Precise Asymptotic Distribution of the Number of Isolated Nodes in Wireless Networks with Lognormal Shadowing

ABSTRACT

In this paper, we study the connectivity of multihop wireless networks under the log-normal shadowing model by investigating the precise distribution of the number of isolated nodes. Under such a realistic shadowing model, all previous known works on the distribution of the number of isolated nodes were obtained only based on simulation studies or by ignoring the important boundary effect to avoid the challenging technical analysis, and thus cannot be applied to any practical wireless networks. It is extremely challenging to take the complicated boundary effect into consideration under such a realistic model because the transmission area of each node is an irregular region other than a circular area. Assume that the wireless nodes are represented by a Poisson point process with densitynover a unit-area disk, and that the transmission power is properly chosen so that the expected node degree of the network equals ln*n* + *ξ* (*n*), where *ξ* (*n*) approaches to a constant *ξ *as n → ∞. Under such a shadowing model with the boundary effect taken into
consideration, we proved that the total number of isolated nodes is
asymptotically Poisson with mean e$ {-*ξ*}. The Brun’s sieve is utilized to derive the precise asymptotic distribution.
Our results can be used as design guidelines for any practical multihop
wireless network where both the shadowing and boundary effects must be taken
into consideration.

In this paper, we study the connectivity of multihop wireless networks under the log-normal shadowing model by investigating the precise distribution of the number of isolated nodes. Under such a realistic shadowing model, all previous known works on the distribution of the number of isolated nodes were obtained only based on simulation studies or by ignoring the important boundary effect to avoid the challenging technical analysis, and thus cannot be applied to any practical wireless networks. It is extremely challenging to take the complicated boundary effect into consideration under such a realistic model because the transmission area of each node is an irregular region other than a circular area. Assume that the wireless nodes are represented by a Poisson point process with densitynover a unit-area disk, and that the transmission power is properly chosen so that the expected node degree of the network equals ln

KEYWORDS

Connectivity, Asymptotic Distribution, Random Geometric Graph, Isolated Nodes, log-Normal Shadowing

Connectivity, Asymptotic Distribution, Random Geometric Graph, Isolated Nodes, log-Normal Shadowing

Cite this paper

Wang, L. , Argumedo, A. and Washington, W. (2014) Precise Asymptotic Distribution of the Number of Isolated Nodes in Wireless Networks with Lognormal Shadowing.*Applied Mathematics*, **5**, 2249-2263. doi: 10.4236/am.2014.515219.

Wang, L. , Argumedo, A. and Washington, W. (2014) Precise Asymptotic Distribution of the Number of Isolated Nodes in Wireless Networks with Lognormal Shadowing.

References

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http://dx.doi.org/10.1093/comjnl/47.4.432

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http://dx.doi.org/10.1145/513800.513811

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[13] Hekmat, R. and Mieghem, P.V. (2006) Connectivity in Wireless Ad-Hoc Networks with a Log-Normal Radio Model. Mobile Networks and Applications, 11, 351-360.

http://dx.doi.org/10.1007/s11036-006-5188-7

[14] Mukherjee, S. and Avidor, D. (2005) On the Probability Distribution of the Minimal Number of Hops between Any Pair of Nodes in a Bounded Wireless Ad-Hoc Network Subject to Fading. International Workshop on Wireless Ad-Hoc Networks (IWWAN), London.

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http://dx.doi.org/10.1214/aop/1022677261

[18] Wang, L. and Baker, A.J. (2013) Critical Power for Vanishing of Isolated Nodes in Wireless Networks with Log-Normal Shadowing. IEEE MASS 2013, Hangzhou.

[19] Bertoni, H. (2000) Radio Propagation for Modern Wireless Systems. Prentice Hall PTR, New Jersey.

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http://dx.doi.org/10.1017/CBO9780511895357

[22] Yi, C.-W., Wan, P.-J., Lin, K.-W. and Huang, C.-H. (2006) Asymptotic Distribution of the Number of Isolated Nodes in Wireless Ad Hoc Networks with Unreliable Nodes and Links. IEEE GLOBECOM.

[23] Alon, N. and Spencer, J.H. (2000) The Probabilistic Method. 2nd Edition, Wiley, New York.

http://dx.doi.org/10.1002/0471722154

[1] Bettstetter, C. (2004) On the Connectivity of Ad Hoc Networks. The Computer Journal, 47, 432-447.

http://dx.doi.org/10.1093/comjnl/47.4.432

[2] Bettstetter, C. (2002) On the Minimum Node Degree and Connectivity of a Wireless Multihop Network. 3rd ACM International Symposium on Mobile Ad Hoc Networking and Computing, Lausanne, 80-91.

http://dx.doi.org/10.1145/513800.513811

[3] Gupta, P. and Kumar, P. (1998) Critical Power for Asymptotic Connectivity in Wireless Networks. Stochastic Analysis, Control, Optimization and Applications, 547-566.

[4] Philips, T.K., Panwar, S.S. and Tantawi, A.N. (1989) Connectivity Properties of a Packet Radio Network Model. IEEE Transactions on Information Theory, 35, 1044-1047.

http://dx.doi.org/10.1109/18.42219

[5] Wan, P.-J. and Yi, C.-W. (2004) Asymptotic Critical Transmission Radius and Critical Neighbor Number for k-Connectivity in Wireless Ad Hoc Networks. MobiHoc 2006, Roppongi, 1-8.

[6] Xue, F. and Kumar, P. (2004) The Number of Neighbors Needed for Connectivity of Wireless Networks. Wireless Networks, 10, 169-181.

http://dx.doi.org/10.1023/B:WINE.0000013081.09837.c0

[7] Yi, C.-W., Wan, P.-J., Li, X.-Y. and Frieder, O. (2006) Asymptotic Distribution of the Number of Isolated Nodes in Wireless Ad Hoc Networks with Bernoulli Nodes. IEEE Transactions on Communications, 54.

[8] Takai, M., Martin, J. and Bagrodia, R. (2001) Effects of Wireless Physical Layer Modeling in Mobile Ad Hoc Networks. ACM International Symposium on Mobile Ad Hoc Networking and Computing (MobiHoc), Long Beach.

[9] Zorzi, M. and Pupolin, S. (1995) Optimum Transmission Ranges in Multihop Packet Radio Networks in the Presence of Fading. IEEE Transactions on Communications, 43, 2201-2205.

http://dx.doi.org/10.1109/26.392962

[10] Bettstetter, C. and Hartmann, C. (2005) Connectivity of Wireless Multihop Networks in a Shadow Fading Environment. Wireless Networks, 11, 571-579.

http://dx.doi.org/10.1007/s11276-005-3513-x

[11] Bernhardt, R.C. (1987) Macroscopic Diversity in Frequency Reuse Systems. IEEE Journal of Selected Areas in Communications SAC, 5, 862-878.

http://dx.doi.org/10.1109/JSAC.1987.1146594

[12] Cox, D.C., Murray, R. and Norris, A. (1984) 800 MHz Attenuation Measured in and around Suburban Houses. AT & T Bell Laboratory Technical Journal, 63, 921-954.

http://dx.doi.org/10.1002/j.1538-7305.1984.tb00030.x

[13] Hekmat, R. and Mieghem, P.V. (2006) Connectivity in Wireless Ad-Hoc Networks with a Log-Normal Radio Model. Mobile Networks and Applications, 11, 351-360.

http://dx.doi.org/10.1007/s11036-006-5188-7

[14] Mukherjee, S. and Avidor, D. (2005) On the Probability Distribution of the Minimal Number of Hops between Any Pair of Nodes in a Bounded Wireless Ad-Hoc Network Subject to Fading. International Workshop on Wireless Ad-Hoc Networks (IWWAN), London.

[15] Stuedi, P., Chinellato, O. and Alonso, G. (2005) Connectivity in the Presence of Shadowing in 802.11 Ad Hoc Networks. IEEE Wireless Communications and Networking Conference, 2225-2230.

[16] Penrose, M. (1997) The Longest Edge of the Random Minimal Spanning Tree. The Annals of Applied Probability, 7, 340-361. http://dx.doi.org/10.1214/aoap/1034625335

[17] Penrose, M. (1999) A Strong Law for the Longest Edge of the Minimal Spanning Tree. The Annals of Applied Probability, 27, 246-260.

http://dx.doi.org/10.1214/aop/1022677261

[18] Wang, L. and Baker, A.J. (2013) Critical Power for Vanishing of Isolated Nodes in Wireless Networks with Log-Normal Shadowing. IEEE MASS 2013, Hangzhou.

[19] Bertoni, H. (2000) Radio Propagation for Modern Wireless Systems. Prentice Hall PTR, New Jersey.

[20] Rappaport, T.S. (2001) Wireless Communications: Principles and Practice. 2nd Edition, Prentice Hall PTR, New Jersey.

[21] Meester, R. and Roy, R. (1996) Continuum Percolation. Cambridge University Press, Cambridge.

http://dx.doi.org/10.1017/CBO9780511895357

[22] Yi, C.-W., Wan, P.-J., Lin, K.-W. and Huang, C.-H. (2006) Asymptotic Distribution of the Number of Isolated Nodes in Wireless Ad Hoc Networks with Unreliable Nodes and Links. IEEE GLOBECOM.

[23] Alon, N. and Spencer, J.H. (2000) The Probabilistic Method. 2nd Edition, Wiley, New York.

http://dx.doi.org/10.1002/0471722154