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Stochastic geometry models of wireless networks
In mathematics and telecommunications, stochastic geometry models of wireless networks refer to mathematical models based on stochastic geometry that are designed to represent aspects of wireless networks. The related research consists of analyzing these models with the aim of better understanding wireless communication networks in order to predict and control various network performance metrics. The models require using techniques from stochastic geometry and related fields including point processes, spatial statistics, geometric probability, percolation theory, as well as methods from more general mathematical disciplines such as geometry, probability theory, stochastic processes, queueing theory, information theory, and Fourier analysis.〔F. Baccelli and B. Błaszczyszyn. ''Stochastic Geometry and Wireless Networks, Volume I — Theory'', volume 3, No 3–4 of ''Foundations and Trends in Networking''. NoW Publishers, 2009.〕〔F. Baccelli and B. Błaszczyszyn. ''Stochastic Geometry and Wireless Networks, Volume II — Applications'', volume 4, No 1–2 of ''Foundations and Trends in Networking''. NoW Publishers, 2009.〕〔W. S. Kendall and I. Molchanov, eds. ''New Perspectives in Stochastic Geometry''. Oxford University Press, 2010.〕〔M. Haenggi. ''Stochastic geometry for wireless networks''. Cambridge University Press, 2012.〕
In the early 1960s a stochastic geometry model 〔E. N. Gilbert. Random plane networks. ''Journal of the Society for Industrial \& Applied Mathematics'', 9(4):533–543, 1961.〕 was developed to study wireless networks. This model is considered to be pioneering and the origin of continuum percolation.〔M. Franceschetti and R. Meester. ''Random networks for communication: from statistical physics to information systems'', volume 24. Cambridge University Press, 2007.〕 Network models based on geometric probability were later proposed and used in the late 1970s 〔L. Kleinrock and J. Silvester. Optimum transmission radii for packet radio networks or why six is a magic number. In ''IEEE National Telecommunications'', pages 4.31–4.35, 1978.〕 and continued throughout the 1980s 〔L. Kleinrock and J. Silvester. Spatial reuse in multihop packet radio networks. ''Proceedings of the IEEE'', 75(1):156–167, 1987.〕〔H. Takagi and L. Kleinrock. Optimal transmission ranges for randomly distributed packet radio terminals. ''IEEE Transactions on Communications'', 32(3):246–257, 1984.〕 for examining packet radio networks. Later their use increased significantly for studying a number of wireless network technologies including mobile ''ad hoc'' networks, sensor networks, vehicular ''ad hoc'' networks, cognitive radio networks and several types of cellular networks, such as heterogeneous cellular networks.〔J. G. Andrews, R. K. Ganti, M. Haenggi, N. Jindal, and S. Weber. A primer on spatial modeling and analysis in wireless networks. ''Communications Magazine, IEEE'', 48(11):156–163, 2010.〕〔M. Haenggi, J. Andrews, F. Baccelli, O. Dousse, and M. Franceschetti. Stochastic geometry and random graphs for the analysis and design of wireless networks. ''IEEE JSAC'', 27(7):1029–1046, september 2009.〕〔S. Mukherjee. ''Analytical Modeling of Heterogeneous Cellular Networks: Geometry, Coverage, and Capacity''. Cambridge University Press, 2014.〕 Key performance and quality of service quantities are often based on concepts from information theory such as the signal-to-interference-plus-noise ratio, which forms the mathematical basis for defining network connectivity and coverage.〔
The principal idea underlying the research of these stochastic geometry models, also known as random spatial models,〔 is that it is best to assume that the locations of nodes or the network structure and the aforementioned quantities are random in nature due to the size and unpredictability of users in wireless networks. The use of stochastic geometry can then allow for the derivation of closed-form or semi-closed-form expressions for these quantities without resorting to simulation methods or (possibly intractable or inaccurate) deterministic models.〔
==Overview==
The discipline of stochastic geometry entails the mathematical study of random objects defined on some (often Euclidean) space. In the context of wireless networks, the random objects are usually simple points (which may represent the locations of network nodes such as receivers and transmitters) or shapes (for example, the coverage area of a transmitter) and the Euclidean space is either 3-dimensional, or more often, the (2-dimensional) plane, which represents a geographical region. In wireless networks (for example, cellular networks) the underlying geometry (the relative locations of nodes) plays a fundamental role due to the interference of other transmitters, whereas in wired networks (for example, the Internet) the underlying geometry is less important.
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