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ALTERNATIVE ANALYSIS OF FINITE-TIME PROBABILITY DISTRIBUTIONS OF RENEWAL THEORY

Published online by Cambridge University Press:  02 January 2014

Percy H. Brill
Percy H. Brill*
Affiliation:
Department of Mathematics and Statistics and Department of Management Science, University of Windsor, Canada. E-mail: brill@uwindsor.ca
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Abstract

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We introduce a level-crossing analysis of the finite time-t probability distributions of the excess life, age, total life, and related quantities of renewal processes. The technique embeds the renewal process as one cycle of a regenerative process with a barrier at level t, whose limiting probability density function leads directly to the time-t quantities. The new method connects the analysis of renewal processes with the analysis of a large class of stochastic models of Operations Research. Examples are given.

Information

Type
Research Article
Information
Probability in the Engineering and Informational Sciences , Volume 28 , Issue 2 , April 2014 , pp. 183 - 201
Copyright
Copyright © Cambridge University Press 2014 

References

1.Feller, W. (1941). On the integral equation of renewal theory. Annals of Mathematical Statistics 12(3): 243–267.Google Scholar
2.Feller, W. (1950). An introduction to probability theory and its applications, Volume I, New York: John WileyGoogle Scholar
3.Feller, W. (1971). An introduction to probability theory and its applications, Volume II, 2nd edn., New York: Wiley.Google Scholar
4.Doob, J.L. (1948). Renewal theory from the point of view of the theory of probability. Transactions of the American Mathematics Society 63(3): 422–438.Google Scholar
5.Smith, W.L. (1958). Renewal theory and its ramifications. Journal of the Royal Statistical Society, Series B (Methodological), 20(2): 243–302.CrossRefGoogle Scholar
6.Cox, D.R. (1970). Renewal theory (reprinted from 1962 edition), London: Methuen.Google Scholar
7.Ross, S.M. (1970). Applied probability with optimization applications, San Francisco: Holden Day.Google Scholar
8.Karlin, S. & Taylor, H.M. (1975). A first course in stochastic processes, 2nd edn., New York: Academic Press.Google Scholar
9.Bartholemew, D.J. (1963). An approximate solution of the integral equation of renewal theory. Journal of the Royal Statistical Society, Series B (Methodological) 25(2): 432–441.Google Scholar
10.Deligonul, S.S. (1985). An approximate solution of the integral equation of renewal theory. Journal of Applied Probability 22: 926–931.CrossRefGoogle Scholar
11.Sigman, K. & Wolff, R.W. (1993). A review of regenerative processes, SIAM Review 35(2): 269–288.Google Scholar
12.Asmussen, S. (2003). Applied probability and queues. New York: Springer–Verlag.Google Scholar
13.Tijms, H.C. (2003). A first course in Stochastic models, New York: WileyGoogle Scholar
14.Brill, P.H. (2009). Compound cycle of a renewal process and applications. Information Systems and Operational Research 47(4): 273–281.Google Scholar
15.Brill, P.H. & Yu, K. (2011). Analysis of risk models using a level crossing technique. Insurance: Mathematics and Economics 49(3): 298–309.Google Scholar
16.Brill, P.H. & Posner, M.J.M. (1977). Level crossings in point processes applied to queues: single-service case. Operations Research 25(4): 662–674.Google Scholar
17.Brill, P.H. & Posner, M.J.M. (1981). The system point method in exponential queues: a level crossing approach. Mathematics of Operations Research 6: 31–49.Google Scholar
18.Cohen, J.W. (1976). On Regenerative Processes in Queueing Theory. Lecture Notes in Economics and Mathematical Systems. (Beckman, M. and Kunzi, H.P. Eds.), New York: Spring-Verlag.Google Scholar
19.Cohen, J.W. (1977). On up- and down-crossings. Journal of Applied Probability 14: 405–410.Google Scholar
20.Brill, P.H. (2000). A brief outline of the level crossing method in Stochastic models, CORS Bulletin Canadian Operational Research Society 34(4): 9–21. (http://www.cors.ca/bulletin/percy.pdf).Google Scholar
21.Brill, P.H. (2008). Level crossing methods in Stochastic models. New York: Springer.Google Scholar
22.Ross, S.M. (2006). Introduction to probability models, 9th edn., San Diego: Academic Press.Google Scholar
23.Pinsky, M. & Karlin, S. (2011). Introduction to Stochastic modelling, 4th edn., San Diego: Academic Press.Google Scholar
24.Wolff, R.W. (1982). Poisson arrivals see time averages, Operations Research 30(2): 223–231.Google Scholar
25.Gerber, H.U. & Shiu, E.S.W. (1997). The joint distribution of the time of ruin, the surplus immediately before ruin, and the deficit at ruin. Insurance: Mathematics and Economics 21: 129–137.Google Scholar
26.Botta, R.F., Marchal, W.D. & Harris, C.M. (1987). Characterizations of generalized hyperexponential distribution functions. Stochastic Models 3: 115–148.Google Scholar
27.Botta, R.F. & Harris, C.M. (1986). Approximation with generalized hyperexponential distributions. Weak convergence results. Queueing Systems 2: 169–190.Google Scholar
28.Harris, C.M., Marchal, W.G. & Botta, R.F. (1992). A note on generalized hyperexponential distributions. Stochastic Models 8: 179–191.Google Scholar
29.Yu, K., Huang, M.L. & Brill, P.H. (2012). An algorithm for fitting heavy-tailed distributions via generalized hyperexponentials. INFORMS J. on Computing 24: 42–52.Google Scholar