Optimal control of a large dam with compound Poisson input and costs depending on water levels

Abstract

This paper studies a discrete model of a large dam where the difference between lower and upper levels, L, is assumed to be large. Passage across the levels leads to damage, and the damage costs of crossing the lower or upper level are proportional to the large parameter L. Input stream of water is described by compound Poisson process, and the water cost depends upon current level of water in the dam. The aim of the paper is to choose the parameters of output stream (specifically defined in the paper) minimizing the long-run expenses that include the damage costs and water costs. The present paper addresses the important question How does the structure of water costs affect the optimal solution? We prove the existence and uniqueness of a solution. A special attention is attracted to the case of linear structure of the costs.As well, the paper contributes to the theory of state-dependent queueing systems. The inter-relations between important characteristics of a state-dependent queueing system are established, and their asymptotic analysis that involves analytic techniques of Tauberian theory and heavy traffic approximations is provided.Contents 1991 Mathematics Subject Classification. 60K30, 40E05, 90B05, 60K25.

Fixed point theorem for an infinite Toeplitz matrix

“…Taking into account that t −n > 0, in cases (a 1 ) and (a 2 ) we easily arrive at the required statements, since the difference z − n τ −n (z) in point z = 0 is negative and in point z = 1 is zero. The derivative of this difference in point z = 1 is equal to 1 − (1/n)τ ′ −n (1). It is nonnegative in case (a 1 ) and strictly negative in case (a 2 ).…”

Regime-switching constrained viscosity solutions approach for controlling dam–reservoir systems

“…The range t  can be determined by the operation purpose of the system. In addition, too large water volume may physically damage the dam (Abramov, 2019) and too small water volume may threaten aquatic species in the reservoir (Kawakami and Tachihara, 2006). In addition, ecologically friendly reservoir operations are preferred if keystone aquatic species are spawning in the reservoir (Li et al, 2020).…”

Fixed Point Theorem for an Infinite Toeplitz Matrix

“…In particular, they describe the probability problems that appear as an extension of the classic ruin and ballot problems and problems on fluctuations of sums of random variables. Further applications of convolution-type recurrence relations are known in queueing theory [8], theory of dams [1] and other areas also considered in [7]. The techniques for the study of (1.2) come from the theory of convolution-type recurrence relations.…”