Skip to main content
SpringerLink
Log in

Graph-Based Mobility Models: Asymptotic and Stationary Node Distribution

  • Conference paper
  • First Online:
Measurement, Modelling and Evaluation of Computing Systems (MMB 2020)

Part of the book series: Lecture Notes in Computer Science ((LNPSE,volume 12040))

Abstract

Under standard assumptions on the stochastic behaviour of mobile nodes in a graph-based mobility model we derive the stationary distribution for the network. This distribution describes as well the asymptotic behaviour of the system. We consider closed (fixed number of moving nodes) as well as open (nodes arrive and depart from the graph-structured area) systems. The stationary state shows that these graph-based models for mobile nodes are separable, i. e. the stationary distribution is for the open system the product of independent coordinate processes and for the closed system holds conditional independence.

I am greatful to Ralf Lehnert (TU Dresden) and Andreas Timm-Giel (TU Hamburg-Harburg) for helpful discussions on mobility models for delay-tolerant networks. – I thank four reviewers for their constructive comments which enhanced the paper.

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. Barbour, A.D.: Networks of queues and the method of stages. Adv. Appl. Probab. 8, 584–591 (1976)

    Article  MathSciNet  Google Scholar 

  2. Bolch, G., Greiner, S., de Meer, H., Trivedi, K.S.: Queueing Networks and Markov Chains, 2nd edn. Wiley, New York (2006)

    Book  Google Scholar 

  3. Bruell, S.C., Balbo, G.: Computational Algorithms for Closed Queueing Networks. North-Holland, New York (1980)

    MATH  Google Scholar 

  4. Baskett, F., Chandy, M., Muntz, R., Palacios, F.G.: Open, closed and mixed networks of queues with different classes of customers. J. Assoc. Comput. Mach. 22, 248–260 (1975)

    Article  MathSciNet  Google Scholar 

  5. Bai, F., Helmy, A.: A survey of mobility models in wireless adhoc networks. In: Wireless Ad-Hoc Networks, chap. 1, pp. 1–30. Kluwer Academic Publisher, Dordrecht (2006)

    Google Scholar 

  6. Bettstetter, C., Resta, G., Santi, P.: The node distribution of the random waypoint mobility model for wireless ad hoc networks. IEEE Trans. Mob. Comput. 2(3), 257–269 (2003)

    Article  Google Scholar 

  7. Daduna, H.: Moving queue on a network. In: Remke, A., Haverkort, B.R. (eds.) MMB&DFT 2016. LNCS, vol. 9629, pp. 40–54. Springer, Cham (2016). https://doi.org/10.1007/978-3-319-31559-1_5

    Chapter  Google Scholar 

  8. Daduna, H., Szekli, R.: Impact of routeing on correlation strength in stationary queueing networks processes. J. Appl. Probab. 45, 846–878 (2008)

    Article  MathSciNet  Google Scholar 

  9. Kelly, F.: Networks of queues. Adv. Appl. Probab. 8, 416–432 (1976)

    Article  MathSciNet  Google Scholar 

  10. Kelly, F.P.: Reversibility and Stochastic Networks. Wiley, Chichester/New York/Brisbane/Toronto (1979)

    MATH  Google Scholar 

  11. Kraaier, J., Killat, U.: Calculating mobility parameters for a predefined stationary user distribution. In: Proceedings of 2004 12th IEEE International Conference on Networks (ICON 2004) (IEEE Cat. No.04EX955), vol. 1, pp. 41–45, November 2004

    Google Scholar 

  12. Kraaier, J., Killat, U.: The random waypoint city model : User distribution in a street-based mobility model for wireless network simulations. In: Proceedings of the 3rd ACM International Workshop on Wireless Mobile Applications and Services on WLAN Hotspots, WMASH 2005, p. 100–103. ACM, New York (2005)

    Google Scholar 

  13. Kraaier, J., Killat, U.: Controlling the stationary distribution of mobile users in wireless network simulations. In: 2007 IEEE 66th Vehicular Technology Conference, pp. 804–808 (2007)

    Google Scholar 

  14. Kriege, J.: Combining mobility models with arrival processes. In: Remke, A., Haverkort, B.R. (eds.) MMB&DFT 2016. LNCS, vol. 9629, pp. 107–121. Springer, Cham (2016). https://doi.org/10.1007/978-3-319-31559-1_10

    Chapter  Google Scholar 

  15. Le Boudec, J., Vojnovic, M.: Perfect simulation and stationarity of a class of mobility models. In: Proceedings IEEE 24th Annual Joint Conference of the IEEE Computer and Communications Societies, vol. 4, pp. 2743–2754, March 2005

    Google Scholar 

  16. Navidi, W., Camp, T.: Stationary distribution for the random waypoint mobility model. IEEE Trans. Mob. Comput. 3(1), 99–108 (2004)

    Article  Google Scholar 

  17. Navidi, W., Camp, T., Bauer, N.: Improving accuracy of random waypoint simulations through steady-state initialization. Technical report, Colorado School of Mines, Golden, CO, June 2006

    Google Scholar 

  18. Nikoletseas, S., Rolim, J.D.P.: Theoretical Aspects of Distributed Computing in Sensor Networks. Monographs in Theoretical Computer Science. An EATCS Series. Springer, Heidelberg (2011). https://doi.org/10.1007/978-3-642-14849-1

    Book  MATH  Google Scholar 

  19. Roy, R.R.: Handbook of Mobile Ad Hoc Networks for Mobility Models. Springer, Heidelberg (2011). https://doi.org/10.1007/978-1-4419-6050-4

    Book  MATH  Google Scholar 

  20. Schassberger, R.: Warteschlangen. Springer, Wien (1973)

    Book  Google Scholar 

  21. Schwamborn, M., Aschenbruck, N.: On modeling and impact of geographic restrictions for human mobility in opportunistic networks. Perform. Eval. 130, 17–31 (2019)

    Article  Google Scholar 

  22. Stepanov, I., Marron, P.J., Rothermel, K.: Mobility modeling of outdoor scenarios for manets. In: Proceedings of the 38th Annual Symposium on Simulation, ANSS 2005, pp. 312–322. IEEE Computer Society, Washington (2005)

    Google Scholar 

  23. Simatos, F., Tibi, D.: Study of a stochastic model for mobile networks. SIGMETRICS Perform. Eval. Rev. 36(2), 122–124 (2008)

    Article  Google Scholar 

  24. Tian, J., Hähner, J., Becker, C., Stepanov, I., Rothermel, K.: Graph-based mobility model for mobile ad hoc network simulation. In: Proceedings of the 35th Annual Simulation Symposium, SS 2002, pp. 337–344. IEEE Computer Society, Washington (2002)

    Google Scholar 

  25. Wang, Y., Dang, H., Wu, H.H.: A survey on analytic studies of delay-tolerant mobile sensor networks. Wirel. Commun. Mob. Comput. 7(10), 1197–1208 (2007)

    Article  Google Scholar 

  26. Wu, H., Wang, Y., Dang, H., Lin, F.: Analytic, simulation, and empirical evaluation of delay/fault-tolerant mobile sensor networks. IEEE Trans. Wirel. Commun. 6(9), 3287–3296 (2007)

    Article  Google Scholar 

  27. Yoon, J., Liu, M., Noble, B.: A general framework to construct stationary mobility models for the simulation of mobile networks. IEEE Trans. Mob. Comput. 5(7), 1–12 (2006)

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Hamburg University, Bundesstr. 55, 20146, Hamburg, Germany

    Hans Daduna

Authors
  1. Hans Daduna
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Hans Daduna .

Editor information

Editors and Affiliations

  1. Saarland University, Saarbrücken, Germany

    Holger Hermanns

Rights and permissions

Reprints and permissions

Copyright information

© 2020 Springer Nature Switzerland AG

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

Daduna, H. (2020). Graph-Based Mobility Models: Asymptotic and Stationary Node Distribution. In: Hermanns, H. (eds) Measurement, Modelling and Evaluation of Computing Systems. MMB 2020. Lecture Notes in Computer Science(), vol 12040. Springer, Cham. https://doi.org/10.1007/978-3-030-43024-5_10

Download citation

  • DOI: https://doi.org/10.1007/978-3-030-43024-5_10

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-030-43023-8

  • Online ISBN: 978-3-030-43024-5

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation