Summary
This paper studies the transient behavior of the maximum level length for general block structured continuous-time Markov chains (CTMC). The approach is presented for the bidimensional case, however, it still holds for multi-dimensional chains. The results can also be easily modified to cover the discrete-time case. This work complements the busy period analysis by Neuts [12] and the asymptotic approach by Serfozo [14]. Some illustrative examples (SIR epidemic model, retrial queue) including numerical implementations are presented.
The author would like to dedicate this paper to the memory of Professor Marisa Menéndez.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Allen, L.J.S.: An Introduction to Stochastic Processes with Applications to Biology. Pretince Hall, New Jersey (2003)
Artalejo, J.R., Chakravarthy, S.R.: Algorithmic analysis of the maximal level length in general-block two-dimensional Markov processes. Math. Probl. Eng., Article ID 53570, 15 (2007)
Artalejo, J.R., Economou, E., López-Herrero, M.J.: Evaluating growth measures in an immigration process subject to binomial and geometric catastrophes. Math. Biosci. Eng. 4, 573–594 (2007)
Artalejo, J.R., Economou, A., López-Herrero, M.J.: The maximum number of infected individuals in SIS epidemic models: Computational techniques and quasi-stationary distributions. J. Comput. Appl. Math. 223, 2563–2574 (2010)
Artalejo, J.R., Gómez-Corral, A.: Retrial Queueing Systems: A Computational Approach. Springer, Berlin (2008)
Artalejo, J.R., López-Herrero, M.J.: Quasi-stationary and ratio of expectations distributions: A comparative study. J. Theor. Biol. 266, 264–274 (2010)
Cohen, A.M.: Numerical Methods for Laplace Transform Inversion. Springer, New York (2007)
Coolen-Schrijner, P., van Doorn, E.A.: Quasi-stationary distributions for a class of discrete-time Markov chains. Methodol. Comput. Appl. Probab. 8, 449–465 (2006)
Kulkarni, V.G.: Modeling and Analysis of Stochastic Systems. Chapman & Hall, London (1995)
Li, Q.L.: Constructive Theory in Stochastic Models with Applications: The RG-Factorizations. Springer, Tsinghua University Press, Berlin, Beijing (2009)
Moler, C., van Loan, C.: Nineteen dubious ways to compute the exponential of a matrix, twenty-five years later. SIAM Rev. 45, 3–49 (2003)
Neuts, M.F.: The distribution of the maximum length of a Poisson queue during a busy period. Oper. Res. 12, 281–285 (1964)
Neuts, M.F., Li, J.M.: An algorithmic study of S-I-R stochastic epidemic models. In: Heyde, C.C., Prohorov, Y.V., Pyke, R., Rachev, S.T. (eds.) Athens Conference on Applied Probability and Time Series, vol. 1, pp. 295–306. Springer, Heidelberg (1996)
Serfozo, R.F.: Extreme values of birth and death processes and queues. Stoch. Process. Appl. 27, 291–306 (1998)
van Doorn, E.A., Pollet, P.K.: Survival in a quasi-death process. Linear Algebra Appl. 429, 776–791 (2008)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2011 Springer-Verlag Berlin Heidelberg
About this chapter
Cite this chapter
Artalejo, J.R. (2011). On the Transient Behavior of the Maximum Level Length in Structured Markov Chains. In: Pardo, L., Balakrishnan, N., Gil, M.Á. (eds) Modern Mathematical Tools and Techniques in Capturing Complexity. Understanding Complex Systems. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-20853-9_26
Download citation
DOI: https://doi.org/10.1007/978-3-642-20853-9_26
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-20852-2
Online ISBN: 978-3-642-20853-9
eBook Packages: EngineeringEngineering (R0)