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Compensator conditions for stochastic ordering of point processes

Published online by Cambridge University Press:  14 July 2016

A. Kwieciński  and
R. Szekli
A. Kwieciński
Affiliation:
University of Wroctaw
R. Szekli*
Affiliation:
University of Wroctaw
*
∗ Postal address for both authors: Mathematical Institute, University of Wroclaw, pl. Grunwaldzki 2/4, 50–384 Wroclaw, Poland.
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Abstract

Sufficient conditions are given under which two simple point processes on the positive half-line can be stochastically compared as random elements of D(0,∞) or R∞+ Using a martingale approach to point processes, the conditions are proposed via a compensator function family. Appropriate versions of the processes being compared are constructed on the same probability space. The results are illustrated by replacement policies and semi-Markov point processes.

Keywords

COMPENSATORREPLACEMENT POLICYSEMI-MARKOV POINT PROCESS
Type
Research Papers
Information
Journal of Applied Probability , Volume 28 , Issue 4 , December 1991 , pp. 751 - 761
Copyright
Copyright © Applied Probability Trust 1991 

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References

Arjas, E. and Lehtonen, T. (1978) Approximating many server queues by means of single server queues. Math. Operat. Res. 3, 205–223.Google Scholar
Barlow, R. and Proschan, F. (1964) Comparison of replacement policies, and renewal theory implications. Ann. Math. Statist. 35, 577–589.Google Scholar
Bremaud, P. (1981) Point Processes and Queues: Martingale Dynamics. Springer-Verlag, New York.Google Scholar
Brown, T. C. and Nair, M. G. (1988) A simple proof of the multivariate random time change theorem for point processes. J. Appl. Prob. 25, 210–214.Google Scholar
Haezendonc, J. and De Vylder, F. (1980) A comparison criterion for explosions in point processes. J. Appl. Prob. 17, 1102–1107.Google Scholar
Jacod, J. (1975) Multivariate point processes: predictable projection, Radon-Nikodym derivatives, representation of martingales. Z. Wahrscheinlichkeitsth. 31, 235–253.Google Scholar
Kamae, T., Krengel, U. and O'Brien, G. L. (1977) Stochastic inequalities on partially ordered spaces. Ann. Prob. 5, 899–912.Google Scholar
Lindvall, T. (1973) Weak convergence of probability measures and random functions in the function space D[0, 8). J. Appl. Prob. 10, 109–121.Google Scholar
Norros, I. (1986) A compensator representation of multivariate life length distributions, with applications. Scand. J. Statist. 13, 99–112.Google Scholar
Rolski, T. and Szekli, R. (1990) Stochastic ordering and thinning of point processes. Stoch. Proc. Appl.Google Scholar
Shaked, M. and Shanthikumar, J. G. (1987a) Multivariate hazard rates and stochastic ordering. Adv. Appl. Prob. 19, 123–137.Google Scholar
Shaked, M. and Shanthirumar, J. G. (1987b) The multivariate hazard construction. Stoch. Proc. Appl. 24, 241–258.Google Scholar
Shanthirumar, J. G. (1987) On stochastic comparison of random vectors. J. Appl. Prob. 24, 123–136.Google Scholar
Stoyan, D. (1983) Comparison Methods for Queues and Other Stochastic Models. Wiley, Chichester.Google Scholar
Strassen, V. (1965) The existence of probability measures with given marginals. Ann. Math. Statist. 36, 423–439.Google Scholar
Whitt, W. (1981) Comparing counting processes and queues. Adv. Appl. Prob. 13, 207–220.Google Scholar