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On the limit behavior of certain quantities in a subcritical storage model

Published online by Cambridge University Press:  01 July 2016

Prem S. Puri and
Eric S. Tollar
Prem S. Puri*
Affiliation:
Purdue University
Eric S. Tollar*
Affiliation:
The Florida State University
*
∗Postal address: Department of Statistics, Purdue University, West Lafayette, IN 47907, USA.
†Postal address: Department of Statistics, The Florida State University, Tallahassee, FL 32306, USA.
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Abstract

The limit behavior of the content of a subcriticai storage model defined on a semi-Markov process is examined. This is achieved by creating a renewal equation using a regeneration point (i 0,0) of the process. By showing that the expected return time to (i 0, 0) is finite, the conditions needed for the basic renewal theorem are established. The joint asymptotic distribution of the content of the storage at time t and the accumulated amount of the unmet (lost) demands during (0,t) is then established by showing the asymptotic independence of these two.

Keywords

TOTAL DEMAND LOSTMARKOV CHAINSSEMI-MARKOV PROCESSESΦ-IRREDUCIBILITYERGODICITYRENEWAL THEORYSTORAGE LEVEL

Information

Type
Research Article
Information
Advances in Applied Probability , Volume 17 , Issue 2 , June 1985, pp. 443 - 459
Copyright
Copyright © Applied Probability Trust 1985 

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Footnotes

Research carried out while the second author was a David Ross Fellow of Purdue University.

These investigations were supported in part by U.S. National Science Foundation Grant No. MCS-8102733.

References

1. Ali Khan, ?. S. and Gani, J. (1968) Infinite dams with inputs forming a Markov chain. J. Appl. Prob. 5, 72–83.Google Scholar
2. Athreya, K. B., Mcdonald, D. and Ney, P. (1978) Limit theorems for semi-Markov processes and renewal theory for Markov chains. Ann. Prob. 6, 788–797.CrossRefGoogle Scholar
3. Balagopal, K. (1979) Some limit theorems for the general semi-Markov storage model. J. Appl. Prob. 16, 607–617.Google Scholar
4. Chung, K. L. (1967) Markov Chains with Stationary Transition Probabilities. Springer-Verlag, New York.Google Scholar
5. Çinlar, E. (1969) Markov renewal theory. Adv. Appl. Prob. 1, 123–187.Google Scholar
6. Çinlar, E. (1969) On semi-Markov processes on arbitrary spaces. Proc. Camb Phil. Soc. 66, 381–392.Google Scholar
7. Hoel, P. C., Port, S. C. and Stone, C. J. (1972) Introduction to Stochastic Processes. Houghton-Mifflin, Boston.Google Scholar
8. Jain, N. and Jamison, B. (1967) Contributions to Doeblin’s theory of Markov processes. Z. Wahrschlichkeitsth. 8, 19–40.Google Scholar
9. Karlin, S. and Taylor, H. M. (1975) A First Course in Stochastic Processes. Academic Press, New York.Google Scholar
10. Lloyd, E. H. (1963) Reservoirs with serially correlated inflows. Technometrics 4, 85–93.Google Scholar
11. Moran, P. A. P. (1954) A probability theory of dams and storage systems. Austral. J. Appl. Sci. 5, 116–124.Google Scholar
12. Moran, P. A. P. (1959) The Theory of Storage. Wiley, New York.Google Scholar
13. Nummelin, E. (1978) A splitting technique for Harris recurrent Markov chains. Z. Wahrschlichkeitsth. 43, 309–318.Google Scholar
14. Orey, S. (1971) Limit Theorems for Markov Chain Transition Probabilities. Van Nostrand Reinhold, London.Google Scholar
15. Prabhu, N. U. (1965) Queues and Inventories. Wiley, New York.Google Scholar
16. 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
17. Puri, P. S. and Woolford, S. W. (1981) On a generalized storage model with moment assumptions. J. Appl. Prob. 18, 473–481.Google Scholar
18. Royden, H. L. (1968) Real Analysis. MacMillan, New York.Google Scholar
19. Senturia, J. and Puri, P. S. (1973) A semi-Markov storage model. Adv. Appl. Prob. 5, 362–378.Google Scholar
20. Senturia, J. and Puri, P. S. (1974) Further aspects of a semi-Markov storage model. Sankhya A 36, 369–378.Google Scholar
21. Tweedie, R. L. (1975) Sufficient conditions for ergodicity and recurrence of Markov chains on a general state space. Stoch. Proc. Appl. 3, 385–403.Google Scholar
22. Woolford, S. W. (1979) On a Generalized Storage Model. Ph.D. Thesis, Purdue University, W. Lafayette, IN.Google Scholar