Hostname: page-component-848d4c4894-x5gtn Total loading time: 0 Render date: 2024-05-13T06:57:05.962Z Has data issue: false hasContentIssue false
  • English
  • Français

Some results for dams with Markovian inputs

Published online by Cambridge University Press:  14 July 2016

R. M. Phatarfod  and
K. V. Mardia
R. M. Phatarfod*
Affiliation:
Monash University
K. V. Mardia
Affiliation:
University of Hull
*
∗ Research partly carried out when this author was holding a Science Research Council Senior Visiting Fellowship at the University of Hull, June-July 1970.
Get access Rights & Permissions [Opens in a new window]

Abstract

The paper considers the dam problem with Markovian inputs, with special reference to the serial correlation coefficient of the input process. An input model is proposed which by giving particular values to the parameters makes the stationary distribution of the inputs one of the standard discrete distributions. The probabilities of first emptiness before overflow are first obtained by using the Markovian analogue of Wald's Identity. From these, the stationary distributions of the dam content are obtained by a duality argument. Both, finite and infinite dams are considered.

Keywords

MARKOVIAN INPUT OF LINEAR REGRESSIVE KINDMARKOVIAN ANALOGUE OF WALD'SIDENTITYPROBABILITIES OF EMPTINESSSTATIONARY DISTRIBUTIONS OF DAM CONTENTDEPENDENCE ON SERIAL CORRELATION COEFFICIENT
Type
Research Papers
Information
Journal of Applied Probability , Volume 10 , Issue 1 , March 1973 , pp. 166 - 180
Copyright
Copyright © Applied Probability Trust 1973 

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] Moran, P. A. P. (1954) A probability theory of dams and storage systems. Aust. J. Appl. Sci. 5, 116–124.Google Scholar
[2] Gani, J. (1957) Problems in the probability theory of storage systems. J. R. Statist. Soc. B19, 181–206.Google Scholar
[3] Prabhu, N. U. (1964) Time-dependent results in storage theory. J. Appl. Prob, 1, 1–46.Google Scholar
[4] Lloyd, E. H. (1963) Reservoirs with correlated inflows. Technometrics 5, 85–93.Google Scholar
[5] Lloyd, E. H. and Odoom, S. (1964) A note on the solution of dam equations. J. R. Statist. Soc. B26, 338–344.Google Scholar
[6] Lloyd, E. H. and Odoom, S. (1965) A note on the equilibrium distribution of levels in a semi-infinite reservoir subject to Markovian inputs and unit withdrawals. J. Appl. Prob. 2, 215–222.Google Scholar
[7] Ali Khan, M. S. and Gani, J. (1968) Infinite dams with inputs forming a Markov chain. J . Appl. Prob. 5, 72–83.CrossRefGoogle Scholar
[8] Ali Khan, M. S. (1970) Finite dams with inputs forming a Markov chain. J. Appl. Prob. 7, 291–303.Google Scholar
[9] Lloyd, E. H. (1963) A probability theory of reservoirs with serially correlated inputs. J. Hydrol. 1, 99–128.Google Scholar
[10] Prabhu, N. U. (1958) Some exact results for the finite dam. Ann. Math. Statist. 29, 1234–1243.Google Scholar
[11] Edwards, C. B. and Gurland, J. (1961) A class of distributions applicable to accidents. J. Amer. Statist. Ass. 56, 503–517.CrossRefGoogle Scholar
[12] Phatarfod, R. M., Speed, T. P. and Walker, A. M. (1971) A note on random walks. J. Appl. Prob. 8, 198–201. 'Google Scholar
[13] Phatarfod, R. M. (1971) Sequential tests for normal Markov sequences. J. Aust. Math . Soc. XII, 433–440.Google Scholar
[14] Phatarfod, R. M. (1971) Some approximate results in renewal and dam theories. J. Aust. Math. Soc. XII, 425–432.Google Scholar
[15] Gani, J. (1969) A note on the first emptiness of dams with Markovian inputs. J. Math. Anal. Appl. 26, 270–274.Google Scholar
[16] Yeo, G. F. (1961) The time-dependent solution for an infinite dam with discrete additive inputs. J. R. Statist. Soc. B23, 173–179.Google Scholar