无线网络中的随机几何理论分析

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"Stochastic Geometry and Wireless Networks Volume I(很清晰)" 是一本专注于无线通信网络的学术著作，由François Baccelli和Bartlomiej Blaszczyszyn合著。该书深入探讨了随机几何学在无线网络理论中的应用。在无线通信网络中，节点可以视为发射器或接收器，例如移动用户、蜂窝网络中的基站或WiFi mesh的接入点。当多个节点同时传输时，每个节点都有自己的接收器。每一对发射器-接收器都需要独立的无线链路。然而，由于信号功率随欧几里得距离以各向同性方式衰减，其他发射器的信号可能会干扰接收器接收到的信号，导致干扰。这种干扰是接收器接收到的所有信号功率之和，除了其自身的发射器信号。信号到干扰和噪声比（SINR）在每个接收器处的大小决定了能否在特定比特率下同时建立这些链接。随机几何学在这个领域起着关键作用，因为它分析了节点位置的几何分布如何影响SINR，并因此影响网络的性能。通过这种方式，可以研究和预测网络容量、覆盖范围以及在给定条件下的服务质量。书中的内容可能涵盖了网络模型的构建、干扰建模、概率论和统计方法的应用，以分析网络的性能特征。此外，还可能涉及如何利用随机过程理论来描述节点的分布和它们之间的相互作用，以及如何优化网络设计以提高效率和可靠性。作者通过HAL（一个多学科开放存取档案）提交了这篇论文，该平台旨在存储和传播来自法国及国外教育和研究机构，以及公共或私人研究中心的科学研究成果。这篇论文是《随机几何与无线网络》系列的第一卷，聚焦于理论方面，为后来的实践应用奠定了基础。引用此版本的参考格式为：Baccelli, F., Blaszczyszyn, B. (2009). 随机几何与无线网络，第一卷 - 理论. Foundations and Trends in Networking, 3(3-4), 249-449. ISBN: 978-1-60198-264-3, 978-1-60198-265-0. <10.1561/1300000006>. <inria-00403039v4> 这本书对于理解无线网络的复杂性和利用随机几何学优化网络设计具有重要意义，是通信领域的宝贵资源。

The most basic objects studied in classical stochastic geometry are multidimensional point processes,

which are covered in Chapter 1, with a special emphasis on the most prominent one, the Poisson point

process. Our default observation space in this part will be the Euclidean space R

of dimension d ≥ 1. Even

if for most of the applications studied later, the plane R

(2D) sufﬁces, it is convenient to formulate some

results in 3D or 4D (e.g. to consider time and space).

Shot noise ﬁelds, which are used quite naturally to represent interference ﬁelds, are studied in Chapter 2.

Chapter 3 is focused on coverage processes, with the particularly important special case of the Boolean

model. Chapter 4 bears on Voronoi tessellations and Delaunay graphs, which are useful in a variety of

contexts in wireless network modeling. These basic tools will be needed for analyzing the SINR models

stemming from wireless communications to be analyzed from Part II on. They will be instrumental for

analyzing spatio-temporal models when combined with Markov process techniques.

Poisson Point Process

Consider the d -dimensional Euclidean space R

. A spatial point process (p.p.) Φ is a random, ﬁnite or

countably-inﬁnite collection of points in the space R

, without accumulation points.

One can consider any given realization φ of a point process as a discrete subset φ = {x

} ⊂ R

the space. It is often more convenient to think of φ as a counting measure or a point measure φ =

where ε

is the Dirac measure at x; for A ⊂ R

, ε

(A) = 1 if x ∈ A and ε

(A) = 0 if x 6∈ A.

Consequently, φ(A) gives the number of “points” of φ in A. Also, for all real functions f deﬁned on R

we have

f(x

) =

f(x) φ(dx). We will denote by M the set of all point measures that do not have

accumulation points in R

. This means that any φ ∈ M is locally ﬁnite, that is φ(A) < ∞ for any bounded

A ⊂ R

(a set is bounded if it is contained in a ball with ﬁnite radius).

Note that a p.p. Φ can be seen as a stochastic process Φ = {Φ(A)}

A∈B

with state space N = {0, 1, . . .} 3

Φ(A) and where the index A runs over bounded Borel subsets of R

. Moreover, as for “usual” stochastic

processes, the distribution of a p.p. is entirely characterized by the family of ﬁnite dimensional distributions

(Φ(A

), . . . , Φ(A

)), where A

, . . . , A

run over the bounded subsets of R

1.1 Deﬁnition and Characterizations

1.1.1 Deﬁnition

Let Λ be a locally ﬁnite non-null measure on R

Deﬁnition 1.1.1. The Poisson point process Φ of intensity measure Λ is deﬁned by means of its ﬁnite-

We do not discuss here the measure-theoretic foundations of p.p. theory; we remark that each time we talk about a subset B of R

or a function

f deﬁned on R

, we understand that they belong to some “nice class of subsets that can be measured” and to some “nice class of functions that

can be integrated”. A similar convention is assumed for subsets of M and functions deﬁned on this space (typically, we want all events of the type

{µ ∈ M : µ(A) = k}, A ⊂ R

, k ∈ N, to be ”measurable”). See (Daley and Vere-Jones 1988) or (Daley and Vere-Jones 2005; Daley and

Vere-Jones 2008) for details.

dimensional distributions:

Φ(A

) = n

, . . . , Φ(A

) = n

i=1



−Λ(A

)

Λ(A

)



for every k = 1, 2, . . . and all bounded, mutually disjoint sets A

for i = 1, . . . , k. If Λ(dx) = λ dx is a

multiple of Lebesgue measure (volume) in R

, we call Φ a homogeneous Poisson p.p. and λ is its intensity

parameter.

It is not evident that such a point process exists. Later we will show how it can be constructed. Suppose

for the moment that it does exist. Here are a few immediate observations made directly from the above

deﬁnition:

•

Φ is a Poisson p.p., if and only if for every k = 1, 2, . . . and all bounded, mutually disjoint

⊂ R

for i = 1, . . . , k, (Φ(A

), . . . , Φ(A

)) is a vector of independent Poisson random

variables of parameter Λ(A

), . . . , Λ(A

), respectively. In particular, E(Φ(A)) = Λ(A), for all

•

Let W be some bounded observation window and let A

, . . . , A

be some partition of this win-

dow: A

∩ A

= ∅ for j 6= i and

= W . For all n, n

, . . . , n

∈ N with

= n,

P{Φ(A

) = n

, . . . , Φ(A

) = n

| Φ(W ) = n } =

! . . . n

Λ(W )

Λ(A

)

. (1.1)

The above conditional distribution is the multinomial distribution. This last property shows that

given there are n points in the window W , these points are independently and identically dis-

tributed (i.i.d.) in W according to the law

Λ(·)

Λ(W )

Example 1.1.2 (Locations of nodes in ad hoc networks). Assume that nodes (users), who are supposed

to constitute an ad hoc network (see Section 25.3.1 in Volume II), arrive at some given region W (a subset

of the plane or the 3D space) and independently take their locations in W at random according to some

probability distribution a(·). This means that each user chooses location dx with probability a(dx); the

uniform distribution corresponds to a “homogeneous” situation and non-uniform distributions allow us to

model e.g. various “hot spots”. Then, in view of what was said above, the conﬁguration of n users of this

ad hoc network coincides in law with the conditional distribution of the Poisson p.p. Φ with intensity Λ(dx)

proportional to a(dx) on W , given Φ(W ) = n.

Suppose now that one does not want to ﬁx a priori the exact number of nodes in the network, but only

the “average” number A(dx) of nodes per dx is known. In such a situation it is natural to assume that the

locations of nodes in W are modeled by the atoms of the (non-conditioned) Poisson process with intensity

Λ(dx) = A(dx).

The observation about conditional distribution suggests a ﬁrst construction of the Poisson p.p. in a

bounded window; sample a Poisson random variable of parameter Λ(W ) and if the outcome is n, sam-

ple n i.i.d. random variables with distribution

Λ(·)

Λ(W )

on W . The extension of the construction to the whole

One can make the story of nodes arriving to W more complete. Assuming a spatio-temporal Poisson arrival process of nodes, independent

Markovian mobility of each node and independent exponential sojourn time of each node in the network before its departure one obtains a spatial

birth-and-death process with migrations, who has Poisson p.p. as its stationary (in time) distribution; see (Serfozo 1999, Ch.9)

space R

can then be done by considering a countable partition of R

into bounded windows and an inde-

pendent generation of the Poisson p.p. in each window. We will return to this idea in Section 1.2. Before

this, we give more terminology and other characterizations of the Poisson p.p.

1.1.2 Characterizations by the Form of the Distribution

•

Say that Φ has a ﬁxed atom at x

if P{Φ({x

}) > 0 } > 0.

•

Call a p.p. Φ simple if P{Φ({x}) = 0 or 1 for all x } = 1; i.e., if with probability 1, Φ =

where the points {x

} are pairwise different.

Proposition 1.1.3. Let Φ be a Poisson p.p. with intensity measure Λ.

•

Φ has a ﬁxed atom at {x

} if and only if Λ has an atom at x

∈ R

(i.e. Λ({x

}) > 0).

•

A Poisson p.p. Φ is simple if Λ is non-atomic, i.e. admits a density with respect to Lebesgue

measure in R

Proof. The ﬁrst part is easy: use Deﬁnition 1.1.1 to write P {Φ({x

}) > 0} = 1 − e

−Λ({x

})

> 0 if and

only if Λ({x

}) > 0.

The second part can be proved using the conditioning (1.1) along the following lines. Let us take a

bounded subset A ⊂ R

P{Φ is simple in A }

∞

n=2

P{Φ(A) = n}P{all n points of Φ are different | Φ(A) = n}

∞

n=2

−Λ(A)

(Λ(A))

. . .

1(x

all different) Λ(dx

) . . . Λ(dx

) = 1 .

We conclude the proof that P{Φ is simple } = 1 by considering an increasing sequence of bounded sets

% R

and using the monotone convergence theorem.

We now give two characterizations of the Poisson p.p. based on the form of the distribution of the variable

Φ(A) for all A.

Theorem 1.1.4. Φ is a Poisson p.p. if and only if there exists a locally ﬁnite measure Λ on R

such that for

all bounded A, Φ(A) is a Poisson random variable (r. v. ) with parameter Λ(A).

Proof. We use the following fact that can be proved using moment generating functions (cf. (Daley and

Vere-Jones 1988, Lemma 2.3.I)): suppose (X, X

, . . . , X

) is a random vector with Poisson marginal dis-

tributions and such that X =

i=1

; then X

, . . . , X

are mutually independent.

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