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Optimal control of a large dam. (English) Zbl 1135.60055

Summary: A large dam model is the object of study of this paper. The parameters \(L^{\text{lower}}\) and \(L^{\text{upper}}\) define its lower and upper levels, \(L =L^{\text{upper}} - L^{\text{lower}}\) is large, and if the current level of water is between these bounds, the dam is assumed to be in a normal state. Passage across one or other of the levels leads to damage. Let \(J_1\) and \(J_2\) denote the damage costs of crossing the lower and, respectively, the upper levels. It is assumed that the input stream of water is described by a Poisson process, while the output stream is state dependent. Let \(L_t\) denote the dam level at time \(t\), and let \(p_1 = \lim_{t\to\infty} P(L_t = L^{\text{lower}})\) and \(p_2 = \lim_{t\to\infty} P\{L_t > L^{\text{upper}}\}\). The long-run average cost, \(J = p_1 J_1+p_2J_2\), is a performance measure. The aim of the paper is to choose the parameter controlling the output stream so as to minimize \(J\).

MSC:

60K25 Queueing theory (aspects of probability theory)
60K30 Applications of queueing theory (congestion, allocation, storage, traffic, etc.)
40E05 Tauberian theorems
90B05 Inventory, storage, reservoirs
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References:

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