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A simple model of contact patterns in delay-tolerant networks

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Abstract

Delay-tolerant networks, consisting of a number of intermittently connected wireless devices, are emerging as a promising application field, able to complement in some situations infrastructure-based service delivery. In this work, we propose a marked point process model for characterizing the pattern of contacts among nodes in such systems. The model can be used for deriving analytical results on various performance indexes. Validation of the model is performed through comparison with experimental traces obtained from real-world DTN deployments.

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Acknowledgement

This work was partially supported by the EC within the framework of the BIONETS project (www.bionets.eu), EU-IST-FET-SAC-FP6-027748.

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Authors and Affiliations

  1. CREATE-NET, via alla Cascata 56/C, 38100, Trento, Italy

    Iacopo Carreras, Daniele Miorandi & Imrich Chlamtac

Authors
  1. Iacopo Carreras
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  2. Daniele Miorandi
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  3. Imrich Chlamtac
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Corresponding author

Correspondence to Iacopo Carreras.

Additional information

Part of this work appeared in the Proceedings of Mobiquitous, San Jose, USA, July 2006 [1].

Appendices

Appendix 1: Proofs

1.1 Proof of Lemma 1

Proof

Let us denote by \(\{T(i,j)_n\}_{n \in {\mathbb{Z}}}\) the sequence of meeting times between nodes i and j. Due to the properties of the M 2 mobility model, the inter-meeting times of i and j are sums of an i.i.d. number of i.i.d. quantities (the intrameeting times), so that they are also i.i.d. The 0-th intermeeting time of nodes i and j, Y 0(i, j) = T 1(i, j) − T 0(i,j) is the sum of η intermeeting times. η is a geometric random variable of parameter π(i,j). By Wald’s lemma, we have \({\mathbb{E}}[Y_0(i,j)]={\mathbb{E}}[\eta]\cdot {\mathbb{E}}[Y_0]={\frac{{\mathbb{E}}[Y_0]}{\pi(i,j)}}.\) Hence if π(i,j) > 0, then the inter-meeting time is finite \({\mathbb{P}}\)-a.s.

1.2 Proof of Corollary 1

Proof

The process \(\{T_n\}_{n \in {\mathbb{Z}}}\) is a renewal process, and the marks \(\{\sigma_n\}_{n \in {\mathbb{Z}}}\) are iid. It follows that the intermeeting time has the same distribution of the time elapsed between a meeting instant \(T_{\hat{n}}\) and the first subsequent meeting between node i and j. The application of Lemma 1 concludes the proof.

Appendix 2: Autocorrelation plots

See Figs. 1, 2, 3, 4, 5 and 6.

Fig. 1
figure 1

Autocorrelation plots (mark and intrameeting times sequences) for the three Intel-Cambridge experiments. a 1st experiment: marks, b 1st experiment: intrameetings, c 2nd experiment: marks, d 2nd experiment: intrameetings, e 3rd experiment: marks, f 3rd experiment: intrameetings

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Fig. 2
figure 2

Autocorrelation plot for marks and intrameeting times sequences for the DieselNet 2152007 experiment. a Marks, b Intrameeting times

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Fig. 3
figure 3

Autocorrelation plot for marks and intrameeting times sequences for the MIT Reality Mining experiment. a Marks, b Intrameeting times

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Fig. 4
figure 4

Autocorrelation plot for marks and intrameeting times sequences for the CREATE-NET experiment. a Marks, b intrameeting times

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Fig. 5
figure 5

Intrameetings ECDF in the case of a INTEL1, b INTEL2, c INTEL3 traces

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Fig. 6
figure 6

Mapping of labels to complex number, σ(x,y) = x + iy for N = 5

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Carreras, I., Miorandi, D. & Chlamtac, I. A simple model of contact patterns in delay-tolerant networks. Wireless Netw 16, 851–862 (2010). https://doi.org/10.1007/s11276-009-0172-3

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