Abstract
One of the most spectacular achievements of Julian Keilson is his development of the perturbation method based on the compensation kernel. The method is mathematically intriguing and also very convenient for practical applications.
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References
Asmussen, S. Applied Probability and Queues. Wiley, New York, 1987.
Bertsimas, D. J., Keilson, J., et al. Transient and busy period analysis of the GI/G/1 queue as a Hilbert factorization problem. J. Appl. Prob. 28, 873–885, 1991.
Blumenthal, R. M., and Getoor, R. K. Markov Processes and Potential Theory. Academic Press, New York, 1968.
Cai, H. On Reviving Markov Processes and Applications. Ph.D. thesis, Department of Mathematics, University of Maryland, 1987.
Feigin, P. D., and Rubinstein, E. Equivalent description of perturbed Markov processes. Stoch. Proc. Appl. 9, 261–272, 1979.
Gerontidis, I.I. Semi-Markov replacement chains. Adv. Appl. Prob. 26, 728–755, 1994.
Hille, E., and Phillips, R. S. Functional Analysis and Semi-Groups. Colloquium, American Mathematics Society, 1957.
Keilson, J. The homogeneous random walk on the half-line and the Hilbert problem. Bull. I. S. I. 113, 33rd Session, paper 113, 1–13, 1961.
Keilson, J. The use of Green’s Functions in the study of the bounded random walks with application to queueing theory. J. Math. Phys. 41, 42–52, 1962.
Keilson, J. An alternative to Wiener—Hopf methods for the study of bounded processes. J. Appl. Prob. 1, 85–120, 1964.
Keilson, J. Green’s Function Methods in Probability Theory. Griffin, 1965.
Keilson, J. The role of Green’s Functions in congestion theory. In: Smith, W. L., and Wilkinson, W. E. (eds), Congestion Theory. University of North Carolina Press, Chapel Hill, NC, 1965, pp. 43–71.
Keilson, J. Markov chain models — rarity and exponentiality. Appl. Math. Sci. 28, 1979.
Keilson, J., and Graves, S. C. The compensation method applied to a one-product production/inventory problem. Math. Oper. Res. 6, 246–262, 1981.
Keilson, J., and Syski, R. Compensation measures in the theory of Markov chains. Stoch. Proc. Appl. 2, 59–72, 1974.
Liu, N. Decomposition Theorems for Standard Processes. Ph.D. thesis, Department of Mathematics, University of Maryland, 1995.
Loeve, M. Probability Theory. Van Nostrand, 1963.
Neuts, M. F. Probability distributions of phase type. In: Liber Amicorum Professor Emeritus H. Florin, Department of Mathematics, University of Louvain, Belgium, 1975, pp. 173–206.
Svoronou, A. Multivariate Markov Processes via the Green’s Function Method. Ph.D. thesis, Department of Statistics, University of Rochester, New York, 1990.
Syski, R. Perturbation models. Stochast. Proc. Appl. 5, 93–129, 1977.
Syski, R. Ergodic potential. Stochast. Proc. Appl. 7, 311–336, 1978.
Syski, R. Phase-type distributions and perturbation model. Appl. Math. Warsaw 17(3), 377–399, 1982.
Syski, R. A note on revival models for Markov chains. Am. J. Math. Mgt. Sci. 10, 159–183, 1990.
Syski, R. Passage Times for Markov Chains. IOS Press, Amsterdam, 1992.
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Syski, R., Liu, N. (1999). Comments on the Perturbation Method. In: Shanthikumar, J.G., Sumita, U. (eds) Applied Probability and Stochastic Processes. International Series in Operations Research & Management Science, vol 19. Springer, Boston, MA. https://doi.org/10.1007/978-1-4615-5191-1_1
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DOI: https://doi.org/10.1007/978-1-4615-5191-1_1
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