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Local conditions for the stochastic comparison of particle systems

Part of: Special processes Markov processes

Published online by Cambridge University Press:  01 July 2016

Rosario Delgado ,
F. Javier López  and
Gerardo Sanz
Rosario Delgado*
Affiliation:
Universitat Autònoma de Barcelona
F. Javier López*
Affiliation:
Universitat Autònoma de Barcelona
Gerardo Sanz*
Affiliation:
Universidad de Zaragoza
*
∗ Postal address: Departamento de Matemáticas, Universitat Autònoma de Barcelona, Edifici C- Campus de la UAB, 08193 Bellaterra (Cerdanyola del Vallès) Barcelona, Spain. Email address: delgado@mat.uab.es
∗∗ Postal address: Departmento Métodos Estadísticos, Facultad de Ciencias, Universidad de Zaragoza, 50009 Zaragoza, Spain.
∗∗ Postal address: Departmento Métodos Estadísticos, Facultad de Ciencias, Universidad de Zaragoza, 50009 Zaragoza, Spain.
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Abstract

We study the stochastic comparison of interacting particle systems where the state space of each particle is a finite set endowed with a partial order, and several particles may change their value at a time. For these processes we give local conditions, on the rates of change, that assure their comparability. We also analyze the case where one of the processes does not have any changes that involve several particles, and obtain necessary and sufficient conditions for their comparability. The proofs are based on the explicit construction of an order-preserving Markovian coupling. We show the applicability of our results by studying the stochastic comparison of infinite-station Jackson networks and batch-arrival, batch-service, and assemble-transfer queueing networks.

Keywords

Stochastic comparisoninteracting particle systemcouplingqueueing network

MSC classification

Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory
Secondary: 60J25: Continuous-time Markov processes on general state spaces
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 36 , Issue 4 , December 2004 , pp. 1252 - 1277
Copyright
Copyright © Applied Probability Trust 2004 

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References

[1] Ahuja, R. K., Magnanti, T. L. and Orlin, J. B. (1993). Network Flows. Prentice-Hall, Englewood Cliffs, NJ.Google Scholar
[2] Chen, M. F. (1992). From Markov Chains to Nonequilibrium Particle Systems. World Scientific, Singapore.Google Scholar
[3] Chen, M. F. and Wang, F. Y. (1993). On order-preservation and positive correlations for multidimensional diffusion processes. Prob. Theory Relat. Fields 95, 421–428.Google Scholar
[4] Economou, A. (2003). Necessary and sufficient conditions for the stochasic comparison of Jackson networks. Prob. Eng. Inf. Sci. 17, 143–151.CrossRefGoogle Scholar
[5] Economou, A. (2003). On the stochastic domination for batch-arrival, batch-service and assemble-transfer queueing networks. J. Appl. Prob. 40, 1103–1120.CrossRefGoogle Scholar
[6] Forbes, F. and François, O. (1997). Stochastic comparison for Markov processes on a product of partially ordered sets. Statist. Prob. Lett. 33, 309–320.Google Scholar
[7] Ford, L. R. and Fulkerson, D. R. (1974). Flows in Networks. Princeton University Press.Google Scholar
[8] Kamae, T., Krengel, G. and O'Brien, G. L. (1977). Stochastic inequalities on partially ordered spaces. Ann. Prob. 5, 899–912.Google Scholar
[9] Liggett, T. M. (1985). Interacting Particle Systems. Springer, New York.CrossRefGoogle Scholar
[10] Liggett, T. M. (1999). Stochastic Interacting Systems: Contact, Voter and Exclusion Processes. Springer, Berlin.Google Scholar
[11] López, F. J. and Sanz, G. (1998). Stochastic comparisons and couplings for interacting particle systems. Statist. Prob. Lett. 40, 93–102.Google Scholar
[12] López, F. J. and Sanz, G. (2002). Markovian couplings staying in arbitrary subsets of the state space. J. Appl. Prob. 39, 197–212.CrossRefGoogle Scholar
[13] López, F. J., Martínez, S. and Sanz, G. (2000). Stochastic domination and Markovian couplings. Adv. Appl. Prob. 32, 1064–1076.Google Scholar
[14] Massey, W. A. (1987). Stochastic orderings for Markov processes on partially ordered spaces. Math. Operat. Res. 12, 350–367.CrossRefGoogle Scholar
[15] Müller, A. and Stoyan, D. (2002). Comparison Methods for Stochastic Models and Risks. John Wiley, Chichester.Google Scholar
[16] Zhang, S. (2000). Existence and application of optimal Markovian coupling with respect to non-negative lower semi-continuous functions. Acta Math. Sin. 16, 261–270.Google Scholar