Hostname: page-component-586b7cd67f-2brh9 Total loading time: 0 Render date: 2024-11-25T16:34:45.520Z Has data issue: false hasContentIssue false
  • English
  • Français

On the limit behavior of a multicompartment storage model with an underlying Markov chain

Published online by Cambridge University Press:  01 July 2016

Eric S. Tollar
Eric S. Tollar*
Affiliation:
The Florida State University
*
∗Postal address: Department of Statistics, The Florida State University, Tallahassee, FL 32306, USA.
Get access Rights & Permissions [Opens in a new window]

Abstract

The present paper considers a multicompartment storage model with one-way flow. The inputs and outputs for each compartment are controlled by a denumerable-state Markov chain. Assuming finite first and second moments, it is shown that the amounts of material in certain compartments converge in distribution while for others they diverge, based on appropriate first-moment conditions on the inputs and outputs. It is also shown that the diverging compartments under suitable normalization converge to functionals of Brownian motion, independent of those compartments which converge without normalization.

Keywords

DUAL MARKOV CHAINAUXILIARY MARKOV CHAIN: MULTIVARIATE BROWNIAN MOTION
Type
Research Article
Information
Advances in Applied Probability , Volume 20 , Issue 1 , March 1988 , pp. 208 - 227
Copyright
Copyright © Applied Probability Trust 1988 

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.)

Footnotes

Research supported by the U.S. Army Research Office under Grant DAAG 29–82-K-0168.

References

Balagopal, K. (1979) Some limit theorems for the general semi-Markov storage model. J. Appl. Prob. 16, 607–617.Google Scholar
Billingsley, P. (1968) Convergence of Probability Measures. Wiley, New York.Google Scholar
Chung, K. L. (1967) Markov Chains with Stationary Transition Probabilities. Springer-Verlag, New York.Google Scholar
ÇInlar, E. (1969) Markov renewal theory. Adv. Appl. Prob. 1, 123–187.CrossRefGoogle Scholar
Donsker, M. (1951) An invariance principle for certain probability limit theorems. Mem. Amer. Math. Soc. 6.Google Scholar
Hoel, P. G., Port, S. C., and Stone, C. J. (1972) Introduction to Stochastic Processes. Houghton-Mifflin, Boston.Google Scholar
Kiefer, J. and Wolfowitz, J. (1956) On the characteristics of the general queueing process with applications to random walk. Ann. Math. Statist. 27, 147–161.CrossRefGoogle Scholar
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
Moran, P. A. P. (1954) A probability theory of dams and storage systems. Aust. J. Appl. Sci. 5, 116–124.Google Scholar
Prabhu, N. U. (1965) Queues and Inventories. Wiley, New York.Google Scholar
Puri, P. S. (1978) A generalization of a formula of Pollaczek and Spitzer as applied to a storage model. Sankhya A 40, 237–252.Google Scholar
Puri, P. S. and Senturia, J. (1972). On a mathematical theory of quantal response assays. Proc. 6th Berkeley Symp. Math. Statist. Prob. 4, 231–247.Google Scholar
Puri, P. S. and Senturia, J. (1975) An infinite depth dam with Poisson input and Poisson release. Scand. Actuarial J., 193–202.Google Scholar
Puri, P. S. and Tollar, E. S. (1985) On the limit behavior of certain quantities in a subcritical storage model. Adv. Appl. Prob. 17, 443–459.Google Scholar
Puri, P. S. and Woolford, S. W. (1981) On a generalized storage model with moment assumptions. J. Appl. Prob. 18, 473–481.Google Scholar