and Reiman, M. (2002). “Designing a call center with impatient customers.” Manufacturing Service Operat. Manag. 4, 208227.
[11] Halfin, S. and Whitt, W. (1981). “Heavy-traffic limits for queues with many exponential servers.” Operat. Res. 29, 567588.
[12] Janssen, A. J. E. M., van Leeuwaarden, J. S. H. and Sanders, J. (2013). “Scaled control in the QED regime.” Performance Evaluation 70, 750769.
[13] Janssen, A. J. E. M., van Leeuwaarden, J. S. H. and Zwart, B. (2011). “Refining square-root safety staffing by expanding Erlang C.” Operat. Res. 59, 15121522.
[14] Jelenković, P., Mandelbaum, A. and Momčilović, P. (2004). “Heavy traffic limits for queues with many deterministic servers.” Queueing Systems 47, 5369.
[15] Maglaras, C. and Zeevi, A. (2004). “Diffusion approximations for a multiclass Markovian service system with ‘guaranteed’ and ‘best-effort’ service levels.” Math. Operat. Res. 29, 786813.
[16] Mandelbaum, A. and Momčilović, P. (2008). “Queues with many servers: the virtual waiting-time process in the QED regime.” Math. Operat. Res. 33, 561586.
[17] Massey, W. A. and Wallace, R. (2005). “An asymptotically optimal design of the M/M/c/K queue for call centers.” Unpublished manuscript.
[18] Nazarathy, Y. (2011). “The variance of departure processes: puzzling behavior and open problems.” Queueing Systems 68, 385394.
[19] Nazarathy, Y. and Weiss, G. (2008). “The asymptotic variance rate of the output process of finite capacity birth–death queues.” Queueing Systems 59, 135156.
[20] Pang, G., Talreja, R. and Whitt, W. (2007). “Martingale proofs of many-server heavy-traffic limits for Markovian queues.” Prob. Surveys 4, 193267.
[21] Reed, J. (2009). “The G/GI/N queue in the Halfin–Whitt regime.” Ann. Appl. Prob. 19, 22112269.
[22] Shiryayev, A. N. (1984). “Probability. Springer, New York.
[23] Whitt, W. (2004). “A diffusion approximation for the G/GI/n/m queue.” Operat. Res. 52, 922941.
[24] Whitt, W. (2005). “Heavy-traffic limits for the G/H 2 */n/m queue.” Math. Operat. Res. 30, 127.
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