Hostname: page-component-586b7cd67f-tf8b9 Total loading time: 0 Render date: 2024-12-02T20:47:55.987Z Has data issue: false hasContentIssue false
  • English
  • Français

Some approximate results in renewal and dam theories

Published online by Cambridge University Press:  09 April 2009

R. M. Phatarfod
R. M. Phatarfod
Affiliation:
Monash UniversityClayton, Victoria
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

It is well known that Wald's Fundamental Identity (F.I.) in sequential analysis can be used to derive approximate (and, sometimes exact) results in most situations wherein we have essentially a random walk phenomenon. Bartlett [2] used it for the gambler's ruin problem and also for a simple renewal problem. Phatarfod [18] used it for a problem in dam theory. It is the purpose of this paper to show how a generalization of the Fundamental Identity to Markovian variables, (Phatarfod [19]) can be used to derive approximate results in some problems in dam and renewal theories where the random variables involved have Markovian dependence. The reason for considering both the theories together is that the models usually proposed for both the theories — input distribution for dam theory, and lifedistribution for renewal theory — are similar, and only a slight modification (to account for the ‘release rules’ in dam theory, plus the fact that we have two barriers) is necessary to derive results in dam theory from those of renewal theory.

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 12 , Issue 4 , November 1971 , pp. 425 - 432
Copyright
Copyright © Australian Mathematical Society 1971

References

[1]Khan, M. S. Ali and Gani, J., ‘Infinite dams with inputs forming a Markov chain’, J. Appl. Prob. 5 (1968), 72–83.CrossRefGoogle Scholar
[2]Bartlett, M. S., An Introduction to Stochastic Processes (Cambridge Univ. Press., 1955).Google Scholar
[3]Birkhoff, G., ‘Extensions of Jentzsch's theorem’, Trans. Amer. Math. Soc. 85 (1957), 219–227.Google Scholar
[4]Chung, K. L., Markov chains with stationary transition probabilities (Springer-Verlag, Berlin, 1960).CrossRefGoogle Scholar
[5]Cox, D. R., ‘Some statistical methods connected with series of events’, J. R. Statist. Soc. B17 (1955), 129–164.Google Scholar
[6]Cox, D. R. and Lewis, P. A. W., The statistical Analysis of Series of Events (Methuen, London, 1966).CrossRefGoogle Scholar
[7]Gani, J.. ‘Problems in the probability theory of storage systems’, J. R. Statist. Soc. B19 (1957), 181–206.Google Scholar
[8]Harris, T. E., The Theory of Branching Processes (Springer-Verlag, Berlin, 1963).CrossRefGoogle Scholar
[9]Jentzsch, R., ‘Über Integralgleichungen mit positivem Kern’, J. Reine Angew. Math. 141 (1912) 235–244.CrossRefGoogle Scholar
[10]Karlin, S., ‘Positive Operators’, J. Math. Mech. 8 (1957), 907–938.Google Scholar
[11]Krein, M. G. and Rutman, M. A., ‘Linear operators leaving invariant a cone in a Banach space’ (Amer. Math. Soc. Translation No. 26., 1950).Google Scholar
[12]Lampard, D., ‘A stochastic process whose successive intervals between events form a first order Markov chain — I’, J. Appl. Prob. 5 (1968), 648–668.CrossRefGoogle Scholar
[13]Lee, A., ‘A stochastic process whose successive intervals between events form a first order Markov chain’ — II, Report No. MEE 66–2 Electrical Engineering Department, Monash University, (1966).Google Scholar
[14]Lloyd, E. H., ‘Reservoirs with correlated inflows’, Technometrics 5 (1963), 85–93.Google Scholar
[15]Lloyd, E. H. and Odoom, S., ‘A note on the solution of dam equations’, J. R. Statist. Soc. B26 (1964), 338–344.Google Scholar
[16]Lloyd, E. H. and Odoom, S., ‘A note on the equilibrium distribution of levels in a semi-infinite reservoir subject to Markovian inputs and unit withdrawals’, J. Appl. Prob., 2 (1965), 215–222.Google Scholar
[17]Moran, P. A. P., ‘A probability theory of dams and storage systems’, Aust. J. Appl. Sci. 5 (1954), 116–124.Google Scholar
[18]Phatarfod, R. M., ‘Application of methods in sequential analysis to dam theory’, Ann. Math. Statist. 34 (1963), 1588–1592.CrossRefGoogle Scholar
[19]Phatarfod, R. M., ‘Sequential tests for normal Markov Sequence’, (to be published).Google Scholar
[20]Prabhu, N. U., ‘Time-dependent results in storage theory’, J. Appl. Prob. 1 (1964), 1–46.CrossRefGoogle Scholar
[21]Vere-Jones, D., ‘Ergodic properties of non-negative matrices - I’, Pacific. J. Math. 22 (1967), 361–386.CrossRefGoogle Scholar
[22]Wold, H., ‘On stationary point processes and Markov chains’, Skand. Aktuar 31 (1948), 229–240.Google Scholar
[23]Wold, H., ‘Sur les processus stationnaires ponctuels’, Colloques Internationaux du C.N.R.S. 13 (1948), 75–86.Google Scholar