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Doubly stochastic Poisson processes and process control

Published online by Cambridge University Press:  01 July 2016

Mats Rudemo
Mats Rudemo*
Affiliation:
Research Institute of National Defence, Stockholm
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Abstract

Consider a Poisson point process with an intensity parameter forming a Markov chain with continuous time and finite state space. A system of ordinary differential equations is derived for the conditional distribution of the Markov chain given observations of the point process. An estimate of the current intensity, optimal in the least-squares sense, is computed from this distribution. Applications to reliability and replacement theory are given. A special case with two states, corresponding to a process in control and out of control, is discussed at length. Adjustment rules, based on the conditional probability of the out of control state, are studied. Regarded as a function of time, this probability forms a Markov process with the unit interval as state space. For the distribution of this process, integro-differential equations are derived. They are used to compute the average long run cost of adjustment rules.

Keywords

POINT PROCESSSTATE ESTIMATIONFORWARD RECURRENCE TIMEBAYESIAN RELIABILITY PREDICTIONDETECTION OF CHANGE POINTREPLACEMENT
Type
Research Article
Information
Advances in Applied Probability , Volume 4 , Issue 2 , August 1972 , pp. 318 - 338
Copyright
Copyright © Applied Probability Trust 1972 

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References

Åström, K. J. (1965) Optimal control of Markov processes with incomplete state information. J. Math. Anal. Appl. 10, 174–205.CrossRefGoogle Scholar
Barlow, R. E. and Proschan, F. (1965) Mathematical Theory of Reliability. John Wiley, New York.Google Scholar
Breakwell, J. and Chernoff, H. (1964) Sequential tests for the mean of a normal distribution II (large t). Ann. Math. Statist. 35, 162–173.Google Scholar
Cox, D. R. (1955) Some statistical methods connected with series events. J. R. Statist. Soc. B 17, 129–164.Google Scholar
Cox, D. R. and Lewis, P. A. W. (1966) The Statistical Analysis of Series of Events. Methuen, London.Google Scholar
Elfving, G. (1962) Quality control for expensive items. Technical Report No. 57. Applied Mathematics and Statistics Laboratory, Stanford Univ. Google Scholar
Feller, W. (1966) An Introduction to Probability Theory and its Applications. Volume II. John Wiley, New York.Google Scholar
Galchuk, L. I. and Rozovskii, B. L. (1971) The “disorder” problem for a Poisson process. Teor. Verojatnost. i Primenen. 16, 729–734.Google Scholar
Gaver, D. P. (1963) Random hazard in reliability problems. Technometrics 5, 211–226.Google Scholar
Girshick, M. A. and Rubin, H. (1952) A Bayes approach to a quality control model. Ann. Math. Statist. 23, 114–125.Google Scholar
Grandell, J. (1971) On stochastic processes generated by a stochastic intensity function. Skand. Aktuarietidsk. 54, No. 3–4.Google Scholar
Grandell, J. (1972) On the estimation of intensities in a stochastic process generated by a stochastic intensity sequence. J. Appl. Prob. 9, 542–556.Google Scholar
Iglehart, D. L. and Taylor, H. M. (1968) Weak convergence of a sequence of quickest detection problems. Ann. Math. Statist. 39, 2149–2153.Google Scholar
Jazwinski, A. H. (1970) Stochastic Processes and Filtering Theory. Academic Press, New York.Google Scholar
Karlin, S. (1966) A First Course in Stochastic Processes. Academic Press, New York.Google Scholar
Lieberman, G. J. (1965) Statistical process control and the impact of automatic process control. Technometrics 7, 283–292.CrossRefGoogle Scholar
Mitchell, A. R. (1969) Computational Methods in Partial Differential Equations. John Wiley, London.Google Scholar
Neuts, M. F. (1971) A queue subject to extraneous phase changes. Adv. Appl. Prob. 3, 78–119.Google Scholar
Pollock, S. M. (1968) A Bayesian reliability growth model. IEEE Trans. Reliability Theory R-17, 187–198.CrossRefGoogle Scholar
Richtmyer, R. D. and Morton, K. W. (1967) Difference Methods for Initial-value Problems. 2nd ed. Interscience, New York.Google Scholar
Ross, S. M. (1970) Applied Probability Models with Optimization Applications. Holden-Day, San Francisco.Google Scholar
Rudemo, M. (1972a) State estimation for partially observed Markov chains. To appear.CrossRefGoogle Scholar
Rudemo, M. (1972b) Point processes generated by transitions of Markov chains. To appear.Google Scholar
Savage, I. R. (1963) Surveillance problems: Poisson models with noise. Technical Report No. 20, Dept. of Statistics, Univ. of Minnesota.Google Scholar
Shiryaev, A. N. (1963) On optimum methods in quickest detection problems. Theor. Probability Appl. 8, 22–46.CrossRefGoogle Scholar
Shiryaev, A. N. (1967) Some new results in the theory of controlled random processes. Trans. Fourth Prague Conf. on Information Theory, Statistical Decision Functions, Random processes, Prague, 1965. Academia, Prague, 1967, 113–203. Also in Selected Translations in Mathematical Statistics and Probability. Amer. Math. Soc, Providence, R. I., 8, 49–130.Google Scholar
Snyder, D. L. (1972a) Filtering and detection for doubly-stochastic Poisson processes. IEEE Trans. Information Theory IT-18, 91–102.Google Scholar
Snyder, D. L. (1972b) Information processing for observed jump processes. Submitted to Information and Control. Google Scholar
Stratonovich, R. L. (1963) Topics in the Theory of Random Noise. Volume I. Gordon and Breach, New York.Google Scholar
Taylor, H. M. (1967) Statistical control of a Gaussian process. Technometrics 9, 29–41.Google Scholar
Wonham, W. M. (1965) Some applications of stochastic differential equations to optimal nonlinear filtering. SIAM J. Control 2, 347–369.Google Scholar
Yashin, A. I. (1970) Filtering of jump processes. Automat. Remote Control 31, 725–730.Google Scholar