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A Note on the Dependence Structure of the Two-State Markovian Arrival Process

Part of: Markov processes Stochastic processes

Published online by Cambridge University Press:  04 February 2016

Pepa Ramírez-Cobo  and
Emilio Carrizosa
Pepa Ramírez-Cobo*
Affiliation:
University of Seville
Emilio Carrizosa*
Affiliation:
University of Seville
*
∗ Postal address: Institute of Mathematics, University of Seville, Avda. Reina Mercedes s/n, 41012 Seville, Spain. Email address: jrcobo@us.es
∗∗ Postal address: Department of Statistics and Operations Research, University of Seville, Avda. Reina Mercedes s/n, 41012 Seville, Spain. Email address: ecarrizosa@us.es
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Abstract

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The Markovian arrival process generalizes the Poisson process by allowing for dependent and nonexponential interarrival times. We study the autocorrelation function of the two-state Markovian arrival process. Our findings show that the correlation structure of such a process has a very specific pattern, namely, it always converges geometrically to zero. Moreover, the signs of the autocorrelation coefficients are either constant or alternating.

Keywords

Markovian arrival processMarkov renewal processautocorrelation function

MSC classification

Primary: 60G55: Point processes
Secondary: 60J05: Discrete-time Markov processes on general state spaces
Type
Research Article
Information
Journal of Applied Probability , Volume 49 , Issue 1 , March 2012 , pp. 295 - 302
Copyright
© Applied Probability Trust 

References

Chakravarthy, S. R. (2001). The batch Markovian arrival process: a review and future work. In Advances in Probability and Stochastic Processes, eds Krishnamoorthy, A., Raju, N. and Ramaswami, V., Notable Publications, NJ, pp. 21–49.Google Scholar
Kang, S. H., Han Kim, Y., Sung, D. K. and Choi, B. D. (2002). An application of Markovian arrival process to modeling superposed ATM cell streams. IEEE Trans. Commun. 50, 633–642.CrossRefGoogle Scholar
Lucantoni, D. M. (1993). The BMAP/G/1 queue: a tutorial. In Performance Evaluation of Computer and Communication Systems (Lecture Notes Comput. Sci. 729), eds Donatiello, L. and Nelson, R. D., Springer, New York, pp. 330–358.CrossRefGoogle Scholar
Montoro-Cazorla, D., Pérez-Ocón, R. and Segovia, M. (2009). Replacement policy in a system under shocks following a Markovian arrival process. Reliab. Eng. System Safety 94, 497–502.Google Scholar
Neuts, M. F. (1979). A versatile Markovian point process. J. Appl. Prob. 16, 764–779.Google Scholar
Ramaswami, V. (1980). The N/G/1 queue and its detailed analysis. Adv. Appl. Prob. 12, 222–261.Google Scholar
Ramírez-Cobo, P., Lillo, R. and Wiper, M. (2010). Bayesian inference for the two-state Markovian arrival process. Working paper 10-43, Statistics and Econometrics Series 26, Universidad Carlos III de Madrid.Google Scholar
Ramírez-Cobo, P., Lillo, R. E. and Wiper, M. P. (2010). Nonidentifiability of the two-state Markovian arrival process. J. Appl. Prob. 47, 630–649.Google Scholar
Rydén, T. (1996). An EM algorithm for estimation in Markov-modulated Poisson processes. Comput. Statist. Data Anal. 21, 431–447.Google Scholar
Scott, S. L. (1999). Bayesian analysis of the two state Markov modulated Poisson process. J. Comput. Graphical Statist. 8, 662–670.Google Scholar