Chapter 10

Dynamic Optimization with Applications to Dynamic Rate Queues

Robert C. Hampshire
School of Public Policy and Management, H. John Heinz III College, Carnegie Mellon University, Pittsburgh, Pennsylvania15213, hamp@andrew.cmu.edu

William A. Massey
Operations Research and Financial Engineering Department, Princeton University, Princeton, New Jersey 08540, wmassey@princeton.edu

Abstract
This tutorial presents recent developments in the management of communications services and applies broadly to services involving the leasing of shared resources. These problems are more realistically modeled by queues with time-varying rates or more simply, dynamic rate queues. We first provide a review and summary of relevant results for various fundamental dynamic rate queues. The focus here is on approximations of these queueing models by low-dimensional dynamical systems.

The dynamic optimization of constrained dynamical systems is based on the calculus of variations and its various incarnations over the past three centuries. We discuss these methods in the context of Lagrangians, Hamiltonians, and Bellman value functions. Finally, we provide examples where we apply these optimization techniques to dynamic rate queues motivated by communications decision problems.

Key words: Bellman value function; calculus of variations; conservation principles; differential equations; dynamical systems; Hamiltonian; Lagrangian; Lagrange multipliers; Legendre transforms; opportunity costs; optimal control; Poisson brackets

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Citation Information:

Hampshire, R. C., W. A. Massey. 2010. Dynamic optimization with applications to dynamic rate queues. J. J. Hasenbein, ed. TutORials in Operations Research, Vol. 7. INFORMS, Hanover, MD, pp. 208–247.