Hostname: page-component-cd9895bd7-mkpzs Total loading time: 0 Render date: 2024-12-26T07:18:08.607Z Has data issue: false hasContentIssue false
  • English
  • Français

Optimality of randomized trunk reservation for a problem with a single constraint

Part of: Special processes

Published online by Cambridge University Press:  01 July 2016

X. Fan-Orzechowski  and
E. A. Feinberg
X. Fan-Orzechowski*
Affiliation:
State University of New York at Stony Brook
E. A. Feinberg*
Affiliation:
State University of New York at Stony Brook
*
∗ Postal address: Department of Applied Mathematics and Statistics, State University of New York at Stony Brook, Stony Brook, NY 11794-3600, USA.
∗ Postal address: Department of Applied Mathematics and Statistics, State University of New York at Stony Brook, Stony Brook, NY 11794-3600, USA.
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We study an optimal admission of arriving customers to a Markovian finite-capacity queue, e.g. an M/M/c/N queue, with several customer types. The system managers are paid for serving customers and penalized for rejecting them. The rewards and penalties depend on customer type. The goal is to maximize the average rewards per unit time subject to the constraint on the average penalties per unit time. We provide a solution to this problem based on Lagrangian optimization. For a feasible problem, we show the existence of a randomized trunk reservation optimal policy with the acceptance thresholds for different customer types ordered according to a linear combination of the service rewards and rejection costs. In addition, we prove that any 1-randomized stationary optimal policy has this structure. In particular, we establish the structure of an optimal policy that maximizes the average rewards per unit time subject to the constraint on the blocking probability of either one of the customer types or a group of customer types pooled together.

Keywords

Queueadmission controlsemi-Markov decision processtrunk reservationLagrangian optimization

MSC classification

Primary: 60K25: Queueing theory
Secondary: 60K15: Markov renewal processes, semi-Markov processes 60K30: Applications (congestion, allocation, storage, traffic, etc.)
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 38 , Issue 1 , March 2006 , pp. 199 - 220
Copyright
Copyright © Applied Probability Trust 2006 

References

Altman, E. (2002). Applications of Markov decision processes in telecommunication networks. In Handbook of Markov Decision Processes, eds Feinberg, E. and Schwartz, A., Kluwer, Boston, MA, pp. 480–536.Google Scholar
Altman, E., Jimenez, T. and Koole, G. (2001). On optimal call admission control in a resource-sharing system. IEEE Trans. Commun. 49, 1659–1668.Google Scholar
Carrizosa, E., Conde, E. and Munoz-Márquez, M. (1998). Admission policies in loss queueing models with heterogeneous arrivals. Manag. Sci. 44, 311–320.Google Scholar
Crabill, T. B., Gross, D. and Magazine, M. J. (1977). A classified bibliography of research on optimal design and control of queues. Operat. Res. 25, 219–232.Google Scholar
Feinberg, E. A. (1994). Constrained semi-Markov decision processes with average rewards. Z. Operat. Res. 39, 257–288.Google Scholar
Feinberg, E. A. (2002). Constrained finite continuous-time Markov decision processes with average rewards. In Proc. IEEE 2002 Conf. Decisions Control (December 2002, Las Vegas), pp. 3805–3810.Google Scholar
Feinberg, E. A. (2004). Continuous time discounted Jump Markov decision processes: a discrete-event approach. Math. Operat. Res. 29, 492–524.CrossRefGoogle Scholar
Feinberg, E. A. (2005). On essential information in sequential decision processes. Math. Meth. Operat. Res. 62, 399–410.CrossRefGoogle Scholar
Feinberg, E. A. and Reiman, M. I. (1994). Optimality of randomized trunk reservation. Prob. Eng. Inf. Sci. 8, 463–489.Google Scholar
Fiacco, A. V. and McCormick, G. P. (1990). Nonlinear Programming: Sequential Unconstrained Minimization Techniques. SIAM, Philadelphia, PA.Google Scholar
Hunt, P. J. and Laws, C. N. (1997). Optimization via trunk reservation in single resource loss systems under heavy traffic. Ann. Appl. Prob. 7, 1058–1079.Google Scholar
Kelly, F. P. (1990). Routing and capacity allocation in networks with trunk reservation. Math. Operat. Res. 15, 771–793.Google Scholar
Key, P. (1990). Optimal control and trunk reservation in loss networks. Prob. Eng. Inf. Sci. 4, 203–242.Google Scholar
Lewis, M. E. (2001). Average optimal policies in a controlled queueing system with dual admission control. J. Appl. Prob. 38, 369–385.Google Scholar
Lewis, M. E., Ayhan, H. and Foley, R. D. (1999). Bias optimality in a queue with admission control. Prob. Eng. Inf. Sci. 13, 309–327.Google Scholar
Lewis, M. E., Ayhan, H. and Foley, R. D. (2002). Bias optimal admission policies for a nonstationary multi-class queueing system. J. Appl. Prob. 39, 20–37.Google Scholar
Lin, K. Y. and Ross, S. M. (2003). Admission control with incomplete information of a queueing system. Operat. Res. 51, 645–654.CrossRefGoogle Scholar
Lin, K. Y. and Ross, S. M. (2004). Optimal admission control for a single-server loss queue. J. Appl. Prob. 41, 535–546.CrossRefGoogle Scholar
Lippman, S. A. (1975). Applying a new device in the optimization of exponential queuing systems. Operat. Res. 23, 687–710.CrossRefGoogle Scholar
Mangasarian, O. L. (1969). Nonlinear Programming. McGraw-Hill, New York.Google Scholar
Miller, B. L. (1969). A queueing reward system with several customer classes. Manag. Sci. 16, 235–245.Google Scholar
Nguyen, V. (1991). Optimality of trunk reservation in overflow process. Prob. Eng. Inf. Sci. 5, 369–390.Google Scholar
Papastavrou, J. D., Rajagopalan, S. and Kleywegt, A.J. (1996). The dynamic and stochastic knapsack problem with deadlines. Manag. Sci. 42, 1706–1718.Google Scholar
Piunovskiy, A. B. (2004). Bicriteria optimization of a queue with a controlled input stream. Queueing Systems 48, 159–184.Google Scholar
Puhalskii, A. A. and Reiman, M. I. (1998). A critically loaded multirate link with trunk reservation. Queueing Systems 28, 157–190.CrossRefGoogle Scholar
Ross, K. W. (1995). Multiservice Loss Models for Broadband Telecommunication Networks. Springer, London.CrossRefGoogle Scholar
Ross, K. W. and Yao, D. D. (1990). Monotonicity properties for the stochastic knapsack. IEEE Trans. Inf. Theory 36, 1173–1179.Google Scholar
Stidham, S. Jr. (1985). On optimal control of admission to a queueing system. IEEE Trans. Automatic Control 30, 705–713.Google Scholar