Hostname: page-component-78c5997874-m6dg7 Total loading time: 0 Render date: 2024-11-18T22:01:03.284Z Has data issue: false hasContentIssue false
  • English
  • Français

Level crossings and stationary distributions for general dams

Published online by Cambridge University Press:  14 July 2016

Michael Rubinovitch  and
J. W. Cohen
Michael Rubinovitch*
Affiliation:
Technion — Israel Institute of Technology
J. W. Cohen*
Affiliation:
University of Utrecht
*
∗Postal address: Faculty of Industrial and Management Engineering, Technion–Israel Institute of Technology, Haifa, Israel. Research supported by Air Force Office of Scientific Research Grant No. 74–2733 while the author was on leave at Northwestern University.
∗∗Postal address: State University of Utrecht, Mathematical Institute, Budapestlaan 6, Utrecht 2506, The Netherlands.
Get access Rights & Permissions [Opens in a new window]

Abstract

Level crossings in a stationary dam process with additive input and arbitrary release are considered and an explicit expression for the expected number of downcrossings (and also overcrossings) of a fixed level, per time unit, is obtained. This leads to a short derivation of a basic relation which the stationary distribution of a general dam must satisfy.

Keywords

LEVEL CROSSINGSADDITIVE INPUTDAMSSTATIONARY DISTRIBUTIONUPCROSSINGSDOWNCROSSINGSREGENERATIVE PROCESSES
Type
Research Papers
Information
Journal of Applied Probability , Volume 17 , Issue 1 , March 1980 , pp. 218 - 226
Copyright
Copyright © Applied Probability Trust 1980 

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.)

References

[1] Brockwell, P. J. (1977) Stationary distributions for dams with additive input and content-dependent release rate. Adv. Appl. Prob. 9, 645–663.CrossRefGoogle Scholar
[2] Çinlar, E. and Pinsky, M. (1971) A stochastic integral in storage theory. Z. Wahrscheinlichkeitsth. 17, 227–240.CrossRefGoogle Scholar
[3] Çinlar, E. and Pinsky, M. (1972) On dams with additive inputs and a general release rule. J. Appl. Prob. 9, 422–429.CrossRefGoogle Scholar
[4] Cohen, J. W. (1976) On Regenerative Processes in Queueing Theory. Lecture Notes in Economics and Mathematical Systems 121, Springer-Verlag, Berlin.CrossRefGoogle Scholar
[5] Cohen, J. W. (1977) On up- and downcrossings. J. Appl. Prob. 14, 405–410.CrossRefGoogle Scholar
[6] Cohen, J. W. and Rubinovitch, M. (1977) On level crossings and cycles in dam processes. Math. Operat. Res. 2, 297–310.CrossRefGoogle Scholar
[7] Cramér, H. and Leadbetter, M. R. (1967) Stationary and Related Stochastic Processes. Wiley, New York.Google Scholar
[8] Harrison, J. M. and Resnick, S. I. (1976) The stationary distribution and first exit probabilities of a storage process with general release rule. Math. Operat. Res. 1, 347–358.CrossRefGoogle Scholar
[9] Shtatland, E. S. (1965) On local propeties of processes with independent increments. Theory Prob. Appl. 10, 317–322.CrossRefGoogle Scholar
[10] Stidham, S. (1972) Regenerative processes in the theory of queues, with applications to the alternating-priority queue. Adv. Appl. Prob. 4, 542–577.CrossRefGoogle Scholar