Abstract
A correlated flow of arrivals is considered in a case, where interarrival times {X n } correspond to the Markov process with the continuous state space R + = (0, ∞ ). The conditional probability density function of X n + 1 given {X n = z} is determined by means of
where {p 1(z),...,p k (z)} is a probability distribution, p 1(z) + ... + p k (z) = 1 for all z ∈ R + ; {h 1(x),...,h k (x)} is a family of probability density functions on R + .
This flow is investigated with respect to stationary case. One is considered as the Semi-Markov process J(t) on the state set {1, ..., k}. Main characteristics are considered: stationary distribution of J and interarrival times X, correlation and Kendall tau (τ) for adjacent intervals, and so on. Further one is considered a Markovian system on which the described flow arrives. Numerical results show that the dependence between interarrival times of the flow exercises greatly influences the efficiency characteristics of considered systems.
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References
Abegaz, F., Naik-Nimbalkar, U.V.: Modelling statistical dependence of Markov chains via copula models. J. of Statistical Planning and Inference 138(4), 1131–1146 (2008)
Breymann, W., Dias, A., Embrechts, P.: Dependence structures for multivariate high-frequency data in finance. J. Quantitative Finance 3(1), 1–16 (2003)
Embrechts, P., Hoeing, A., Juri, A.: Using Copulae to bind the Value-at-Risk for function of dependent risk. J. Finance & Stochasti 7(2), 145–167 (2003)
Embrechts, P., Pucetti, G.: Bounds for function of dependent risk. J. Finance & Stochastic 10, 341–352 (2006)
Embrechts, P., Lindskog, F., McNeil, A.: Modelling Dependence with Copulas and Applications to Risk Management. Handbook of Heavy Tailed Distributions in Finance, vol. ch. 8, pp. 329–384. Elsevier, Amsterdam (2003)
Hernandez-Campos, F., Jeffay, K.F., Park, C., Marron, J.S., Resnick, S.I.: Extremal dependence: Internet traffic applications. In: Stochastic Models, vol. 22(1), pp. 1–35. J. Taylor & Francis, Taylor (2005)
Kijima, M.: Markov Processes for Stochastic Modelling. Chapman and Hall, London (1997)
Lan, K.C., Hedemann, J.: On the correlation of Internet flow characteristics. Technical report ISI-TR-574, USC/Information Sciences Institute (July 2003)
Limnios, N., Ouhbi, B.: Nonparametric Estimation of Integral Functionals for Semi-Markov Processes with Application in Reliability. In: Vonta, F., Nikulin, M., Limnios, N., Huber-Carol, C. (eds.) Statistical Models and Methods for Biomedical and Technical Systems, pp. 405–418. Birkhauser, Boston-Basel-Berlin (2008)
Nelsen Roger, B.: An Introduction to Copulas, 2nd edn. Springer, New York (2006)
Pacheco, A., Tang, L.C., Prabhu, U.N.: Markov-Modulated process and Semiregenerative Phenomena. World Scientific, New Jersey (2009)
Zhang, Y., Breslay, L., Paxson, V., Shenker, S.: On the Characteristics and Origins of Internet Flow Rates. In: ACM SIGCOMM, Pittsburgh, PA (2002)
Pettere, G.: Stochastic Risk Capital Model for Insurance Company. In: Applied Stochastic Models and Data Analysis: Proceedings of the XIIth International Conference, Chania, Crete, Greece, May 29-31 and June 1 (2007)
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Andronov, A., Revzina, J. (2011). Simple Correlated Flow and Its Application. In: Al-Begain, K., Balsamo, S., Fiems, D., Marin, A. (eds) Analytical and Stochastic Modeling Techniques and Applications. ASMTA 2011. Lecture Notes in Computer Science, vol 6751. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-21713-5_21
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DOI: https://doi.org/10.1007/978-3-642-21713-5_21
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