Hostname: page-component-cb9f654ff-9b74x Total loading time: 0 Render date: 2025-08-19T18:21:51.060Z Has data issue: false hasContentIssue false
  • English
  • Français

Level Crossing Ordering of Skip-Free-to-the-Right Continuous-Time Markov Chains

Part of: Distribution theory - Probability Markov processes Special processes

Published online by Cambridge University Press:  14 July 2016

Fátima Ferreira  and
António Pacheco
Fátima Ferreira*
Affiliation:
CEMAT and Universidade de Trás os Montes e Alto Douro
António Pacheco*
Affiliation:
CEMAT and Instituto Superior Técnico, Universidade Técnica de Lisboa
*
∗ Postal address: Departamento de Matemática, Universidade de Trás os Montes e Alto Douro, Quinta dos Prados, Apartado 1013, 5001-911 Vila Real, Portugal. Email address: mmferrei@utad.pt
∗∗ Postal address: Departamento de Matemática, Instituto Superior Técnico, Universidade Técnica de Lisboa, Av. Rovisco Pais, 1049-001 Lisboa, Portugal. Email address: apacheco@math.ist.utl.pt
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

As proposed by Irle and Gani in 2001, a process X is said to be slower in level crossing than a process Y if it takes X stochastically longer to exceed any given level than it does Y. In this paper, we extend a result of Irle (2003), relative to the level crossing ordering of uniformizable skip-free-to-the-right continuous-time Markov chains, to derive a new set of sufficient conditions for the level crossing ordering of these processes. We apply our findings to birth-death processes with and without catastrophes, and M/M/s/c systems.

Keywords

Birth-death process with catastrophecontinuous-time Markov chainlevel crossing orderingM/M/s/c systemstochastic ordering

MSC classification

Secondary: 60E15: Inequalities; stochastic orderings 60J27: Continuous-time Markov processes on discrete state spaces 60J80: Branching processes (Galton-Watson, birth-and-death, etc.) 60K25: Queueing theory

Information

Type
Research Papers
Information
Journal of Applied Probability , Volume 42 , Issue 1 , March 2005 , pp. 52 - 60
Copyright
© Applied Probability Trust 2005 

References

Brockwell, P. J., Gani, J. and Resnick, S. I. (1982). Birth, immigration and catastrophe processes. Adv. Appl. Prob. 14, 709–731.CrossRefGoogle Scholar
Chao, X. and Zheng, Y. (2003). Transient analysis of immigration birth–death processes with total catastrophes. Prob. Eng. Inf. Sci. 17, 83–106.CrossRefGoogle Scholar
Economou, A. and Fakinos, D. (2003). A continuous-time Markov chain under the influence of a regulating point process and applications in stochastic models with catastrophes. Europ. J. Operat. Res. 149, 625–640.CrossRefGoogle Scholar
Gross, D. and Harris, C. (1998). Fundamentals of Queueing Theory, 3rd edn. John Wiley, New York.Google Scholar
Irle, A. (2003). Stochastic ordering for continuous-time processes. J. Appl. Prob. 40, 361–375.CrossRefGoogle Scholar
Irle, A. and Gani, J. (2001). Stochastic ordering and the detection of words. In Probability, Statistics and Seismology (J. Appl. Prob. Spec. Vol. 38A), ed. Daley, D. J., Applied Probability Trust, Sheffield, pp. 66–77.Google Scholar
Kirstein, B. M. (1976). Monotonicity and comparability of time-homogeneous Markov processes with discrete state space. Math. Operationsforsch. Statist. 7, 151–168.CrossRefGoogle Scholar
Kulkarni, V. (1995). Modeling and Analysis of Stochastic Systems. Chapman and Hall, London.Google Scholar
Müller, A. and Stoyan, D. (2002). Comparison Methods for Stochastic Models and Risks. John Wiley, Chichester.Google Scholar
Resnick, S. (1992). Adventures in Stochastic Processes. Birkhäuser, Boston, MA.Google Scholar
Shaked, M. and Shanthikumar, J. (1994). Stochastic Orders and Their Applications. Academic Press, San Diego, CA.Google Scholar
Singh, H. and Jain, K. (1989). Preservation of some partial orderings under Poisson shock models. Adv. Appl. Prob. 21, 713–716.CrossRefGoogle Scholar