## Stochastic Systems

### Optimal Admission Control for Many-Server Systems with QED-Driven Revenues

#### Abstract

We consider Markovian many-server systems with admission control operating in a Quality-and-Efficiency-Driven (QED) regime, where the relative utilization approaches unity while the number of servers grows large, providing natural Economies-of-Scale. In order to determine the optimal admission control policy, we adopt a revenue maximization framework, and suppose that the revenue rate attains a maximum when no customers are waiting and no servers are idling. When the revenue function scales properly with the system size, we show that a nondegenerate optimization problem arises in the limit. Detailed analysis demonstrates that the revenue is maximized by nontrivial policies that bar customers from entering when the queue length exceeds a certain threshold of the order of the typical square-root level variation in the system occupancy. We identify a fundamental equation characterizing the optimal threshold, which we extensively leverage to provide broadly applicable upper/lower bounds for the optimal threshold, establish its monotonicity, and examine its asymptotic behavior, all for general revenue structures. For linear and exponential revenue structures, we present explicit expressions for the optimal threshold.

#### Article information

Source
Stoch. Syst., Volume 7, Number 2 (2017), 315-341.

Dates
Accepted: September 2017
First available in Project Euclid: 24 February 2018

https://projecteuclid.org/euclid.ssy/1519441216

Digital Object Identifier
doi:10.1287/stsy.2017.0004

Mathematical Reviews number (MathSciNet)
MR3741356

Zentralblatt MATH identifier
06849804

#### Citation

Sanders, Jaron; Borst, S. C.; Janssen, A. J. E. M.; Leeuwaarden, J. S. H. van. Optimal Admission Control for Many-Server Systems with QED-Driven Revenues. Stoch. Syst. 7 (2017), no. 2, 315--341. doi:10.1287/stsy.2017.0004. https://projecteuclid.org/euclid.ssy/1519441216

#### References

• Armony, M., Maglaras, C. (2004) On customer contact centers with a call-back option: Customer decisions, routing rules, and system design. Oper. Res. 52(2):271–292.
• Atar, R. (2005a) Scheduling control for queueing systems with many servers: Asymptotic optimality in heavy traffic. Ann. Appl. Probab. 15(4):2606–2650.
• Atar, R. (2005b) A diffusion model of scheduling control in queueing systems with many servers. Ann. Appl. Probab. 15(1B):820–852.
• Atar, R., Mandelbaum, A., Reiman, MI. (2004) A Brownian control problem for a simple queueing system in the Halfin–Whitt regime. Systems and Control Lett. 51(3–4):269–275.
• Atar, R., Mandelbaum, A., Shaikhet, G. (2006) Queueing systems with many servers: Null controllability in heavy traffic. Ann. Appl. Probab. 16(4):1764–1804.
• Avram, F., Janssen, AJEM., Leeuwaarden, JSHV. (2013) Loss systems with slow retrials in the Halfin–Whitt regime. Adv. Appl. Probab. 45(1): 274–294.
• Bekker, R., Borst, SC. (2006) Optimal admission control in queues with workload-dependent service rates. Probab. Engrg. Inform. Sci. 20(4): 543–570.
• Borgs, C., Chayes, JT., Doroudi, S., Harchol-Balter, M., Xu, K. (2014) The optimal admission threshold in observable queues with state dependent pricing. Probab. Engrg. Inform. Sci. 28(1):101–119.
• Çil, EB., Örmeci, EL., Karaesmen, F. (2009) Effects of system parameters on the optimal policy structure in a class of queueing control problems. Queueing Systems 61(4):273–304.
• Chen, H., Frank, MZ. (2001) State dependent pricing with a queue. IIE Trans. 33(10):847–860.
• Corless, RM., Gonnet, GH., Hare, DEG., Jeffrey, DJ., Knuth, DE. (1996) On the Lambert W function. Adv. Comput. Math. 5(1):329–359.
• De Waal, P. (1990) Overload control of telephone exchanges. Ph.D. thesis, Katholieke Universiteit Brabant, Tilburg.
• Garnett, O., Mandelbaum, A., Reiman, M. (2002) Designing a call center with impatient customers. Manufacturing Service Oper. Management 4(3):208–227.
• Ghosh, AP., Weerasinghe, AP. (2007) Optimal buffer size for a stochastic processing network in heavy traffic. Queueing Systems 55(3):147–159.
• Ghosh, AP., Weerasinghe, AP. (2010) Optimal buffer size and dynamic rate control for a queueing system with impatient customers in heavy traffic. Stochastic Processes and Their Appl. 120(11):2103–2141.
• Gurvich, I., Whitt, W. (2009) Queue-and-idleness-ratio controls in many-server service systems. Math. Oper. Res. 34(2):363–396.
• Halfin, S., Whitt, W. (1981) Heavy-traffic limits for queues with many exponential servers. Oper. Res. 29(3):567–588.
• Jagerman, DL. (1974) Some properties of the Erlang loss function. Bell System Tech. J. 53(3):525–551.
• Janssen, AJEM., van, Leeuwaarden JSH., Sanders, J. (2013) Scaled control in the QED regime. Performance Evaluation 70(10):750–769.
• Koçağa, YL., Ward, AR. (2010) Admission control for a multi-server queue with abandonment. Queueing Systems 65(3):275–323.
• Kumar, S., Randhawa, RS. (2010) Exploiting market size in service systems. Manufacturing Service Oper. Management 12(3):511–526.
• Kushner, HJ., Dupuis, P. (2001) Numerical Methods for Stochastic Control Problems in Continuous Time (Springer, New York).
• Lippman, SA. (1975) Applying a new device in the optimization of exponential queuing systems. Oper. Res. 23(4):687–710.
• Luenberger, DG. (1969) Optimization by Vector Space Methods (John Wiley & Sons, New York).
• Maglaras, C., Zeevi, A. (2003) Pricing and capacity sizing for systems with shared resources: Approximate solutions and scaling relations. Management Sci. 49(8):1018–1038.
• Maglaras, C., Zeevi, A. (2005) Pricing and design of differentiated services: Approximate analysis and structural insights. Oper. Res. 53(2): 242–262.
• Maglaras, C., Yao, J., Zeevi, A. (2017) Optimal price and delay differentiation in large-scale queueing systems. Management Sci, ePub ahead of print February 22, 2017, https://doi.org/10.1287/mnsc.2016.2713.
• Massey, W., Wallace, R. (2005) An asymptotically optimal design of M/M/c/k queue. Technical report, Princeton University.
• Meyn, SP. (2008) Control Techniques for Complex Networks (Cambridge University Press, New York).
• Naor, P. (1969) The regulation of queue size by levying tolls. Econometrica 37(1):15–24.
• Olver, FWJ. (2010) NIST Handbook of Mathematical Functions (Cambridge University Press, New York).
• Rudin, W. (1987) Real and Complex Analysis (McGraw-Hill, New York).
• Stidham, S. (1985) Optimal control of admission to a queueing system. IEEE Trans. Automatic Control 30(8):705–713.
• Szegö, G. (1939) Orthogonal Polynomials (American Mathematical Society, Providence, RI).
• Teschl, G. (2014) Topics in Real and Functional Analysis. Unpublished, available at http://www.mat.univie.ac.at/~gerald/.
• Ward, AR., Kumar, S. (2008) Asymptotically optimal admission control of a queue with impatient customers. Math. Oper. Res. 33(1):167–202.
• Weerasinghe, A., Mandelbaum, A. (2013) Abandonment versus blocking in many-server queues: Asymptotic optimality in the QED regime. Queueing Systems 75(2–4):279–337.
• Whitt, W. (2004) A diffusion approximation for the $G/GI/n/m$ queue. Oper. Res. 52(6):922–941.
• Whitt, W. (2005) Heavy-traffic limits for the $G/H^*_2/n/m$ queue. Math. Oper. Res. 30(1):1–27.
• Yildirim, U., Hasenbein, JJ. (2010) Admission control and pricing in a queue with batch arrivals. Oper. Res. Lett. 38(5):427–431.