Skip to main content
SpringerLink
Log in

Joint Probability Density of the Intervals Length of Modulated Semi-synchronous Integrated Flow of Events in Conditions of a Constant Dead Time and the Flow Recurrence Conditions

  • Conference paper
  • First Online:
Information Technologies and Mathematical Modelling - Queueing Theory and Applications (ITMM 2015)

Part of the book series: Communications in Computer and Information Science ((CCIS,volume 564))

Abstract

This paper is focused on studying the modulated semi-synchronous integrated flow of events which is one of the mathematical models for incoming streams of events (claims) in computer communication networks and is related to the class of doubly stochastic Poisson processes (DSPPs). The flow is considered in conditions of its incomplete observability, when the dead time period of a constant duration T is generated after every registered event. In this paper we propose a technique for obtaining the formulas for calculation the probability density of the interval length between two neighboring flow events and the joint probability density of the length of two successive intervals. Also we find the conditions of the flow recurrence.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. Cox, D.R.: Some statistical methods connected with series of events. J. R. Stat. Soc. B 17, 129–164 (1955)

    MathSciNet  MATH  Google Scholar 

  2. Kingman, Y.F.C.: On doubly stochastic poisson process. Proc. Camb. Phylosophical Soc. 60(4), 923–930 (1964)

    Article  MATH  Google Scholar 

  3. Cox, D.R., Isham, V.: Point Processes. Chapman & Hall, London (1980)

    MATH  Google Scholar 

  4. Bremaud, P.: Point Processes and Queues: Martingale Dynamics. Springer-Verlag, New York (1981)

    Book  MATH  Google Scholar 

  5. Last, G., Brandt, A.: Marked Point Process on the Real Line: The Dynamic Approach. Springer-Verlag, New York (1995)

    MATH  Google Scholar 

  6. Basharin, G.P., Kokotushkin, V.A., Naumov, V.A.: Method of equivalent substitutions for calculating fragments of communication networks for digital computer. Eng. Cybern. 17(6), 66–73 (1979)

    MathSciNet  MATH  Google Scholar 

  7. Neuts, M.F.: A versatile Markov point process. J. Appl. Probab. 16, 764–779 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  8. Lucantoni, D.M.: New results on the single server queue with a batch markovian arrival process. Commun. Stat. Stoch. Models 7, 1–46 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  9. Lucantoni, D.M., Neuts, M.F.: Some steady-state distributions for the MAP/SM/1 queue. Commun. Stat. Stoch. Models 10, 575–598 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  10. Breuer, L.: An EM algorithm for batch Markovian arrival processes and its comparison to a simpler estimation procedure. Ann. Oper. Res. 112, 123–138 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  11. Telek, M., Horvath, G.: A minimal representation of Markov arrival processes and a moments matching method. Perform. Eval. 64, 1153–1168 (2007)

    Article  Google Scholar 

  12. Okamura, H., Dohi, T., Trivedi, K.S.: Markovian arrival process parameter estimation with group data. IEEE/ACM Trans. Networking 17(4), 1326–1339 (2009)

    Article  Google Scholar 

  13. Horvath, A., Horvath, G., Telek, M.: A joint moments based analysis of networks of MAP/MAP/1 queues. Perform. Eval. 67(9), 759–778 (2010)

    Article  Google Scholar 

  14. Bushlanov, I.V., Gortsev, A.M., Nezhelskaya, L.A.: Estimating parameters of the synchronous twofold-stochastic flow of events. Autom. Remote Control 69(9), 1517–1533 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  15. Gortsev, A.M., Nezhelskaya, L.A.: Estimation of the dead-time period and parameters of a synchronous alternating flow of events. Bull. Tomsk State Univ. 6, 232–239 (2003). (in Russian)

    Google Scholar 

  16. Gortsev, A.M., Nezhelskaya, L.A.: Parameters estimation of a synchronous doubly stochastic flow of events using method of moments. Bull. Tomsk State Univ. 1, 24–29 (2002). (in Russian)

    Google Scholar 

  17. Gortsev, A.M., Nezhelskaya, L.A.: An asynchronous double stochastic flow with initiation of superfluous events. Discrete Math. Appl. 21(3), 283–290 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  18. Gortsev, A.M., Nissenbaum, O.V.: Estimation of the dead time period and parameters of an asynchronous alternative flow of events with unextendable dead time period. Russ. Phys. J. 48(10), 1039–1054 (2005). (in Russian)

    Google Scholar 

  19. Gortsev, A.M., Nezhelskaya, L.A., Shevchenko, T.I.: States estimation of the MC flow of events in the presence of measurement errors. Russ. Phys. J. 12, 67–85 (1993). (in Russian)

    Google Scholar 

  20. Gortsev, A.M., Nezhelskaya, L.A.: Estimation of the dead-time period and parameters of a semi-synchronous double-stochastic stream of events. Meas. Tech. 46(6), 536–545 (2003)

    Article  Google Scholar 

  21. Gortsev, A.M., Nezhelskaya, L.A.: Semi-synchronous doubly stochastic flow of events in condition of prolonged dead time. Comput. Technol. 13(1), 31–41 (2008) (in Russian)

    Google Scholar 

  22. Gortsev, A.M., Nezhelskaya, L.A.: Parameters estimation of a semi-synchronous doubly stochastic flow of events using method of moments. Bull. Tomsk State Univ. 1, 18–23 (2002). (in Russian)

    Google Scholar 

  23. Gortsev, A.M., Nezhelskaya, L.A.: On connection of MC flows and MAP flows of events. Bull. Tomsk State Univ. Control, Comput. Eng. Inform. 1(14), 13–21 (2011). (in Russian)

    Google Scholar 

  24. Adamu, A., Gaidamaka, Y., Samuylov, A.: Discrete Markov chain model for analyzing probability measures of P2P streaming network. In: Balandin, S., Koucheryavy, Y., Hu, H. (eds.) NEW2AN 2011 and ruSMART 2011. LNCS, vol. 6869, pp. 428–439. Springer, Heidelberg (2011)

    Chapter  Google Scholar 

  25. Bouzas, P.R., Valderrama, M.J., Aguilera, A.M., Ruiz-Fuentes, N.: Modelling the mean of a doubly stochastic poisson process by functional data analysis. Comput. Stat. Data Anal. 50(10), 2655–2667 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  26. Centanni, S., Minozzo, M.: A Monte Carlo approach to filtering for a class of marked doubly stochastic poisson processes. J. Am. Stat. Assoc. 101, 1582–1597 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  27. Dubois, J.-P.: Traffic estimation in wireless networks using filtered doubly stochastic point processes (conference paper). In: Proceedings - 2004 International Conference on Electrical, Electronic and Computer Engineering, ICEEC 2004, pp. 116–119 (2004)

    Google Scholar 

  28. Hossain, M.M., Lawson, A.B.: Approximate methods in Bayesian point process spatial models. Comput. Stat. Data Anal. 53(8), 2831–2842 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  29. Snyder, D.L., Miller, M.I.: Random Point Processes in Time and Space. Springer-Verlag, Heidelberg (1991)

    Book  MATH  Google Scholar 

  30. Gortsev, A.M., Nezhelskaya, L.A., Solovev, A.A.: Optimal state estimation in MAP event flows with unextendable dead time. Autom. Remote Control 73(8), 1316–1326 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  31. Bakholdina, M.A.: Optimal estimation of the states of modulated semi-synchronous integrated flow of events. Bull. Tomsk State Univ. Control Comput. Eng. Inf. 2(23), 10–21 (2013). (in Russian)

    Google Scholar 

  32. Bakholdina, M.A., Gortsev, A.M.: Optimal estimation of the states of modulated semi-synchronous integrated flow of events in condition of a constant dead time. Bull. Tomsk State Univ. Control Comput. Eng. Inf. 1(26), 13–24 (2014) (in Russian)

    Google Scholar 

  33. Bakholdina, M.A., Gortsev, A.M.: Optimal estimation of the states of modulated semi-synchronous integrated flow of events in condition of its incomplete observability. Appl. Math. Sci. 9(29), 1433–1451 (2015)

    Article  Google Scholar 

  34. Gortsev, A.M., Nezhelskaya, L.A.: Estimation of the dead time period and intensity of synchronous doubly stochastic flow of events. Radio Eng. 10, 8–16 (2004). (in Russian)

    Google Scholar 

  35. Vasileva, L.A., Gortsev, A.M.: Estimation of parameters of a double-stochastic flow of events under conditions of its incomplete observability. Autom. Remote Control 63(3), 511–515 (2002). (in Russian)

    Google Scholar 

  36. Gortsev, A.M., Zavgorodnyaya, M.E.: States estimation of the alternating flow of events in condition of its incomplete observability. Atmos. Ocean Opt. 10(3), 273–280 (1997). (in Russian)

    Google Scholar 

  37. Gortsev, A.M., Klimov, I.S.: Intensity estimation of the Poisson flow of events in condition of its incomplete observability. Radio Eng. 12, 3–7 (1991). (in Russian)

    Google Scholar 

  38. Normey-Rico, J.E.: Control of dead-time processes (Advanced textbooks in control and signal processing). Springer, London (2007)

    Google Scholar 

  39. Gortsev, A.M., Kalyagin, A.A., Nezhelskaya, L.A.: Optimal states estimation of integrated semi-synchronous flow of events. Bull. Tomsk State Univ. Control Comput. Eng. Inf. 2(11), 66–81 (2010). (in Russian)

    Google Scholar 

  40. Bakholdina, M., Gortsev, A.: Joint probability density of the intervals length of the modulated semi-synchronous integrated flow of events and its recurrence conditions. In: Dudin, A., Nazarov, A., Yakupov, R., Gortsev, A. (eds.) ITMM 2014. CCIS, vol. 487, pp. 18–25. Springer, Heidelberg (2014)

    Google Scholar 

Download references

Acknowledgments

The work is supported by Tomsk State University Competitiveness Improvement Program.

Author information

Authors and Affiliations

  1. Department of Operations Research, Faculty of Applied Mathematics and Cybernetics, National Research Tomsk State University, 36 Lenina Avenue, Tomsk, 634050, Russia

    Maria Bakholdina & Alexander Gortsev

Authors
  1. Maria Bakholdina
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Alexander Gortsev
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Maria Bakholdina .

Editor information

Editors and Affiliations

  1. Applied Mathematics and Computer Science, Belarusian State University, Minsk, Russia

    Alexander Dudin

  2. Tomsk State University, Tomsk, Russia

    Anatoly Nazarov

  3. Kemerovo State University, Anzhero-Sudzhensk, Russia

    Rafael Yakupov

Rights and permissions

Reprints and permissions

Copyright information

© 2015 Springer International Publishing Switzerland

About this paper

Cite this paper

Bakholdina, M., Gortsev, A. (2015). Joint Probability Density of the Intervals Length of Modulated Semi-synchronous Integrated Flow of Events in Conditions of a Constant Dead Time and the Flow Recurrence Conditions. In: Dudin, A., Nazarov, A., Yakupov, R. (eds) Information Technologies and Mathematical Modelling - Queueing Theory and Applications. ITMM 2015. Communications in Computer and Information Science, vol 564. Springer, Cham. https://doi.org/10.1007/978-3-319-25861-4_2

Download citation

  • DOI: https://doi.org/10.1007/978-3-319-25861-4_2

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-25860-7

  • Online ISBN: 978-3-319-25861-4

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation