在廣用無線通道模型上無線網路內的孤立節點數

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資訊科學與工程研究所

資訊科學與工程研究所

資訊科學與工程研究所

資訊科學與工程研究所

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在廣用無線通道模型上無線網路內的孤立節點數

The Number of Isolated Nodes in Wireless Networks with

Generalized Channel Models

研 究 生：郭哲瑋

指導教授：易志偉 教授

中 華 民 國 九 十 七 年 十 月

在廣用無線通道模型上無線網路內的孤立節點數

The Number of Isolated Nodes in Wireless Networks with

Generalized Channel Models

研 究 生：郭哲瑋 Student：Zhe-Wei Kuo

指導教授：易志偉 Advisor：Chih-Wei Yi

國 立 交 通 大 學

資 訊 科 學 與 工 程 研 究 所

碩 士 論 文

Submitted to Institute of Network Engineering College of Computer Science

National Chiao Tung University in partial Fulfillment of the Requirements

for the Degree of Master

In

Computer Science

October 2008

Hsinchu, Taiwan, Republic of China

i 

在廣用無線通道模型上無線網路內的孤立節點數

學生:郭哲瑋 指導教授：易志偉

國立交通大學

資訊科學與工程研究所

摘要

∫

ii 

關鍵字

連通性，孤立節點，普瓦松點過程，圓盤圖，對數常態分佈，傳遞耗損模型，慢速 衰減通道，快速衰減通道，中上衰減通道

The Number of Isolated Nodes in Wireless Networks with

Generalized Channel Models

Student: Zhe-Wei Kuo

Advisor: Dr. Chih-Wei Yi

National Chiao Tung University

Institute of Computer Science and Engineering

Abstract

In large-scale wireless ad hoc networks the exist of isolated nodes can be used as an indicated of network connectivity. In real world, we often use different network channel models to construct networks by considering the effect and restriction of different environment. Evaluating the reliability and connectivity base on the network channel model which be chosen. In this thesis, we will analysis the expected distribution of isolated nodes in networks.

Assume wireless nodes are deployed by a Poisson point process with density

λ

over a deployment region D=[0, ]l 2, and every node has the same transmission power that is fixed and will not change with l. Let f r be the probability of the event that ( ) two nodes have a link if they are apart from each other by r . Assume f r is a ( ) decreasing function on [0, )∞ with bounded supports. Let 2

0 ( )2 R a=

∫

π

λ= + +ζ for some constant ζ , and let ω =ζ +ln a a. In our work, we

proved that the expected number of isolated nodes in a network is e−ω

l→ ∞, the number of isolated nodes is asymptotically Poisson with mean e−ω

. In order to verity our theorem, we also simulate several popular network channel models and observe the variation of the number of network’s nodes in different l while considering the connectivity property.

Keywords

connectivity, isolated nodes, Poisson point processes, disk graphs, log-normal distribution, path-loss model, slow fading channel, fast fading channel, Nakagami fading channel.

Contents

1 Introduction 1

1.1 Wireless Ad Hoc Network . . . 1

1.2 Research Issue and Motivation . . . 2

1.3 Related Work . . . 3

2 Wireless Channels 5 2.1 The Disk Model . . . 5

2.2 The Bernoulli Model . . . 6

2.3 The Slow Fading Model . . . 6

2.4 The Fast Fading Model . . . 8

2.5 The Nakagami Fading Model . . . 10

3 The Expected Number of Isolated Nodes 14 3.1 The Expected Number . . . 15

3.2 Proof of The Expected Number Theorem . . . 15

3.3 Simulations . . . 19

4.1 Probability Distribution . . . 21

4.2 Proof of Probability Distribution Theorem . . . 22

5 Connectivity Analysis by Simulations 31 5.1 Simulation Models and Environments . . . 32

5.2 Examples of Network Topology . . . 32

5.3 PDF and CDF of Connectivity . . . 36

5.3.1 Toroidal Metrics . . . 37

5.3.2 Euclidean Metrics . . . 39

5.3.3 With Toroidal Metrics But Without Considering R2 . . . 41

List of Figures

2.1 In a 15 × 15 square region, in slow fading model, nodes that have links with the centered node are shown. . . 7 2.2 The probability density function of slow fading model. . . 7

2.3 The Cumulative density function of slow fading model. . . 8

2.4 In a 40 × 40 square region, in fast fading model, nodes that have links with the centered node are shown. . . 9 2.5 The probability density function of fast fading model. . . 10 2.6 The Cumulative density function of fast fading model. . . 10 2.7 In a 30×30 square region, in nakagami fading model, nodes that have links

with the centered node are shown. . . 11 2.8 The probability density function of nakagami fading model. . . 12 2.9 The Cumulative density function of nakagami fading model. . . 12 3.1 The average number of isolated nodes in networks with l = 500 over slow

fading channel. . . 19 3.2 The distribution of the number of isolated nodes. . . 20 4.1 θ (ρ, r) is the angle of the arc of ∂Br(x2) not contained in BR2(x1). . . 24

4.2 A annulus with center at x1. . . 25

5.1 The graph of f (r) in each link model. . . 33 5.2 In a 30 × 30 square region, given R1 = 0 and R2 = 10 in disk model, nodes

that have links with the centered node are shown. . . 34 5.3 In a 30 × 30 square region, given R1 = 0 and R2 = 11.2 in Bernoulli model,

nodes that have links with the centered node are shown. . . 34 5.4 In a 30 × 30 square region, given R1 = 0 and R2 = 13.18597 in slow fading

model, nodes that have links with the centered node are shown. . . 35 5.5 In a 30 × 30 square region, given R1 = 0 and R2 = 28.3 in Nakagami fading

model, nodes that have links with the centered node are shown. . . 36

5.6 In a 45 × 45 square region, given R1 = 0 and R2 = 38.1 in fast fading

model, nodes that have links with the centered node are shown. . . 36

5.7 An instance of networks with R1 = 0, R2 = 13.18597 and λ = 0.02 in a

square of l = 100 in slow fading model. . . 37 5.8 An instance of network with R1 = 0, R2 = 10 and λ = 0.02 in a square of

l = 100 in disk model. . . 38

5.9 An instance of network with R1 = 0, R2 = 11.2 and λ = 0.02 in a square

of l = 100 in Bernoulli model. . . 38 5.10 An instance of network with R1 = 0, R2 = 38.1 and λ = 0.02 in a square

of l = 100 in fast fading model. . . 39 5.11 An instance of network with R1 = 0, R2 = 23.8 and λ = 0.02 in a square

5.12 The cdfs of Diso, Dcon and Dth in the squares of l = 200, l = 500 and

l = 800 in Bernoulli model with R1 = 0, R2 = 11.2 and f (x) = 0.8. . . 40

5.13 The cdfs of Diso, Dcon and Dth in the squares of l = 200, l = 500 and

l = 800 over slow fading channels with R1 = 0, R2 = 13.18597 and f (r) =

1 2 ³ 1 + erf ³ 5−5 log r 2√2 ´´ . . . 40 5.14 The cdfs of Diso, Dcon and Dth in the squares of l = 200, l = 500 and

l = 800 over disk channels with R1 = 0, R2 = 13.18597 and f (r) = 1. . . . 41

5.15 The cdfs of Diso, Dconand Dthin the squares of l = 200, l = 500 and l = 800

over fast fading channels with R1 = 0, R2 = 38.1 and f (r) = exp

³

−r2

100

´

. . 41 5.16 The cdfs of Diso, Dcon and Dth in the squares of l = 200, l = 500 and

l = 800 over Nakagami fading channels with R1 = 0, R2 = 23.8 and

f (r) = ³ r2 50 + 1 ´ e−r2 50. . . 42

5.17 The cdfs of Diso, Dcon and Dth in the squares of l = 200 and l = 800 in

Bernoulli model with R1 = 0, R2 = 11.2 and f (x) = 0.8 without torus. . . 43

5.18 The cdfs of Diso, Dcon and Dth in the squares of l = 200 and l = 800 in fast

fading model with R1 = 0, R2 = 38.1 and f (x) = exp

³

−r2

100

´

without torus. 43 5.19 The cdfs of Diso, Dcon and Dth in the squares of l = 200 and l = 800 in

Disk model with R1 = 0, R2 = 10 and f (x) = 1 without torus. . . 43

5.20 The cdfs of Diso, Dcon and Dth in the squares of l = 200 and l =

800 in Slow fading model with R1 = 0, R2 = 13.18597 and f (x) =

1 2 ³ 1 + erf ³ 5−5 log r 2√2 ´´ without torus. . . 44

5.21 The cdfs of Diso, Dcon and Dth in the squares of l = 200 and l = 800 in fast

fading model with R1 = 0, R2 = 38.1 and f (x) = exp

³

−r2

100

´

List of Tables

Chapter 1

Introduction

1.1

Wireless Ad Hoc Network

In recent years, there has been wide interest in wireless ad hoc networks. A wireless ad hoc network is a collection of wireless devices distributed in a geographic region, that forms a decentralized wireless network. ’Ad hoc’ represents that each node is willing to forward data for other nodes, and so the decision on which nodes forward data is made dynamically based on the network connectivity. Each device in the network can communicate through either direct links with nearby nodes or multi-hop communication sessions with far apart nodes. Due to the lack of fixed infrastructures, wireless ad hoc networks can be flexibly deployed with low cost for various missions. Wireless ad hoc networks can be further classified by their application.

Mobile ad hoc networks : A self-configuring network of mobile routers. Nodes can move randomly and organize themselves arbitrarily. Hence, network topology may change significantly in short time period.

Wireless mesh networks : All radio nodes are static but has highly reliable and redun-dancy. When one node fails, the rest of network nodes can still communicate with each other. Node has self form and self heal characteristic.

Wireless sensor networks : A wireless sensor network consists of distributed au-tonomous sensors to acquire environment status, such as temperature, luminosity, pres-sure, and motion.

1.2

Research Issue and Motivation

In wireless ad hoc network, without infrastructures, connectivity, a vital prop-erty to many applications, becomes a major concern [1], [2], [3], [4]. If a network is not connected, the network will be composed of several independent components, and mem-bers belong to different components cannot exachange information with each other. This situation does not satistying the basic function : communication of a network.

Research on the connectivity problem often focus on two issues : the number of isolated nodes in the network and the connected property of a network. Node is called isolated if it does not have a link to other nodes. Network is connected if any two nodes in the network can communicate through one or multi-hops.

Based on the requirement of connectivity, we are interested in the number of isolated nodes in a network in our work. Thus, we have theoretical derivation to estimate the number of isolated nodes in networks and also simulate several popular network models to verify our theoretical results.

1.3

Related Work

In the past, most theoretical studies of connectivity were based on a disk model in which two nodes have a link if and only if the distance between them is no more than r [5]. This model is too simple to precisely depict behaviors of networks over wireless channels. In this work, we study the connectivity of randomly-deployed wireless networks in a generalized channel model.

In many applications, such like wireless sensor networks, wireless devices are deployed in a large volume and distributed in a random manner. Therefore, it is natural to represent wireless nodes by a random point set. No existing of isolated nodes is a prerequisite of connectivity. In 1989, it was proved in [6] that if n nodes with transmission radius

q

ln n+ξ

πn

for some constant ξ are independently distributed over a unit-area square, the network asymptotically has no isolated nodes with probability exp¡−e−ξ¢_{. In 1998, it was proved}

in [7] that if n nodes are distributed over a unit-area disk, the network is asymptotically connected if ξ → ∞ and asymptotically not connected if ξ → −∞. However, both works did not consider the possibility of node or link failures. In [8], assuming each node has the same probability p1 to be failed and each link has the same probability p2 to be down,

authors proved that if the transmission radius is given by q

ln n+ξ

πp1p2n, the total number of isolated active nodes is asymptotically Poisson with mean p1e−ξ. But in the real world,

the probability of existence of a link between two nodes is highly related to the distance between them instead of a constant. In [9], [10], [11], the impact on connectivity due to link failures caused by shadowing or fading was investigated.

deployment region is fixed and the transmission radius is expressed as a function of the number of nodes. Instead, we assume wireless nodes are represented by a Poisson point process with density λ over a deployment region D = [0, l]2. Here λ is a function of l,

but for convenience, we always suppress the parameter l of λ. Every node has the same transmission power that is fixed and won’t change with l. For any two nodes, let f (r) be the probability of the event that these two nodes have a link if they are apart from each other by r. Without loss of generality, we assume f (r) is a decreasing function on [0, ∞) with bounded supports, and exists two constants 0 ≤ R1 ≤ R2 < ∞ such that f (r) = 1

if 0 ≤ r ≤ R1 and f (r) = 0 if r > R2. To avoid tedious argument about boundary effect

and simplify the calculation, we apply the torus convention, for example, described in [12], [13], [14]. Hence, instead of Euclidean distance, we use toroidal distance [15], [16]. Let d (u, v) denote the (toroidal) distance between nodes u and v. In addition, in what follows, let a =RR2

0 f (r) 2πrdr.

As we mentioned before, the vanishment of isolated nodes is not only a prerequisite but also a good indication for network connectivity. So∇, we first derive the expected number of isolated nodes. Then, we further prove that the probability distribution of the number of isolated nodes asymptotically follows Poisson distribution. The explicit formulas given in this work allow people to control the expected number of isolated nodes by tuning the node density or even transmission power. Thus, desired level of connectivity can be expected. We run extensive simulations to verify our probability analysis. As a remark, we conjecture that no isolated nodes asymptotically imply connectivity. This conjecture is also verified through simulations.

Chapter 2

Wireless Channels

Here, we introduce several link models, including disk model, Bernoulli link model, slow fading channel model, fast fading channel model and Nakagami fading channel model. We will introduce the parameters in Chapter 5.

Let α denote the path loss exponent and Sref (dB) be the signal power measured at a

reference distance d0. For convenience, we assume the reference distance d0 = 1. Due to

the path loss, the signal measured at a distance r apart is Sref − 10α log r. Let Sthr (dB)

be the minimum signal power that a signal can be decoded.

2.1

The Disk Model

In the disk model, we assume links exist whenever signals do not decay below Sthr. Due to the monotonic properties, let RD denote the transmission radius. Based on

log-distance path loss model in [17], we have

Therefore, RD = 10

Sref −Sthr 10α .

2.2

The Bernoulli Model

In a realistic environment, signal strength is not the only factor for the existence of links. To capture the uncertainty, the Bernoulli link model is a variation of disk model in which two nodes may have a link with some constant probability p if they are within the transmission range each other. So, in the Bernoulli link model, R1 = 0 and f (r) = p.

Let RB denote the R2 in the Bernoulli link model.

2.3

The Slow Fading Model

In the slow fading channel model [17], a signal measured at distance r apart from the transmitter is with strength Sref− 10α log r − LSF. Here LSF is a log-normal random

variable due to slow fading. The pdf and cdf of a normal distribution with mean µ and standard deviation σ are

n (x) = 1 σ√2π exp à −(x − µ) 2 2σ2 ! and N (x) = 1 2 µ 1 + erf µ x − µ σ√2 ¶¶ respectively, where erf (x) = √2 π Z _x 0 e−t2 dt.

Figure. 2.1 is a slow fading model’s snapshot of distribution of links. We consider a 15 × 15 grid with center at the origin and assume there is a workstation on each grid

point. If a workstation has a link to the one at the origin, the gird is marked. Figure. 2.2 is the probability density function of slow fading distribution with different parameters σ and µ . The x-axis is the variable x of slow fading from 0 to 3, and the y-axis is the corresponding probability over the variation of variable x with each association σ, and µ = 0. And Figure. 2.3 is the cumulative density function of slow fading.

−15 −10 −5 0 5 10 15 −15 −10 −5 0 5 10 15

Figure 2.1: In a 15 × 15 square region, in slow fading model, nodes that have links with the centered node are shown.

0 0.5 1 1.5 2 2.5 3 0 0.5 1 1.5 2 2.5 3 3.5 4 σ=10 σ=3/2 σ=1 σ=1/2 σ=1/4 σ=1/8

Figure 2.2: The probability density function of slow fading model.

0 0.5 1 1.5 2 2.5 3 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 σ=3/2 σ=1 σ=1/2 σ=1/4 σ=1/8

Figure 2.3: The Cumulative density function of slow fading model. strength must be larger than Sthr. Therefore,

f (r) = Pr (LSF ≤ Sref − Sthr− 10α log r) = N (Sref − Sthr− 10α log r) = 1 2 µ 1 + erf µ Sref − Sthr− 10α log r σ√2 ¶¶ .

Let RS denote the R2 in the slow fading model. To keep the mean node degree the same,

we have πR2 D = 2π Z _RS 0 1 2 µ 1 + erf µ Sref − Sthr− 10α log r σ√2 ¶¶ rdr.

2.4

The Fast Fading Model

The Rayleigh model is usually used to characterize the fast fading phenomenon [18]. Rayleigh model assumes the amplitude of a signal will vary according to a Rayleigh distribution. The pdf and cdf of a Rayleigh distribution with standard deviation σ are

r (x) = x σ2 exp µ − x 2 2σ2 ¶ , (0 ≤ x ≤ ∞)

and R (x) = 1 − exp µ − x 2 2σ2 ¶ respectively.

Figure. 2.4 is a slow fading model’s snapshot of distribution of links. We consider a 40×40 grid with center at the origin and assume there is a workstation on each grid point. If a workstation has a link to the one at the origin, the gird is marked. Figure. 2.5 is the probability density function of slow fading distribution with different parameter σ. The x-axis is the variable x of slow fading from 0 to 10, and the y-axis is the corresponding probability over the variation of variable x with each σ. And Figure. 2.6 is the cumulative density function of slow fading.

−40 −20 0 20 40 −40 −30 −20 −10 0 10 20 30 40

Figure 2.4: In a 40 × 40 square region, in fast fading model, nodes that have links with the centered node are shown.

Here 2σ2 _{is equal to the mean square value E (x}2_{). After some derivation, we have}

f (r) = exp µ

− rα

10Sref −Sthr10 ¶

. To keep the mean node degree the same, let RF denote the

R2 in the fast fading model and satisfy

πR2 D = 2π Z _RF 0 exp à − rα 10Sref −Sthr10 ! rdr.

0 2 4 6 8 10 0 0.2 0.4 0.6 0.8 1 1.2 1.4 σ=0.5 σ=1.0 σ=2.0 σ=3.0 σ=4.0

Figure 2.5: The probability density function of fast fading model.

0 2 4 6 8 10 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 σ=0.5 σ=1.0 σ=2.0 σ=3.0 σ=4.0

Figure 2.6: The Cumulative density function of fast fading model.

2.5

The Nakagami Fading Model

The Nakagami fading model suggests the magnitude of the received envelope obeys a Nakagami distribution. The pdf of the Nakagami distribution is

p (x) = 2³ µ ω ´_{µ 1} Γ (µ)x 2µ−1_e−_ωµx2 ,

where µ is the shape parameter, denoting the severity of fading, and ω is the scale

pa-rameter, equaling E (x2_{). In the case µ = 1, the distribution reduces to a Rayleigh}

distribution. If x has a Nakagami distribution with parameters µ and ω, then x2 _{has a}

gamma distribution with shape parameter µ and scale parameter ω

a gamma function are g (x) = xµ−1 e −xω µ ³ ω µ ´_µ Γ (µ) and G (x) = γ ³ µ, x ω µ ´ Γ (µ) .

Figure. 2.7 is a slow fading model’s snapshot of distribution of links. Where consider a 30 × 30 grid with center at the origin and assume there is a workstation on each grid point. If a workstation has a link to the one at the origin, the gird is marked. Figure. 2.8 is the probability density function of slow fading distribution with different parameters ω and µ. The x-axis is the variable x of slow fading from 0 to 3, and the y-axis is the corresponding probability over the variation of variable x with each association ω and µ. And Figure. 2.9 is the cumulative density function of slow fading.

−30 −20 −10 0 10 20 30 −30 −20 −10 0 10 20 30

Figure 2.7: In a 30 × 30 square region, in nakagami fading model, nodes that have links with the centered node are shown.

The average received power at distance r is 10Sref −10α log r10 , so ω = 10

Sref −10α log r

There-0 0.5 1 1.5 2 2.5 3 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 µ=0.5, ω=1=2 µ=1, ω=1 µ=1, ω=2 µ=1, ω=3 µ=2, ω=1 µ=2, ω=2 µ=5, ω=1

Figure 2.8: The probability density function of nakagami fading model.

0 0.5 1 1.5 2 2.5 3 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 µ=0.5, ω=1=2 µ=1, ω=1 µ=1, ω=2 µ=1, ω=3 µ=2, ω=1 µ=2, ω=2 µ=5, ω=1

Figure 2.9: The Cumulative density function of nakagami fading model. fore, f (r) = Pr ³ x ≥ 10Sthr10 ´ = 1 − G ³ 10Sthr10 ´ = 1 − γ  µ, 10Sthr10 10Sref −10α log r10 µ   Γ (µ) .

Let RN denote the R2 in the Nakagami distribution, then RN satisfies

πR2_D = 2π Z _RN 0         1 − γ  µ, 10Sthr10 10Sref −10α log r10 µ   Γ (µ)         rdr.

γ ³ 2,r2 50 ´ Γ (2) = Z r2 50 0 te−tdt = − µ r2 50+ 1 ¶ e−r250 + 1, and 50 = Z _RN 0 r µ r2 50+ 1 ¶ e−r2_50dr = −1 2 ¡ 100 + R2 N ¢ e−R2N 50 + 50.

Chapter 3

The Expected Number of Isolated

Nodes

In this Chapter we will introduce our first main theorem and its proof. The disap-pearance of isolated nodes is a precondition of network connectivity. It was proved that if a random geometric graph has no isolated nodes, the graph will be asymptotically almost surely connected. In [19], the probability of no isolated nodes was considered as a tight upper bound for the probability of connectivity. The number of isolated nodes can be a good index of connectivity level. To study the connectivity of wireless networks over generalized communication channels, we investigate the number of isolated nodes. The following theorem gives the expected number of isolated nodes in a network.

Remind that in what follows, λ = 1

3.1

The Expected Number

Theorem 1 The expected number of isolated nodes is e−ω_.

The proof of Theorem 1 will be given in next section. Here we call a =RR2

0 f (r) 2πrdr

the effective transmission area, the equivalent transmission range in which signals can be clearly received. λa = ln l2_{+ ln ln l}2_{+ ξ can be regarded as average number of neighbors,}

which is the so-called mean node degree. According to Theorem 1, we can control the average number of isolated nodes by tuning the parameter ω that depends only on ξ and a.

Divide RR2

0 f (r) 2πrdr by πR22. The ratio is between R

2 1

R2

2 and 1, and is the conditional probability that two nodes have a link between them if they are within distance R2. If

f (r) = 1, the ratio is equal to 1. So, any two nodes within distance R2 always have a

link, and this is the traditional random geometric graph model.

3.2

Proof of The Expected Number Theorem

In this section, we use Palm theory [20] to derive the expected number of isolated nodes. Some lemmas will be used to help the proof of theorem. For completeness, we give the Palm theory in the form used in [20].

Lemma 2 Let ω = ξ + ln a. For any positive integer k,

λk Z (x1,··· ,xk)∈Ckk e−kλa à k Y i=1 dxi ! ∼¡e−ω¢k_.

Proof. This can be proved by straightforward calculation. λk Z (x1,··· ,xk)∈Ckk e−kλa à _k Y i=1 dxi ! = µ 1 a ¡ ln l2+ ln ln l2+ ξ¢ ¶_kÃ_Yk i=1 ¡ l2− (i − 1) πR₂2¢ ! e−k(ln l2+ln ln l2+ξ) = µ 1 a ¶_k e−kξ_· (ln l2+ ln ln l2+ ξ) k (ln l2₎k · Q_k i=1(l2− (i − 1) πR22) (l2₎k ∼ µ 1 ae −ξ ¶_k =¡e−ω¢k_.

Theorem 3 (Palm theory) Let Pλ be a Poisson point set with mean λ. Suppose j ∈ N,

and suppose h (Y, X ) is a bounded measurable function defined on all pairs of the form (Y, X ) with X a finite subset and Y a subset of X , satisfying h (Y, X ) = 0 except when Y has j elements. Then

E   X Y⊆Pλ h (Y, Pλ)   = λj j!E [h (Xj, Xj∪Pλ)] ,

where the sum on the left-hand side is over all subsets Y of the random Poisson point

set Pλ, and on the right-hand side the set Xj is a binomial process with j elements,

independent of Pλ.

Let Pλ(D) denote a Poisson point process with density λ over D. Let X be a random

point with uniform distribution over D and independent of Pλ(D). Let h (Y, X ) for Y ⊆ X

in X ; otherwise, h (Y, X ) = 0. So, we have

The expected number of isolated nodes in Pλ(D)

= X {X0_}⊆P λ(D) Pr (X0 _{is isolated) =} X {X0_}⊆P λ(D) E [h ({X0_{} , P} λ(D))] = E   X {X0_}⊆Pλ(D) h ({X0} , Pλ(D))   =¡λl2¢E [h ({X} , {X} ∪ Pλ(D))] = λl2Pr (X is isolated) . (3.1)

The equality in the 3rd line is based on Palm theory, and λl2 _{is the expected total number}

of nodes. The probability Pr (X is isolated) can be calculated by Pr (X is isolated) = Z x∈D Pr (X is isolated| X = x) Pr (X = x) dx = 1 l2 Z x∈D Pr (X is isolated| X = x) dx. (3.2)

To calculate Pr (X is isolated| X = x), we apply the partition technique used in elemen-tary calculus. The disk BR2(x) is divided by k concentric circles with center at x and radii r1 < r2 < · · · < rk = R2 into k annuli. For convenience, let r0 = 0. For 1 ≤ i ≤ k,

let 4ri = ri− ri−1, and the annulus with radii ri−1 and ri is called the i-th annulus. The

area of the i-th annulus can be approximated by 2πri4ri. If a node is in the i-th annulus,

denote the number of nodes in the i-th annulus. Then, Pr    

X doesn’t have links with nodes in the i-th annulus

¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ X = x     = ∞ X j=0 Pr    

All links between X and nodes in the i-th annulus fail

¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ Ni = j     Pr (Ni = j | X = x) ∼ ∞ X j=0 (1 − f (ri))j à (λ2πri4ri)j j! e −λ2πri4ri ! = e−λ2πri4ri à ∞ X j=0 (1 − f (ri))j (λ2πri4ri)j j! ! = e−λ2πri4ri_e(1−f (ri))λ2πri4ri _{= e}−f (ri)λ2πri4ri_. Therefore, Pr (X is isolated| X = x) = Pr         For all 1 ≤ i ≤ k, X doesn’t have links with

nodes in the i-th annulus ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ X = x         = lim k→∞ k Y i=1 Pr    

X doesn’t have links with nodes in the i-th annulus

¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ X = x     = lim k→∞ k Y i=1 e−f (ri)λ2πri4ri = lim k→∞e −λPki=1f (ri)2πri4ri = e−λR0R2f (r)2πrdr = e−λa. (3.3)

Put (3.1), (3.2), and (3.3) together, and we have

The expected number of isolated nodes in Pλ(D) = λ

Z

x∈D

e−λadx ∼ e−ω.

3.3

Simulations

Here we shows the simulation result of first main theorem. We choose slow fading as the sample network channel model. Fig. 3.1 depicts the number of isolated nodes w.r.t. ξ in networks with l = 500 over slow fading channels. In Fig. 3.1, the solid line represents expected number of isolated nodes and the dotted line represents the average number of isolated nodes with simulations. The comparison between the simulation results and theoretical results verifies the accuracy of Theorems 1

−6.5 −6 −5.5 1 2 3 4 5 ξ

number of isolated nodes

Theorem Simulation Result

Figure 3.1: The average number of isolated nodes in networks with l = 500 over slow fading channel.

Next, we run another simulation. Fig. 3.2 depicts the distribution of the number of isolated nodes over slow fading channels with ξ = −5.8. The blue line marked with triangle is the theoretical distribution. The black line marked with star and the red line marked with circle respectively are the experimental distribution corresponding to l = 500 and l = 1000 (the corresponding densities are 0.0291 and 0.339). From this simulation, we can observe that the both the behavior of distribution of l = 500 and l = 1000 is similar to Poisson distribution. Hence, we have a conjecture that the distribution of number

of isolated nodes may approximate to the distribution of Poisson distribution, and this conjecture is proved in next chapter.

0 1 2 3 4 5 6 7 8 9 10 0 0.1 0.2 0.3 0.4 0.5

number of isolated nodes

probability

l=500 l=1000 Poisson

Chapter 4

Asymptotic Distribution of the

Number of Isolated Nodes

In chapter 3, we have a conjecture by observing the simulation result. Thus in this chapter, we will have theoretical derivation to prove the conjecture. And more simulation results are shown in chapter 5 later.

4.1

Probability Distribution

Theorem 4 The number of isolated nodes is asymptotically Poisson with mean e−ω_.

Theorem 4 will be proved in Chapter 4. According to Theorem 4, we know that a network without isolated nodes is asymptotically with probability exp (−e−ω_{). This}

probability is a bound for the probability of connectivity. We conjecture that even with link failures, a network without isolated nodes is almost surely connected. If this is true, then Pr (a network is connected) ∼ exp (−e−ω_{). This conjecture will be verified through}

simulations in Chapter 5.

4.2

Proof of Probability Distribution Theorem

This section is dedicated to the proof of Theorem 4. We will apply Brun’s sieve in the form, for example, used in [21] to derive the asymptotic distribution of the number of isolated nodes.

Theorem 5 (Brun’s Sieve) Assume m (n) is a non-negative integer random variable.

Let B1, · · · , Bm(n) be events and Y be the number of Bi that hold, and

S(j)₌ X

{i1,··· ,ij}⊆{1,··· ,m(n)}

Pr¡Bi1 ∧ · · · ∧ Bij

¢ . Suppose there is a constant µ such that for every fixed j,

E£S(j)¤∼ 1

j!µ

j_.

Then Y is also asymptotically Poisson with mean µ.

For convenience, let n be the total number of nodes. Let Bi for i = 1, · · · , n be the

event that the node Xi is isolated, and Y be the number of Bi that hold. So, Y is the

total number of isolated nodes. To prove Theorem 4 by applying Brun’s Sieve, we will need to show that for every fixed k,

E   X {i1,··· ,ik}⊆{1,··· ,n} Pr (Bi1 ∧ · · · ∧ Bik)   ∼ 1 k! ¡ e−ω¢k_{. (4.1)}

To apply Palm theory (Theorem 3) to prove (4.1), let Xk = {X1, · · · , Xk} denote a

event that Xi is isolated in the network of Xk∪ Pλ(D). Then, E   X {i1,··· ,ik}⊆{1,··· ,n} Pr (Bi1 ∧ · · · ∧ Bik)   = nk k! Pr (B1∧ · · · ∧ Bk) = 1 k!n k_{Pr (B} 1∧ · · · ∧ Bk) .

Compared with (4.1), we can see if nk_{Pr (B}

1∧ · · · ∧ Bk) ∼ (e−ω)k, the proof is complete.

In the rest of this section, we are going to prove this.

Lemma 6 For any integer k ≥ 2 and (x1, x2, · · · , xk) ∈ Ckk, we have

Pr         k ^ i=1 Bi ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ X1 = x1 ... Xk = xk         = e−kλa_.

Proof. For any (x1, · · · , xk) ∈ Ckk, since BR2(x1) , · · · , BR2(xk) are pairwise disjoint,

we have Pr         k ^ i=1 Bi ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ X1 = x1 ... Xk= xk         = k Y i=1 Pr (Bi|Xi = xi) = e−kλa_.

Lemma 7 For any two integers k ≥ 2 and 1 ≤ m ≤ k −1, there exists a positive constant

c such that for any (x1, · · · , xk) ∈ Ckm,

Pr         k ^ i=1 Bi ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ X1 = x1 ... Xk= xk         ≤ e−(m+c)λa_.

Proof. First, we prove the inequality for k = 2 and m = 1. Consider the case in which d (x1, x2) ≥ 1₂R2. Let B02 be the event that X2 doesn’t have links to nodes in

BR2(x2) − BR2(x1). Then, Pr    B1 ∧ B2 ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ X1 = x1 X2 = x2     ≤ Pr (B1|X1 = x1) Pr    B02 ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ X1 = x1 X2 = x2     .

Since it is known from (3.3) that Pr (B1|X1 = x1) = e−λa, we only need to show that

there exists a positive constant c1 such that

Pr    B02 ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ X1 = x1 X2 = x2     ≤ e−c1λa. Let ρ = d (x1, x2). For any ρ ∈

£₁

2R2, R2

¤

and r ∈ [0, R2], let θ (ρ, r) denote the angle of

the arc of ∂Br(x2) not contained in BR2(x1). See Fig. 4.1. Since θ (ρ, r) is increasing

r

ρ

( , )r

1

x x₂ θ ρ

Figure 4.1: θ (ρ, r) is the angle of the arc of ∂Br(x2) not contained in BR2(x1).

w.r.t. ρ and f (r) ≥ 0 for r ∈ [0, R2], we have

Z _R2 0 f (r) θ (ρ, r) rdr ≥ Z _R2 1 2R2 f (r) θ µ 1 2R2, r ¶ rdr, and let c1 = R_R2 1 2R2f (r) θ ¡₁ 2R2, r ¢ rdr R_R2 0 f (r) θ (ρ, r) rdr .

Applying the same approach in deriving the probability Pr (X is isolated| X = x) in Chapter 3, we have Pr    B02 ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ X1 = x1 X2 = x2     = e−λ R_R2 0 f (r)θ(ρ,r)rdr ≤ e−λ R_R2 1 2 R2 f (r)θ(1 2R2,r)rdr = e−c1λR0R2f (r)2πrdr = e−c1λa_. Therefore, if 1 2R2 ≤ d (x1, x2) ≤ R2, we have Pr    B1∧ B2 ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ X1 = x1 X2 = x2     ≤ e−(1+c1)λa.

Now, consider the case in which 0 ≤ d (x1, x2) ≤ 1₂R2. For this case, we only consider

nodes in BR2(x1) and divide BR2(x1) by h concentric circles with center at x1 and radii

r1 < r2 < · · · < rh = R2 as illustrated in Fig. 4.2. Since f (r) is a decreasing function, we

x ₂ i x i r 1 ρ

have Pr    

X1 and X2 doesn’t have links

with nodes in the i-th annulus ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ X1 = x1 X2 = x2     ≤ ∞ X j=0 Ã (λ2πri4ri)j j! e −λ2πri4ri ! (1 − f (ri))j(1 − f (ri+ ρ))j = e−(f (ri)+f (ri+ρ)−f (ri)f (ri+ρ))λ2πri4ri_.

Note that the inequality still holds for annuli not fully contained in BR2(x2). So,

Pr    B1∧ B2 ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ X1 = x1 X2 = x2     ≤ lim_k→∞ k Y i=1 e−(f (ri)+f (ri+ρ)−f (ri)f (ri+ρ))λ2πri4ri = e−λR0R2(f (r)+f (r+ρ)−f (r)f (r+ρ))2πrdr = e−λ R_R2 0 f (r)2πrdr−λ R_R2 0 (f (r+ρ)−f (r)f (r+ρ))2πrdr. Since RR2

0 f (r) 2πrdr = a, if we can prove that there exists a positive constant c2 such

that RR2

0 (f (r + ρ) − f (r) f (r + ρ)) 2πrdr ≥ c2a, then this case is also proved. For any

r ∈£1 8R2,14R2 ¤ , we have f (r + ρ) ≥ f¡3 4R2 ¢ and 1 − f (r) ≥ 1 − f¡1 8R2 ¢ . Let c2 = R1 4R2 1 8R2 f¡3 4R2 ¢ ¡ 1 − f¡1 8R2 ¢¢ 2πrdr R_R2 0 f (r) 2πrdr . Then, Z _R2 0 f (r + ρ) (1 − f (r)) 2πrdr ≥ Z 1 4R2 1 8R2 f (r + ρ) (1 − f (r)) 2πrdr ≥ Z 1 4R2 1 8R2 f µ 3 4R2 ¶ µ 1 − f µ 1 8R2 ¶¶ 2πrdr = c2a. So, if 0 ≤ d (x1, x2) ≤ 1₂R2, we have Pr    B1∧ B2 ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ X1 = x1 X2 = x2     ≤ e−(1+c2)λa. Choose

there always exist two overlapping disks in the component S_{i=1,··· ,k}BR2(xi), it is not hard to see that the inequality is still correct. For any k ≥ 3 and 2 ≤ m ≤ k − 1, if

(x1, · · · , xk) ∈ Ckm, then {x1, · · · , xk} is partitioned into m sets K1, K2, · · · , Km such

that for each j = 1, · · · m, S_x∈KjBR2(x) is a maximal component. Let nj = |Kj| be the number of elements in Kj. Suppose Kj =

© xj1, · · · , xjnj ª . Then, Pr         ^ xi∈Kj Bi ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ Xj1 = xj1 ... Xjnj = xjnj         = e−λa if nj = 1; and Pr         ^ xi∈Kj Bi ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ Xj1 = xj1 ... Xjnj = xjnj         ≤ e−(1+c)λa if nj > 1.

Since there exists at least one component contains more than one node, we have

Pr         k ^ i=1 Bi ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ X1 = x1 ... Xk = xk         = m Y j=1 Pr         ^ xi∈Kj Bi ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ Xj1 = xj1 ... Xjnj = xjnj         ≤ e−(m+c)λa.

So, the lemma is proved.

Lemma 8 For any two integers k ≥ 2 and 1 ≤ m ≤ k − 1,

λk Z (x1,x2,··· ,xk)∈Ckm e−(m+c)λa à k Y i=1 dxi ! = o (1) .

Proof. First, consider m = 1. This can be validated by straightforward calculation. λk Z (x1,x2,··· ,xk)∈Ck1 e−(1+c)λa à k Y i=1 dxi ! ≤ λk_l2 à k Y i=2 ¡ π (2 (i − 1) R2)2 ¢! e−(1+c)λa = O (1)¡ln l2_{+ ln ln l}2_{+ ξ}¢k_l2_e−(1+c)(ln l2_{+ln ln l}2_+ξ) = o (1) .

Next, we consider 2 ≤ m ≤ k − 1. If (x1, · · · , xk) ∈ Ckm, and then {x1, · · · , xk} can be

partitioned into m sets K1, K2, · · · , Km such that for each j = 1, · · · m,

S

x∈KjBR2(x)

is a maximal connected component. Let nj = |Kj| be the number of elements in Kj,

and suppose Kj =

©

xj1, · · · , xjnj

ª

. For fixed k and m, the number of m-partition of

{x1, · · · , xk} are constant. Then,

λk Z (x1,x2,··· ,xk)∈Ckm e−(m+c)λa à _k Y i=1 dxi ! = O (1) e−cλa m Y j=1 à λnj Z (xj1,··· ,x_jnj)∈C_{nj 1} e−λa Ã_n j Y j=1 dxi !! = o (1) .

Because the the last equality is one nj > 1 at least. So, the lemma is proved.

Lemma 9 For any integer k ≥ 2, (λl2₎k_{Pr (B}

1∧ · · · ∧ Bk) ∼ (e−ω)k.

Proof. Let Gr(x1, · · · , xk) denote the r-disk graph over x1, · · · , xk in which there is

an edge between two nodes if and only if their distance is at most r. For any positive integers k and m with 1 ≤ m ≤ k, let Ckm denote the set of (x1, · · · , xk) ∈ Dk

sat-isfy that G2R2(x1, · · · , xk) has exactly m connected components. We partition D

Ck1, Ck2· · · , Ckk. ¡ λl2¢k_{Pr (B} 1∧ · · · ∧ Bk) =¡λl2¢k Z (x1,x2,··· ,xk)∈Dk Pr         k ^ i=1 Bi ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ X1 = x1 ... Xk = xk         Pr         X1 = x1 ... Xk= xk         Ã _k Y i=1 dxi ! = λk Z (x1,x2,··· ,xk)∈Dk Pr         k ^ i=1 Bi ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ X1 = x1 ... Xk = xk         Ã _k Y i=1 dxi ! = k X i=1 λk Z (x1,x2,··· ,xk)∈Cki Pr         k ^ i=1 Bi ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ X1 = x1 ... Xk = xk         Ã _k Y i=1 dxi ! . (4.2)

For the integral over Ckm with 1 ≤ m ≤ k − 1, according to Lemmas 7 and 8, we have

λk Z (x1,x2,··· ,xk)∈Ckm Pr         k ^ i=1 Bi ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ X1 = x1 ... Xk = xk         à _k Y i=1 dxi ! ≤ λk Z (x1,x2,··· ,xk)∈Ckm e−(m+c)λa à _k Y i=1 dxi ! = o (1) . (4.3)

For the integral over Ckk, according to Lemmas 6 and 2, we have

λk Z (x1,x2,··· ,xk)∈Ckk Pr         k ^ i=1 Bi ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ X1 = x1 ... Xk = xk         à k Y i=1 dxi ! = λk Z (x1,x2,··· ,xk)∈Ckk e−kλa à k Y i=1 dxi ! ∼¡e−ω¢k_{. (4.4)}

After putting (4.2), (4.3), and (4.4) together, and then we have ¡

λl2¢kPr (B1∧ · · · ∧ Bk) ∼

¡

Chapter 5

Connectivity Analysis by

Simulations

In this chapter, we will show the simulation results of network models that men-tioned previous. These models will be the same with Chapter 2. In order to simulate the network models, we need to figure out the node’s transmission radius of each model. Thus, we will need to set the values of each model’s parameters. According to each model’s characteristic and reflect the reality, we choose the values from each model’s suggested range by their author. And without loss of generality and fairness, we assume that each node has the same transmission power and expect each node to have the same number of network neighbors.

Section 5.1 involves each network models’ parameters we used in our simulations. In Section 5.2, several snapshots will be shown as samples of network topology. And in Section 5.3, we will use charts of probability distribution to discuss about the connectivity issue.

5.1

Simulation Models and Environments

Based on the condition of each node has the same number of network neighbors. We select suitable parameter’s value and then we can derive to obtain the node’s transmission radius in different link models. Thus, we consider transmission radius R1 and R2 here. If

the distance between any two nodes is smaller than R1, they will be directly connected. If

the distance bwtween any two nodes is more than R1 but smaller than R2, there will be a

probability to be connected or disconnected which is determined by each models’ property. If the distance between any two nodes is more than R2, they will be disconnected. So,

in this chapter, while simulating in the disk model, we let f (r) = 1 and assume α = 2, Sref = 35 dBm, and Sthr = 15 dBm, and then we can obtain R1 = R2 = 10. To have a fair

comparison in the simulation, the mean node degree in each model should be the same as that in the naive disk model. In the Bernoulli model let p = 0.8, so R1 = 0 and R2 = 11.2.

In the slow fading model, we set σ = 8 dB. Therefore, f (r) = 1

2 ³ 1 + erf ³ 5−5 log r 2√2 ´´ , R1 = 0, and R2 = 13.18597. In the fast fading model, we set f (r) = exp

³

−r2

100

´

, R1 = 0,

and R2 = 38.1. And in the Nakagami fading model, we set µ = 2 in our simulation,

f (r) = 1 − γ  2,r2₅₀ Γ(2) = ³ r2 50 + 1 ´ e−r2_{50, R} 1 = 0, and R2 = 28.3.

5.2

Examples of Network Topology

In this section, we would like to give some snapshots of network topology, and our simulation will be considered the of effect toroidal metrics in this section, where toroidal metrics is a concept that the network nodes nearby the boundary of network deployment region will still have communication probability with opposite nodes which are also nearby

the boundary of network deployment region. Then we regard two opposite boundary as the same one. If two opposite nodes are inside the transmission radius with each other, then they will have probability to direct communication which is determined by the network link model.

As defined earlier, f (r) is the probability of the event that two nodes have a link if they are apart from each other by r. Fig. 5.1 depicts f (r) in each link model introduced in Chapter 2. 0 10 20 30 40 0 0.2 0.4 0.6 0.8 1 Distance Probability Disk Bernoulli Slow fading Fast fading

Figure 5.1: The graph of f (r) in each link model.

Next, we would like to give some snapshot of our models. Unlike the snapshots we show in the chapter 2, we consider transmission radius R1 and R2 in this chapter. We

consider a grid with center at the origin. Assume there is a workstation on each grid point. Fig. 5.2 is a disk model’s snapshot of distribution of links with R1 = 10, R2 = 10

and f (x) = 1. If a workstation has a link to the one at the origin, the gird is marked. Fig. 5.3 is a Bernoulli model’s snapshot of distribution of links with R1 = 0, R2 = 11.2

and f (x) = 0.8. Because of the characteristic of Bernoulli model, each grid node inside radius R2 will has the same probability to has a connection with the original node. The

original node. −15 −10 −5 0 5 10 15 −15 −10 −5 0 5 10 15

Figure 5.2: In a 30 × 30 square region, given R1 = 0 and R2 = 10 in disk model,

nodes that have links with the centered node are shown.

−15 −10 −5 0 5 10 15 −15 −10 −5 0 5 10 15

Figure 5.3: In a 30 × 30 square region, given R1 = 0 and R2 = 11.2 in Bernoulli

model, nodes that have links with the centered node are shown.

Fig. 5.4 is a slow fading model’s snapshot of distribution of links with R1 = 0,

R2 = 13.18597 and f (r) = 1₂ ³ 1 + erf ³ 5−5 log r 2√2 ´´

. We can observe that the grid nodes are closed to the original node have higher probability to communication directly, Even the grid nodes are far away from the original node still not have very low probability to communicate with original node if they still inside the R2 of original node.

Fig. 5.5 is a snapshot of Nakagami fading model with R1 = 0, R2 = 23.8 and

f (r) = ³ r2 50+ 1 ´ e−r2

50. From the snapshot, we could obtain that marked area is highly the

aggregation in the nearby region of original node. In other words, in the Nakagami fading model, if the grid nodes are closed to the original node, it will have highly probability to have connection with original node. However, if the grid nodes are far away from the original nodes, they will have low probability to become an effectively communication

−15 −10 −5 0 5 10 15 −15 −10 −5 0 5 10 15

Figure 5.4: In a 30 × 30 square region, given R1 = 0 and R2 = 13.18597 in slow fading

model, nodes that have links with the centered node are shown. area. There is a little difference to slow fading model.

Fig. 5.6 is a snapshot of fast fading model R1 = 0, R2 = 38.1 and f (r) = exp

³

−r2

100

´ . From the figure, we can observe the effectively communication areas are also aggregated around the original node in the fast fading model.

Next, we would like to show the network connection state of each link model in a small deployment region.

Fig. 5.7 is an instance of wireless networks with R1 = 0, R2 = 13.18597 and λ = 0.02

in a square of l = 100 over slow fading channels. The solid line between two nodes represents that it exists a link between these two nodes. And the dotted line denotes there is a link between nodes in opposite sides due to the toroidal metric. Fig. 5.8 is an instance with R1 = 0, R2 = 10 and λ = 0.02 in a square of l = 100 over disk channels.

Fig. 5.9 is an instance with R1 = 0, R2 = 11.2 and λ = 0.02 in a square of l = 100

over Bernoulli channels. Fig. 5.10 is an instance with R1 = 0, R2 = 38.1 and λ = 0.02

−30 −20 −10 0 10 20 30 −30 −20 −10 0 10 20 30

Figure 5.5: In a 30 × 30 square region,

given R1 = 0 and R2 = 28.3 in

Nak-agami fading model, nodes that have links with the centered node are shown.

−40 −20 0 20 40 −40 −30 −20 −10 0 10 20 30 40

Figure 5.6: In a 45 × 45 square region, given R1 = 0 and R2 = 38.1 in fast

fad-ing model, nodes that have links with the centered node are shown.

R2 = 23.8 and λ = 0.02 in a square of l = 100 over Nakagami fading channels.

5.3

PDF and CDF of Connectivity

In this section, we will show the simulation results of probability distribution that compares with the theoretical results on connectivity issue. We will show simulations of network channel models with considering the toroidal metrics property and without the toroidal metrics property. And we will also simulate the network channel models with considering R2.

For its convenience, let random variable Diso be the node density for a network without

isolated nodes, random variable Dcon be the node density for a network being connected

and random variable Dth be the theoretical node density leading to a connected network.

0 50 100 0

50 100

Figure 5.7: An instance of networks with R1 = 0, R2 = 13.18597 and λ = 0.02 in a square

of l = 100 in slow fading model.

are no isolated nodes in the network to get Diso for 400 iterations. In a similar way, we

can get Dcon.

5.3.1

Toroidal Metrics

First, we simulate network channel models with toroidal metrics property.

Fig. 5.12 shows, in Bernoulli model, with R1 = 0, R2 = 11.2 and f (x) = 0.8, the cdfs of

Diso, Dcon and Dthin the squares of l = 200, l = 500 and l = 800 respectively. The results

in slow fading model with R1 = 0, R2 = 13.18597 and f (r) = 1₂

³ 1 + erf ³ 5−5 log r 2√2 ´´ , are shown in Fig. 5.13, for l = 200, l = 500 and l = 800. Fig.5.14 shows disk model with R1 = 0, R2 = 10 and f (r) = 1, for l = 200, l = 500 and l = 800.

Fig.5.15 shows fast fading model with R1 = 0, R2 = 38.1 and f (r) = exp

³ −r2 100 ´ , for l = 200, l = 500 and l = 800.

0 50 100 0

50 100

Figure 5.8: An instance of network with

R1 = 0, R2 = 10 and λ = 0.02 in a

square of l = 100 in disk model.

0 50 100

0 50 100

Figure 5.9: An instance of network with

R1 = 0, R2 = 11.2 and λ = 0.02 in a

square of l = 100 in Bernoulli model.

Fig.5.16 shows Nakagami fading model with R1 = 0, R2 = 23.8 and f (r) =

³

r2

50 + 1

´

e−r2_{50, for l = 200, l = 500 and l = 800.}

In disk, Nakagami fading and fast fading models, the cdfs resemble those in Bernoulli and slow fading models. Therefore, we can have the insight from these figures that the higher the value of l is, the closer the cdfs of Diso, Dcon and Dth are. These results also

verify our theoretical analysis.

For grasping the convergence of graphs, Table 5.1 lists the differences between Diso

and Dcon for p = 0.5 as l = 200, l = 500 and l = 800. It can be observed that the

gap decreases as l increases. We can expect that Pr (a network without isolated nodes) ∼ Pr (a network is connected) as l → ∞. This verifies our conjecture in Chapter 3.

0 50 100 0

50 100

Figure 5.10: An instance of network with R1 = 0, R2 = 38.1 and λ = 0.02 in

a square of l = 100 in fast fading model.

0 50 100

0 50 100

Figure 5.11: An instance of network

with R1 = 0, R2 = 23.8 and λ = 0.02

in a square of l = 100 in Nakagami fad-ing model.

In our simulations, we also simulate our each channel models without toroidal

metrics property. Fig. 5.17 shows, in Bernoulli model, with R1 = 0, R2 = 11.2 and

f (x) = 0.8, the cdfs of Diso, Dcon and Dth in the squares of l = 200 and l = 800

respectively. Fig. 5.18 shows the fast fading model, with R1 = 0, R2 = 38.1 and f (r) =

exp³− r2

100

´

, for l = 200 and l = 800.

Generally speaking, without considering the toroidal metrics means that network need to be deployed more nodes to achieve network’s connectivity. This intuition reflects on the simulations. From these simulations, we can observe that the c.d.f . on the models without torodial metrics is right shift to the c.d.f . on the models with toroidal metrics. Also, we can observe that the behavior of ’not isolated’ and ’connected’ is closer to each adapter with the growth of network scale. This indicate torus constrain may lead to need to be deployed more nodes when we discuss network connectivity, but it doesn’t

0.010 0.02 0.03 0.04 0.05 0.06 0.2 0.4 0.6 0.8 1 l=200 density probability l=500 l=800 Not isolated Connected Theoretical

Figure 5.12: The cdfs of Diso, Dcon and Dth in the squares of l = 200, l = 500 and l = 800

in Bernoulli model with R1 = 0, R2 = 11.2 and f (x) = 0.8.

0.010 0.02 0.03 0.04 0.05 0.06 0.2 0.4 0.6 0.8 1 l=200 density probability l=500 l=800 Not isolated Connected Theoretical

Figure 5.13: The cdfs of Diso, Dcon and Dth in the squares of l = 200, l = 500 and l = 800

over slow fading channels with R1 = 0, R2 = 13.18597 and f (r) = 1₂

³ 1 + erf ³ 5−5 log r 2√2 ´´ . effect the convergence relationship of ’not isolated’ and ’connected’ while network scale is growth. Fig. 5.19 shows the disk model, with R1 = 0, R2 = 10 and f (r) = 1, for l = 200

and l = 800. Fig.5.20 shows the slow fading model, with R1 = 0, R2 = 13.18597 and

f (r) = 1 2 ³ 1 + erf ³ 5−5 log r 2√2 ´´

, for l = 200 and l = 800.The convergent property in both models also holds.

0.010 0.02 0.03 0.04 0.05 0.06 0.2 0.4 0.6 0.8 1 l=200 density probability l=500 l=800 Not isolated Connected Theoretical

Figure 5.14: The cdfs of Diso, Dcon and Dth in the squares of l = 200, l = 500 and l = 800

over disk channels with R1 = 0, R2 = 13.18597 and f (r) = 1.

0.010 0.02 0.03 0.04 0.05 0.06 0.2 0.4 0.6 0.8 1 l=200 density probability l=500 l=800 Not isolated Connected Theoretical

Figure 5.15: The cdfs of Diso, Dcon and Dth in the squares of l = 200, l = 500 and l = 800

over fast fading channels with R1 = 0, R2 = 38.1 and f (r) = exp

³

−r2

100

´ .

5.3.3

With Toroidal Metrics But Without Considering R

In the above simulations, we give bounded transmission radius R2 in each models

and force communication probability to zero while two nodes’ distance exceeds R2.

How-ever, in the fast fading channel model and slow fading channel model, the radio signal strength decays with distance, therefore. There may be weakly radio signal while two

nodes are far away more than R2. Next we simulate model without consider R2. Fig.

5.21 shows the fast fading model, with R1 = 0, R2 = 38.1 and f (r) = exp

³

−r2

100

´ , for

0.010 0.02 0.03 0.04 0.05 0.06 0.2 0.4 0.6 0.8 1 l=200 density probability l=500 l=800 Not isolated Connected Theoretical

Figure 5.16: The cdfs of Diso, Dcon and Dth in the squares of l = 200, l = 500 and l = 800

over Nakagami fading channels with R1 = 0, R2 = 23.8 and f (r) =

³

r2

50+ 1

´

e−r2_50.

Table 5.1: The density differences between Diso and Dcon for p = 0.5.

l = 200 l = 500 l = 800

Disk model 0.0022 0.00170 0.001100

Bernoulli model 0.0013 0.00038 0.000338

Slow fading model 0.0006 0.00011 0.000016

Fast fading model 0.0002 0.00013 0.000038

l = 200, l = 500and l = 800 without consider R2. Compared with the above simulation

results, we can observe that removing R2 constraint doesn’t cause drastic change on the number of deploy nodes. Although it still has weakly signal strength while two nodes ’ distance exceeds R2, the signal strength is not strong enough to support two nodes’

0.010 0.02 0.03 0.04 0.05 0.06 0.2 0.4 0.6 0.8 1 l=200 density probability l=800 Not isolated Connected Theoretical

Figure 5.17: The cdfs of Diso, Dcon and Dth in the squares of l = 200 and l = 800 in

Bernoulli model with R1 = 0, R2 = 11.2 and f (x) = 0.8 without torus.

0.010 0.02 0.03 0.04 0.05 0.06 0.2 0.4 0.6 0.8 1 l=200 density probability l=800 Not isolated Connected Theoretical

Figure 5.18: The cdfs of Diso, Dcon and Dth in the squares of l = 200 and l = 800 in fast

fading model with R1 = 0, R2 = 38.1 and f (x) = exp

³ − r2 100 ´ without torus. 0.010 0.02 0.03 0.04 0.05 0.06 0.2 0.4 0.6 0.8 1 l=800 density probability l=200 Not isolated Connected Theoretical

Figure 5.19: The cdfs of Diso, Dcon and Dth in the squares of l = 200 and l = 800 in Disk

0.010 0.02 0.03 0.04 0.05 0.06 0.2 0.4 0.6 0.8 1 l=800 density probability l=200 Not isolated Connected Theoretical

Figure 5.20: The cdfs of Diso, Dcon and Dth in the squares of l = 200 and l = 800 in Slow

fading model with R1 = 0, R2 = 13.18597 and f (x) = 1₂

³ 1 + erf ³ 5−5 log r 2√2 ´´ without torus. 0.010 0.02 0.03 0.04 0.05 0.06 0.2 0.4 0.6 0.8 1 l=200 density probability l=500 l=800 Not isolated Connected Theoretical

Figure 5.21: The cdfs of Diso, Dcon and Dth in the squares of l = 200 and l = 800 in fast

fading model with R1 = 0, R2 = 38.1 and f (x) = exp

³

− r2

100

´

Chapter 6

Conclusions

In this work, we assume that the distribution of wireless nodes is modeled by a Poisson point process with mean density over a deployment region instead of a uniform deployment. Also, a generalized link model is proposed. By deploying nodes with ade-quate mean density, we can get the expected number of isolated nodes and the distribution of the number of isolated nodes. Theoretical derivation and simulations are elaborately demonstrated. In conclusion, the expected number of isolated nodes is e−ω _{with certain}

constant ω and the distribution of the number of isolated nodes is asymptotically Poisson

with mean e−ω_{. This information can help developer to design a wireless ad hoc and}

sensor network or to make decisions based on the demand of performance. Hopefully, the future research hopefully can loosen the restrictions on the deployment method and the deployment region.