Abstract
In this chapter, we introduce the basic mathematical model for the random locations of many point-like objects in the Euclidean space, the Poisson point process (PPP) . This process will be used for modeling the places of devices, additional boxes (supporting devices) and/or base stations in space. Apart from this interpretation in telecommunication, the PPP is universally applicable in many situations and is fundamental for the theory of stochastic geometry. The main assumption is a high degree of statistical independence of all the random points, which leads to many explicit and tractable formulas and to the validity of many properties that make a mathematical treatment simple. For these reasons, the PPP is the initial method of choice practically in any spatial telecommunication modeling and the most obvious starting point for a mathematical analysis.
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Notes
- 1.
Note that the measure μ ⊗ K is defined by \(\mu \otimes K(B)=\int _{B^{{\scriptscriptstyle {({1}})}}}\mu ({\operatorname {d}} x)\, K(x,B_x^{{\scriptscriptstyle {({2}})}})\), where \(B^{{\scriptscriptstyle {({1}})}}=\{x\in D\colon \exists y\in {\mathcal M }\colon (x,y)\in B\}\) and \(B_x^{{\scriptscriptstyle {({2}})}}=\{y\in {\mathcal M }\colon (x,y)\in B\}\).
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Jahnel, B., König, W. (2020). Device Locations: Point Processes. In: Probabilistic Methods in Telecommunications. Compact Textbooks in Mathematics. Birkhäuser, Cham. https://doi.org/10.1007/978-3-030-36090-0_2
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DOI: https://doi.org/10.1007/978-3-030-36090-0_2
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