Drift rate control of a Brownian processing system

Details Drift rate control of a Brownian processing system Drift rate control of a Brownian processing system

2005-05-11

arxiv.org/abs/math/0505210v1

Annals of Applied Probability 2005, Vol. 15, No. 2, 1145-1160

Mathematics

Probability

Published at http://dx.doi.org/10.1214/105051604000000855 in the Annals of Applied Probability (http://www.imstat.org/aap/) by

Scientific paper

10.1214/105051604000000855

A system manager dynamically controls a diffusion process Z that lives in a finite interval [0,b]. Control takes the form of a negative drift rate \theta that is chosen from a fixed set A of available values. The controlled process evolves according to the differential relationship dZ=dX-\theta(Z) dt+dL-dU, where X is a (0,\sigma) Brownian motion, and L and U are increasing processes that enforce a lower reflecting barrier at Z=0 and an upper reflecting barrier at Z=b, respectively. The cumulative cost process increases according to the differential relationship d\xi =c(\theta(Z)) dt+p dU, where c(\cdot) is a nondecreasing cost of control and p>0 is a penalty rate associated with displacement at the upper boundary. The objective is to minimize long-run average cost. This problem is solved explicitly, which allows one to also solve the following, essentially equivalent formulation: minimize the long-run average cost of control subject to an upper bound constraint on the average rate at which U increases. The two special problem features that allow an explicit solution are the use of a long-run average cost criterion, as opposed to a discounted cost criterion, and the lack of state-related costs other than boundary displacement penalties. The application of this theory to power control in wireless communication is discussed.

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