Hostname: page-component-89b8bd64d-n8gtw Total loading time: 0 Render date: 2026-05-07T21:36:45.394Z Has data issue: false hasContentIssue false
  • English
  • Français

Two-Stage Release Rule Procedure in a Regenerative Dam

Published online by Cambridge University Press:  27 July 2009

Shaul K. Bar-Lev  and
David Perry
Shaul K. Bar-Lev
Affiliation:
Department of Statistics, University of Haifa, Haifa 31905, Israel
David Perry
Affiliation:
Department of Statistics, University of Haifa, Haifa 31905, Israel
Get access Rights & Permissions [Opens in a new window]

Abstract

We consider a dam model with a compound Poisson input having positive jumps in which the release rule can be dynamically controlled between two release functions, r1 and r2. The dam process has two switchover levels, a and b, 0 < a < b < ∞, which are prescribed to switch the release rule from r2 to r1 only when the dam downcrosses levels and switch from r1 to r2 only when the dam upcrosses level b. We derive integral equations whose solutions determine the stationary distribution of the dam content and demonstrate such a determination explicitly for the case of exponential jumps. We also compute an expression for the average number of switches from r1 to r2, and vice versa; consider an approximation to the general case (with b − a being sufficiently large with respect to r1 and r2 and the jump sizes), which allows the use of asymptotic results from renewal theory; and study the wet period analysis for the case of exponential jumps.

Information

Type
Research Article
Information
Probability in the Engineering and Informational Sciences , Volume 7 , Issue 4 , October 1993 , pp. 571 - 588
Copyright
Copyright © Cambridge University Press 1993

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

1.Attia, F.A. & Brockwell, P.J. (1982). The control of a finite dam. Journal of Applied Probability 19: 815–825.CrossRefGoogle Scholar
2.Brockwell, P.J. (1977). Stationary distributions for dams with additive input and content-dependent release rate. Advances in Applied Probability 9: 645–663.CrossRefGoogle Scholar
3.Brockwell, P.J. & Chung, K.L. (1975). Emptiness times of a dam with stable input and general release rule. Journal of Applied Probability 12: 212–217.CrossRefGoogle Scholar
4.Cinlar, E. & Pinsky, M. (1972). On dams with additive input and general release rule. Journal of Applied Probability 9: 422–427.CrossRefGoogle Scholar
5.Cohen, J.W. (1977). On up and down crossings. Journal of Applied Probability 14: 405–410.CrossRefGoogle Scholar
6.De Kok, A.G., Tijms, H.C., & van der Duyn Schouten, F.A. (1984). Approximation for the single product production-inventory model with compound Poisson demand and service level constraints. Advances in Applied Probability 16: 378–401.CrossRefGoogle Scholar
7.Faddy, M.J. (1974). Optimal control of finite dams: Discrete (2-stage) output procedure. Journal of Applied Probability 11: 111–121.CrossRefGoogle Scholar
8.Harrison, J.M. & Resnick, S.I. (1976). The stationary distribution of fire exist probabilities of a storage process with general release rule. Mathematics of Operations Research 1: 347–358.CrossRefGoogle Scholar
9.Harrison, J.M. & Resnick, S.I. (1978). The recurrence classification of risk and storage process. Mathematics of Operations Research 3: 57–67.CrossRefGoogle Scholar
10.Kaspi, H. & Perry, D. (1989). On a duality between a non-Markovian storage/production process with state dependent input and output. Journal of Applied Probability 27: 835–844.CrossRefGoogle Scholar
11.Moran, P.A.P. (1969). A theory of dams with continuous input and a general release rule. Journal of Applied Probability 6: 88–98.CrossRefGoogle Scholar
12.Perry, D. & Posner, M.J.M. (1989). Control policies for two classes of inventory systems via a duality-equivalence relationship. Probability in the Engineering and Informational Sciences 3: 561–579.CrossRefGoogle Scholar
13.Perry, D. & Posner, M.J.M. (1990). Control of input and demand rates in inventory systems of perishable commodities. Naval Research Logistics 37: 85–97.3.0.CO;2-F>CrossRefGoogle Scholar
14.Posner, M.J.M. & Zuckerman, D. (1984). A replacement model for an additive damage model with restoration. Operations Research Letters 3: 141–148.CrossRefGoogle Scholar
15.Smith, N.M.H. & Yeo, G.F. (1981). On a general storage problem and its approximate solution. Advances in Applied Probability 13: 567–602.CrossRefGoogle Scholar
16.Tijms, H.C. & van der Duyn Schouten, F.A. (1978). Inventory control with two switchover levels for the class of M/G/1 queueing systems with variable arrival and service rate. Stochastic Processes and Their Applications 6: 213–222.CrossRefGoogle Scholar
17.Wolff, R. (1989). Stochastic modelling and the theory of queues. Englewood Cliffs, NJ: Prentice-Hall.Google Scholar
18.Yeo, G.F. (1975). A finite dam with variable release rate. Journal of Applied Probability 12: 205–211.CrossRefGoogle Scholar
19.Zuckerman, D. (1977). Two-stage output procedure of a finite dam. Journal of Applied Probability 14: 421–425.CrossRefGoogle Scholar
20.Zuckerman, D. (1978). Optimal replacement policy for the case where damage process is a one-sided Lèvi process. Stochastic Processes and Their Applications 7: 141–151.CrossRefGoogle Scholar