Abstract
We consider a single-stage queuing system where arrivals and departures are modeled by point processes with stochastic intensities. An arrival incurs a cost, while a departure earns a revenue. The objective is to maximize the profit by controlling the intensities subject to capacity limits and holding costs. When the stochastic model for arrival and departure processes are completely known, then a threshold policy is known to be optimal. Many times arrival and departure processes can not be accurately modeled and controlled due to lack of sufficient calibration data or inaccurate assumptions. We prove that a threshold policy is optimal under a max–min robust model when the uncertainty in the processes is characterized by relative entropy. Our model generalizes the standard notion of relative entropy to account for different levels of model uncertainty in arrival and departure processes. We also study the impact of uncertainty levels on the optimal threshold control.
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References
Bertsekas, D.P.: Dynamic Programming and Optimal Control, vol. II. Athena Scientific, Belmont (1995)
Bremaud, P.: Point Processes and Queues: Martingale Dynamics. Springer, New York (1981)
Chen, H., Yao, D.: Optimal intensity control of a queuing system with state-dependent capacity limit. IEEE Trans. Autom. Control, 35(4) (1990)
Durrett, R.: Probability: Theory and Examples. Duxbury, N. Scituate (2003)
Hansen, L.P., Sargent, T.J., Turmuhambetova, G.A., Williams, N.: Robust control and model misspecification. J. Econ. Theory 128(1), 45–90 (2006)
Jacod, J.: Calcul stochastique et problèmes de martingales. In: Lecture Notes in Math., vol. 714. Springer, Berlin (1979)
Li, L.: A stochastic theory of the firm. Math. Oper. Res. 13, 447–466 (1988)
Lim, A.E.B., Shanthikumar, J.G.: Relative entropy, exponential utility, and robust dynamic pricing. Oper. Res. 55(2), 198–214 (2007)
Lim, A.E.B., Shanthikumar, J.G., Watewai, T.: Robust multi-product pricing and a new class of risk-sensitive control problems. Working Paper (2009)
Petersen, I.R., James, M.R., Dupuis, P.: Minimax optimal control of stochastic uncertain systems with relative entropy constraints. IEEE Trans. Autom. Control 45, 398–412 (2000)
Pra, P. Dai, Meneghini, L., Runggaldier, W.J.: Connections between stochastic control and dynamic games. Math. Control Signals Syst. 9, 303–326 (1996)
Serfozo, R.: Optimal control of random walks, birth and death processes, and queues. Adv. Appl. Probab. 13, 61–83 (1981)
Shapley, L.S.: Stochastic games. Proc. Natl. Acad. Sci. 39, 1095–1100 (1953)
Stidham, S.: Optimal control of admission to a queueing system. IEEE Trans. Autom. Control 30, 705–713 (1985)
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This work is supported in part by an NSF CAREER Award DMI-0348746 (Lim) and the NSF Grant DMI-0500503 (Lim and Shanthikumar). Nevertheless, the opinions, findings, conclusions and recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation. Support from the Coleman Fung Chair in Financial Modelling (Lim) is also acknowledged.
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Jain, A., Lim, A.E.B. & Shanthikumar, J.G. On the optimality of threshold control in queues with model uncertainty. Queueing Syst 65, 157–174 (2010). https://doi.org/10.1007/s11134-010-9172-3
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DOI: https://doi.org/10.1007/s11134-010-9172-3