Two-stage output procedure of a finite dam

The input of water into a finite dam is a Wiener process with positive drift. Water may be released at either of two possible rates 0 or M. At any time the output rate can be increased from 0 to M with cost KM, (K ≧ 0), or decreased from M to 0 with zero cost. There is a reward of A monetary units for each unit of output, (A > 0). We will consider the problem of specifying an optimal control output policy under the following optimal criteria: (a)

Resolvent operators of Markov processes and their applications in the control of a finite dam

The resolvent operators of the following two processes are obtained: (a) the bivariate Markov process W = (X, Y), where Y(t) is an irreducible Markov chain and X(t) is its integral, and (b) the geometric Wiener process G(t) = exp{B(t} where B(t) is a Wiener process with non-negative drift μ and variance parameter σ2. These results are then used via a limiting procedure to determine the long-run average cost per unit time of operating a finite dam where the input process is either X(t) or G(t). The system is controlled by a policy (Attia [1], Lam [6]).

The control of a finite dam

The long-run average cost per unit time of operating a finite dam controlled by a PlM policy (Faddy (1974), Zuckerman (1977)) is determined when the cumulative input process is (a) a Wiener process with drift and (b) the integral of a Markov chain. It is shown how the cost for (a) can be obtained as the limit of the costs associated with a sequence of input processes of the type (b).

The control of a finite dam

The long-run average cost per unit time of operating a finite dam controlled by a Pl M policy (Faddy (1974), Zuckerman (1977)) is determined when the cumulative input process is (a) a Wiener process with drift and (b) the integral of a Markov chain. It is shown how the cost for (a) can be obtained as the limit of the costs associated with a sequence of input processes of the type (b).

Characterization of the optimal class of output policies in a control model of a finite dam

In this paper we characterize the optimal class of output policies in a control model of a dam having a finite capacity. The input of water into the dam is determined by a Wiener process with positive drift. Water may be released at either of two possible rates 0 or M. At any time the output rate can be increased from 0 to M with a cost of K, (K ≧ 0) or decreased from M to 0 with zero cost, any such changes taking effect instantaneously. There is a reward of A monetary units for each unit of output, (A ≧ 0). The problem is to formulate an optimal output policy which maximizes the long-run average net reward per unit time.

Extreme values, range and weak convergence of integrals of Markov chains

A study is made of the maximum, minimum and range on [0, t] of the integral process where S is a finite state-space Markov chain. Approximate results are derived by establishing weak convergence of a sequence of such processes to a Wiener process. For a particular family of two-state stationary Markov chains we show that the corresponding centered integral processes exhibit the Hurst phenomenon to a remarkable degree in their pre-asymptotic behaviour.

Two-stage output procedure of a finite dam

The input of water into a finite dam is a Wiener process with positive drift. Water may be released at either of two possible rates 0 or M. At any time the output rate can be increased from 0 to M with cost KM, (K ≧ 0), or decreased from M to 0 with zero cost. There is a reward of A monetary units for each unit of output, (A > 0). We will consider the problem of specifying an optimal control output policy under the following optimal criteria: (a)

The control of a multipurpose reservoir

A technique known as potential cost, used by Faddy [3] for assessing the operation of a dam is seen to be capable of extension to allow for

(i) a very general cost function, as is required for a multipurpose reservoir (the norm nowadays) and

(ii) the use of discounting of future costs, a very widespread accounting procedure.

Numerical results are obtained for an optimal policy based on such an assessment, and demonstrate the need for an accurate specification of the costs associated with the operation of a reservoir. As a by-product a very full description of the steady-state stochastic behaviour of the dam is obtained.

Optimal control of finite dams: continuous output procedure

The problem discussed is that of controlling optimally the release of water from a finite dam, when the release rate may vary continuously and the optimality is defined in terms of a cost structure imposed on the operation of the dam. A diffusion model is suggested and by considering a family of plausible output policies, the control problem is reduced to the solution of a free boundary problem associated with a certain partial differential equation. A set of necessary conditions for the optimal choice of these boundaries is established and a method of solution is suggested. By using this method, together with well-established computational techniques, numerical solutions are obtained. These numerical solutions indicate that this optimal policy does not result in very much improvement over a much simpler policy where the output rate is constrained to take only two values.

Optimal Control of a Large Dam

A large dam model is the object of study of this paper. The parametersLlowerandLupperdefine its lower and upper levels,L=Lupper-Lloweris 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. LetJ1andJ2denote 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. LetLtdenote the dam level at timet, and letp1= limt→∞P{Lt=Llower} andp2= limt→∞P{Lt>Lupper} exist. The long-run average cost,J=p1J1+p2J2, is a performance measure. The aim of the paper is to choose the parameter controlling the output stream so as to minimizeJ.

Optimization under the P M λ,τ policy of a finite dam with both continuous and jumpwise inputs

We consider a finite dam under the policy, where the input of water is formed by a Wiener process subject to random jumps arriving according to a Poisson process. The long-run average cost per unit time is obtained after assigning costs to the changes of release rate, a reward to each unit of output, and a penalty that is a function of the level of water in the reservoir.

P λ M-policy for a dam with input formed by a compound Poisson process

An infinite dam with input formed by a compound Poisson process is considered. As an output policy, we adopt the P λ M -policy. The stationary distribution and expectation of the level of water in the reservoir are obtained.

The control of a finite dam with penalty cost function: Markov input rate

The long-run average cost per unit time of operating a finite dam controlled by a policy (Lam Yeh (1985)) is determined when the cumulative input process is the integral of a Markov chain. A penalty cost which accrues continuously at a rate g(X(t)), where g is a bounded measurable function of the content, is also introduced. An example where the input rate is a two-state Markov chain is considered in detail to illustrate the computations.

Extreme values, range and weak convergence of integrals of Markov chains

A study is made of the maximum, minimum and range on [0,t] of the integral processwhereSis a finite state-space Markov chain. Approximate results are derived by establishing weak convergence of a sequence of such processes to a Wiener process. For a particular family of two-state stationary Markov chains we show that the corresponding centered integral processes exhibit the Hurst phenomenon to a remarkable degree in their pre-asymptotic behaviour.