• Bernoulli
  • Volume 22, Number 4 (2016), 2143-2176.

Connection times in large ad-hoc mobile networks

Hanna Döring, Gabriel Faraud, and Wolfgang König

Full-text: Open access


We study connectivity properties in a probabilistic model for a large mobile ad-hoc network. We consider a large number of participants of the system moving randomly, independently and identically distributed in a large domain, with a space-dependent population density of finite, positive order and with a fixed time horizon. Messages are instantly transmitted according to a relay principle, that is, they are iteratively forwarded from participant to participant over distances smaller than the communication radius until they reach the recipient. In mathematical terms, this is a dynamic continuum percolation model.

We consider the connection time of two sample participants, the amount of time over which these two are connected with each other. In the above thermodynamic limit, we find that the connectivity induced by the system can be described in terms of the counterplay of a local, random and a global, deterministic mechanism, and we give a formula for the limiting behaviour.

A prime example of the movement schemes that we consider is the well-known random waypoint model. Here, we give a negative upper bound for the decay rate, in the limit of large time horizons, of the probability of the event that the portion of the connection time is less than the expectation.

Article information

Bernoulli, Volume 22, Number 4 (2016), 2143-2176.

Received: January 2015
Revised: March 2015
First available in Project Euclid: 3 May 2016

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

ad-hoc networks connectivity dynamic continuum percolation large deviations random waypoint model


Döring, Hanna; Faraud, Gabriel; König, Wolfgang. Connection times in large ad-hoc mobile networks. Bernoulli 22 (2016), no. 4, 2143--2176. doi:10.3150/15-BEJ724.

Export citation


  • [1] Asmussen, S. (2003). Applied Probability and Queues, 2nd ed. Applications of Mathematics (New York) 51. New York: Springer.
  • [2] Bettstetter, C. and Wagner, C. (2002). The spatial node distribution of the random waypoint mobility model. WMAN, 41–58.
  • [3] Bettstetter, C., Hartenstein, H. and Pérez-Costa, X. and (2004). Stochastic properties of the random waypoint mobility model. ACM/Kluwer Wireless Networks 6, Special Issue on Modeling and Analysis of Mobile Networks 10 555–567.
  • [4] Bryc, W. and Dembo, A. (1996). Large deviations and strong mixing. Ann. Inst. Henri Poincaré Probab. Stat. 32 549–569.
  • [5] Camp, T., Boleng, J. and Davies, V. (2002). A survey of mobility models for ad-hoc network research WCMC: Special Issue on Mobile ad-hoc Networking: Research, Trends and Applications 2 483–502.
  • [6] Clementi, A.E.F., Pasquale, F. and Silvestri, R. (2009). MANETS: High mobility can make up for low transmission power. In Automata, Languages and Programming. Part II. Lecture Notes in Computer Science 5556 387–398. Berlin: Springer.
  • [7] Dembo, A. and Zeitouni, O. (2010). Large Deviations Techniques and Applications. Stochastic Modelling and Applied Probability 38. Berlin: Springer. Corrected reprint of the second (1998) edition.
  • [8] Hyytia, E., Lassila, P. and Virtamo, J. (2006). Spatial node distribution of the random waypoint mobility model with applications. IEEE Trans. Mob. Comput. 5 680–694.
  • [9] Kaspi, H. and Mandelbaum, A. (1994). On Harris recurrence in continuous time. Math. Oper. Res. 19 211–222.
  • [10] Kesten, H. (1974). Renewal theory for functionals of a Markov chain with general state space. Ann. Probab. 2 355–386.
  • [11] Le Boudec, J.-Y. (2007). Understanding the simulation of mobility models with palm calculus. Perform. Eval. 64 126–146.
  • [12] Le Boudec, J.-Y. and Vojnovic, M. (2006). The random trip model: Stability, stationary regime, and perfect simulation. IEEE/ACM Transactions on Networking 14 1153–1166.
  • [13] Meester, R. and Roy, R. (1996). Continuum Percolation. Cambridge Tracts in Mathematics 119. Cambridge: Cambridge Univ. Press.
  • [14] Penrose, M.D. (2003). Random Geometric Graphs. Oxford Studies in Probability 5. Oxford: Oxford Univ. Press.
  • [15] Penrose, M.D. (1991). On a continuum percolation model. Adv. in Appl. Probab. 23 536–556.
  • [16] Penrose, M.D. (1995). Single linkage clustering and continuum percolation. J. Multivariate Anal. 53 94–109.
  • [17] Peres, Y., Sinclair, A., Sousi, P. and Stauffer, A. (2013). Mobile geometric graphs: Detection, coverage and percolation. Probab. Theory Related Fields 156 273–305.
  • [18] Quintanilla, J.A. and Ziff, R.M. (2007). Asymmetry in the percolation thresholds of fully penetrable disks with two different radii. Phys. Rev. E 76 051115.
  • [19] Roy, R.R. (2011). Handbook of Mobile Ad Hoc Networks for Mobility Models. New York: Springer.
  • [20] Sarkar, A. (1997). Continuity and convergence of the percolation function in continuum percolation. J. Appl. Probab. 34 363–371.