Wireless sensor network scheduling schemes

Wireless Sensor Network

Scheduling Schemes

Lakehead

UNIVERSITY

Contents

List of Tables ... i List of Figures ... ii Abstract ... v Acknowledgements ... vi Chapter 1 Introduction ... 1

Chapter 2 Research Background ... 4

Chapter 3 Reviews of Sensor Scheduling Schemes for WSN ... 17

Chapter 4 New Proposal: A Clique Based Sensor Scheduling Scheme with Guaranteed Connectivity ... 48

Chapter 5 Second Proposal: An Efficient Node Scheduling Scheme Based on Combinatorial Assignment Code ... 81

Chapter 6 Conclusions and Future Work ... 98

List of Figures

𝑟_{𝑡𝑟𝑎𝑛𝑠} = 20𝑚 𝑟_{𝑠𝑒𝑛𝑠𝑖𝑛𝑔} = 14.4𝑚 𝑟𝑡𝑟𝑎𝑛𝑠 = 15𝑚 𝑟_{𝑠𝑒𝑛𝑠𝑖𝑛𝑔} = 10𝑚 𝑅 = {𝑟1 ,𝑟2,𝑟3} 𝑆 = {𝑠₁, 𝑠_1,𝑠₁, 𝑠₁} 𝑆1 = {𝑠1, 𝑠2} 𝑆2 = {𝑠2, 𝑠3} 𝑆₃ = {𝑠₁, 𝑠₃} 𝑆₄ = {𝑠₄}

ℵ 𝑁_𝑔 = 20 ℵ 𝑁_𝑔 = 20 ℵ 𝑁_𝑔 = 20 𝑁_𝑔 ℵ ℵ 𝑁_𝑔 = 20

Abstract

Chapter 1

Chapter 2

Research Background

2.1

Wireless Sensor Deployment

𝑆_{𝑢𝑛𝑖𝑡} 𝜋×𝑟𝑠2 𝑆𝑢𝑛𝑖𝑡 𝑟_𝑠 𝑑 ≥ 𝑆𝑢𝑛𝑖𝑡 𝜋×𝑟𝑠2 𝑛 = 150, 175, 200, 225, . . . , 400 100 × 100 𝑚2 20𝑚 15𝑚 14.4𝑚 10𝑚 𝑛 = 200 𝑟_{𝑡𝑟𝑎𝑛𝑠} = 20𝑚 𝑟_{𝑠𝑒𝑛𝑠𝑖𝑛𝑔} = 14.4𝑚 𝑟_{𝑡𝑟𝑎𝑛𝑠} = 15𝑚 𝑟_{𝑠𝑒𝑛𝑠𝑖𝑛𝑔} = 10𝑚

𝑟_{𝑡𝑟𝑎𝑛𝑠} = 20𝑚 𝑟_{𝑠𝑒𝑛𝑠𝑖𝑛𝑔}= 14.4𝑚

𝑟_{𝑡𝑟𝑎𝑛𝑠} = 15𝑚 𝑟_{𝑠𝑒𝑛𝑠𝑖𝑛𝑔} = 10𝑚

𝑃_𝐵 𝐼 𝐹 𝐴 𝑆 𝑃𝐵 𝐼 𝐹 𝑝 𝑃_𝐵 𝑝 𝑃𝐵 𝑆 𝑃_𝐵 𝑃_𝐵 𝑆 𝐴 𝐴 𝐴 𝐼 𝐹

𝑃𝑆 𝑃𝑆 𝑃_𝐵 𝑃_𝑆 𝑆 𝑃𝑆 𝑂(𝑛 𝑙𝑜𝑔 𝑛) 𝑂(𝑚) 𝑂(𝑛) 𝑂(𝑛2₎ 𝑂(𝑙𝑜𝑔 𝑟𝑎𝑛𝑔𝑒) 𝑂(𝑛2 _{𝑙𝑜𝑔 𝑛)} 𝑂(𝑛 𝑙𝑜𝑔 𝑛) 𝑃_𝐵 𝑃_𝑆 𝐼 𝐹 𝐼 𝐹 𝑃_𝐵

2.3

Location Awareness of Wireless Sensor

Network

2.4

Energy Consumption Analysis for Wireless

Sensor Network

   

2.5

Power-saving Strategies for Wireless

Sensor network

Chapter 3

Reviews of Sensor Scheduling

Schemes for WSN

3.2

Energy -Efficient Target Coverage

Scheduling Scheme for WSN

3.2.1 MSC Problem Definition 𝑛 𝑠₁, 𝑠₂, … … , 𝑠_𝑛 𝑟1, 𝑟2, … … , 𝑟𝑚 𝑠₁, 𝑠₂, 𝑠₃, 𝑠₄ 𝑟1, 𝑟2, 𝑟3 𝑠₁ = {𝑟₁, 𝑟₂} 𝑠₂ = {𝑟₂, 𝑟₃} 𝑠₃ = {𝑟₁, 𝑟₃} 𝑠₄ = {𝑟₁, 𝑟₂, 𝑟₃} 𝑅 = {𝑟₁, 𝑟₂, 𝑟₃} 𝑆 = {𝑠₁, 𝑠₂, 𝑠₃, 𝑠₄} 𝑆1 = {𝑠1, 𝑠2} 𝑆₂ = {𝑠₃, 𝑠₄} 𝑆1 = {𝑠1, 𝑠2}

𝑆2 = {𝑠2, 𝑠3} 𝑆3 = {𝑠1, 𝑠3} 𝑆4 = {𝑠4} 𝐶 𝑅 𝑆1, … , 𝑆𝑝 𝑡1, … , 𝑡𝑝 0 ≤ 𝑡_𝑖 ≤ 11 𝑡₁+ … + 𝑡_𝑝 𝑠 𝐶 𝑠 𝑆₁, … , 𝑆_𝑝 𝑆₁= {𝑠₁, 𝑠₂} 𝑆₂ = {𝑠₂, 𝑠₃} 𝑆3 = {𝑠1, 𝑠3} 𝑆4 = {𝑠4} 𝑆₁= {𝑠₁, 𝑠₂} 𝑆₂= {𝑠₂, 𝑠₃} 𝑆₃= {𝑠₁, 𝑠₃} 𝑆₄= {𝑠₄} C S₁, … , S_p S_i , i = 1, … , p

t1+ … + tp

t_j , j = 1, … , p S_j

3.2.2 Solutions to Compute the Maximum Set Covers and

Runtime Complexity Analysis

𝑂(𝑝3_𝑛3₎

𝑂(𝑖𝑚2_{𝑛) 𝑖}

𝑖 𝑑/𝑤

𝑤

𝑑 < 𝑛 𝑂(𝑑𝑚2_𝑛)

3.2.3 Simulation Result for the Heuristics

  

𝑟 = 250𝑚

  

3.2.4 Limitation of the Target Coverage Scheme

3.3

A Coverage-preserving Node Scheduling

Scheme for Large WSN

𝑖 𝑆(𝑖) 𝑖 𝑁 𝑖 = 𝑛 ∈ ℵ 𝑑 𝑖, 𝑗 ≤ 𝑟, 𝑛 ≠ 𝑖 } ℵ 𝑑 𝑖, 𝑗 𝑖 𝑗 𝑖 𝑆 𝑗 𝑗 ∈𝑁 𝑖 ⊇ 𝑆(𝑖) 𝑗 ∈𝑁 𝑖 ( 𝑆 𝑗 ∩ 𝑆 𝑖 ) ⊇ 𝑆(𝑖) 𝑆 𝑗 ∩ 𝑆 𝑖 𝑖 𝑖 𝑗 𝑖 𝑗 𝑖 𝑆_{𝑗 →𝑖} 𝑆_{𝑗 →𝑖} 𝑗 ∈𝑁 𝑖 ⊇ 𝑆(𝑖) 𝑗 ∈𝑁 𝑖 ( 𝑆 𝑗 ∩ 𝑆 𝑖 ) ⊇ 𝑆(𝑖) 𝑆_{𝑗 →𝑖} 𝑗 ∈𝑁 𝑖 ⊇ 𝑆(𝑖)

𝑆𝑗 →𝑖 𝑗 ∈𝑁 𝑖 ⊇ 𝑆(𝑖) 𝑖 𝜃𝑗 →𝑖 = 2𝑎𝑟𝑐𝑐𝑜𝑠 𝑑(𝑖,𝑗 )_2𝑟 0 < 𝑑 𝐼, 𝑗 ≤ 𝑟 𝜃_{𝑗 →𝑖} [120, 180)

3.3.2 Sponsored Coverage Calculation for extension model

𝜃_{𝑗 →𝑖} ∅_{𝑗 →𝑖}

𝜃_{𝑗 →𝑖} [120, 180)

𝜃_{𝑗 →𝑖} ∅𝑗 →𝑖

𝑊 1/𝑊 𝑊 𝑇_𝑊 𝑇_𝑊 3.3.4 Simulation results

ℵ 𝑁𝑔 = 20

ℵ 𝑁_𝑔 = 20

ℵ 𝑁_𝑔 = 20

3.3.5 Limitations of the Coverage-Preserving Node Scheduling

Scheme

3.4

A joint Scheduling Scheme: Random

Coverage with Guaranteed Connectivity

3.4.1 Network Model

3.4.3 Joint Scheduling: Random Coverage with Guaranteed Connectivity

𝑖

𝑖

𝑖

200𝑚 × 200𝑚

10𝑚 𝑚𝑠

3.4.5 Simulation evaluation

10𝑚

Chapter 4

New Proposal: A Clique Based

Sensor Scheduling Scheme with

Guaranteed Connectivity

𝑘(𝑘 ≤ 𝑚) {0, 1, … , 𝑘 − 1}

𝑚 𝑚 𝑚 𝑚 𝑘(𝑘 ≤ 𝑚) {0, 1, … , 𝑘 − 1} 𝑚

4.1

Network Model and Definitions

4.1.1 Network Model 𝑟_𝑠 𝑟_𝑐 2𝑟_𝑠 00 … 001 n 11 … 111 n 00 … 000 n 00 … 001 n 11 … 111 n 00 … 000 n 4.1.2 Definitions 𝑡 𝑝₁, 𝑝₂, … , 𝑝_𝑡 { 𝑝1, 𝑝2, … , 𝑝𝑡} 𝑡 𝐶1 𝐶2 𝐶1 𝐶₂ 𝐶₁ ⊆ 𝐶₂ 𝐶₁ ∈ 𝐶₂ 𝐶₁ 𝐶₂ 𝐶₁ 𝐶₂ 𝐶₁ 𝐶₂

𝑝1 𝑝2 𝑝1

𝑝2 𝑝1 > 𝑝2

4.2

Clique Based Node Scheduling Algorithm

𝑟_𝑐 ≥ 2𝑟_𝑠 𝑝 𝑡 𝑡 𝑝 𝑝₁, 𝑝₂, … , 𝑝_𝑡 𝑝_𝑗 𝑟_𝑠 𝑝𝑗 ∈ 𝑝1, 𝑝2, … , 𝑝𝑡 𝑑 𝑝𝑗, 𝑝 ≤ 𝑟𝑠 𝑑 𝑝𝑖, 𝑝𝑗 ≤ 𝑑 𝑝𝑖, 𝑝 + 𝑑 𝑝𝑗, 𝑝 ≤ 2𝑟𝑠 ≤ 𝑟𝑐 𝑝𝑖 𝑝𝑗 { 𝑝1, 𝑝2, … , 𝑝𝑡 } 𝑝_𝑖 𝑝_𝑗 t ≥ 𝑚 t ≥ 𝑚 𝑝1, 𝑝2, … , 𝑝𝑡 𝑃1, 𝑃2, … , 𝑃𝑡 𝑘 𝑘 ≤ 𝑚 { 0, 1, … , 𝑘 − 1 } 𝑃₁ 𝑚𝑖𝑛 { 𝑃₁, 𝑃₂, … , 𝑃_𝑡 } 𝑝1 { 𝑝1, 𝑝2, … , 𝑝𝑡 } 𝑎 (𝑎 < 𝑡) { 𝑝₁, 𝑝₂, … , 𝑝_𝑎 } { 𝑝₁, 𝑝₂, … , 𝑝_𝑡 } { 𝑔₁ , 𝑔₂ , … , 𝑔_𝑎 } 𝑔1, 𝑔2, … , 𝑔𝑏 { 𝑔1 , 𝑔2 , … , 𝑔𝑎 } 𝑏 ≤ 𝑎 𝑏 = 𝑘 𝑝_𝑗 ∈ { 𝑝_𝑎+1, 𝑝_𝑎+2, … , 𝑝_𝑡 } 𝑝_𝑗 𝑗 ∈ { 0, 1, … , 𝑘 − 1 } 𝑗 𝑝𝑗 𝑏 < 𝑘

𝑈 = 0, 1, … , 𝑘 − 1 \ 𝑔1, 𝑔2, … , 𝑔𝑏 = 𝑢0, 𝑢1, … , 𝑢𝑘−𝑏−1 𝑡 − 𝑎 ≥ 𝑘 − 𝑏 𝑝1 𝑡 − 𝑎 − (𝑘 − 𝑏) 𝑉 = 𝑣0, 𝑢1, … , 𝑢 𝑡−𝑎 −(𝑘−𝑏)−1 { 0, 1, … , 𝑘 − 1 } 𝑈 ∪ 𝑉 𝑝𝑎+1, 𝑝𝑎+2, … , 𝑝𝑡 𝑡 − 𝑎 < 𝑘 − 𝑏 𝑝₁ 𝑡 − 𝑎 𝑣₀, 𝑢₁, … , 𝑢_{𝑡−𝑎−1} 𝑈 𝑝𝑎+1, 𝑝𝑎+2, … , 𝑝𝑡 { 𝑝₁, 𝑝₂, … , 𝑝_𝑎 } { 𝑝₁, 𝑝₂, … , 𝑝_𝑡 } 𝑡 − 𝑎 𝑝_𝑎+1, 𝑝_𝑎+2, … , 𝑝_𝑡 𝑏 { 𝑝₁, 𝑝₂, … , 𝑝_𝑎 } 𝑈 { 𝑝₁, 𝑝₂, … , 𝑝_𝑎 } 𝑡 − 𝑎 ≥ 𝑘 − 𝑏 𝑡 − 𝑎 < 𝑘 − 𝑏

𝑘 = 4 {0, 1, 2, 3} {001,010,011,011,101,111} 𝑎 = 𝑏 = 0 𝑈 = {0, 1, 2, 3} {0, 1, 2, 3} 𝑉 = {3} 𝑈 ∪ 𝑉 𝑘 𝑡 ≥ 𝑘 𝑡 𝑝₁, 𝑝₂, … , 𝑝_𝑡 { 𝑝₁, 𝑝₂, … , 𝑝_𝑡 } 𝑗 𝑗 𝐼𝐷𝑁𝑁 ∪ {𝑁𝑁𝐿𝑁} {001,011,101,111} {𝐼𝐷𝑁𝑁} ∪ {𝑁𝑁𝐿𝑁}

001,010,011,101,110,111 ∩ 001,010,011,101,111 ∩ 001,011,101,111 ∩ 001,011,101,111 = {001,011,101,111} 𝑚 𝐼𝐷𝑁𝑁 ∪ {𝑁𝑁𝐿𝑁} 𝑡 ≥ 𝑚 𝑝1 𝑡 𝑠 𝑚 𝑠 𝑚 𝑡

𝑡 ≥ 𝑚 𝑝2, 𝑝3, … , 𝑝𝑠 𝑝₁ 𝑃₁, 𝑃₂, … , 𝑃_𝑠 𝑝1, 𝑝2, … , 𝑝𝑠 𝑁1, 𝑁2, … , 𝑁𝑠 𝑁₁ , 𝑁 , … , 𝑁2 𝑠 𝐼𝐷𝑁𝑁 ∪ {𝑁𝑁𝐿𝑁} 𝑝1, 𝑝2, … , 𝑝𝑠 𝑡 = 𝑠; 𝑡 ≥ 𝑚; 𝑡 − − 𝑝1 𝑠 − 1_{𝑡 − 1} 𝑆1, 𝑆2, … , 𝑆 𝑠 − 1 𝑡 − 1 { 𝑝₁, 𝑝₂, … , 𝑝_𝑠 } 𝐶𝑥 𝑥 (𝑡 < 𝑥 ≤ 𝑠) 𝑝₁ 𝑗 = 1; 𝑗 ≤ 𝑠 − 1_{𝑡 − 1} ; 𝑗 + + 𝑥 ∈ 𝐶𝑥 𝑆𝑗 ⊆ 𝑥 𝑆𝑗 = 𝑆𝑗 ∪ {𝑝1} 𝑁𝑖 𝑖∈𝑆𝑗 𝐶 = 𝑡 List 4.2-1 110, 010, 011, 101, 111 𝑠 = 6 𝑚 = 3 {001,011,111,101} {110, 010, 011, 101, 111

𝐼𝐷𝑁𝑁 ∪ {𝑁𝑁𝐿𝑁} 𝑛 𝑘 𝑘 ≤ 𝑚 { 0, 1, … , 𝑘 − 1 } 𝑝_𝑖 (𝑚 ≤ 𝑡 ≤ 𝑠) 𝑠 𝑝_𝑖 𝑡 𝐶_𝑡 𝑡 = 𝑠; 𝑡 ≥ 𝑚; 𝑡 − − 𝑡 ∈ 𝐶𝑡 𝑝𝑖 𝑡 𝑝_𝑖 𝐶_𝑡 { 0, 1, … , 𝑘 − 1 } 𝑝₁ 𝑝1 𝑝₁ 𝑝1 𝑝₁ 𝑘 (𝑘 ≤ 𝑚) 𝑝_𝑖 𝑝_𝑗 𝑝𝑖 𝑝𝑖 𝑝𝑖 𝐻𝑖 𝐻𝑗 𝐻𝑖 = 𝐻𝑗 + 1 𝑝𝑗 𝑝𝑖 𝑝𝑖

𝑝𝑗 𝐻𝑖 = 𝐻𝑗 𝑝𝑗 𝑝_𝑖 𝑝_𝑗 𝑝_𝑖 𝑝_𝑗 𝑝_𝑖 𝑝_𝑖 𝑝_𝑗 𝑝_𝑖 𝐿_𝑖 𝑔 ∈ 𝐿_𝑖 𝑝_𝑖 𝑝𝑗 𝑝_𝑗 𝑝𝑖 𝑝_𝑗 𝑝_𝑗 𝑝𝑗 𝑝_𝑖 𝑝𝑗 𝑝𝑖 𝑝𝑗 𝑔 𝑝𝑥 𝑝_𝑥

𝐻𝑖 𝐻_𝑖 𝐻𝑖 𝑝_𝑖 𝑝𝑖 𝑝_𝑖 𝑝_𝑖 𝑝𝑗 𝑝𝑗 𝑝_𝑖 𝑝𝑗 𝑝𝑖 𝑝𝑖 𝑝𝑜1 𝑝𝑖 𝑝𝑜1 𝑝𝑥 𝑝𝑥 𝑝_𝑜2 𝑝_𝑜1 𝑝_𝑜𝑛 𝐻_𝑖 𝐻_𝑖 = 𝑑 + 1 𝑝_𝑖 𝑝_𝑖 𝑝_𝑖 𝑑 𝑝𝑗 𝑝𝑖 𝑑 𝑝𝑖 𝑝_𝑖 𝑝_𝑜1 𝑝𝑖 𝑝𝑜1

𝑚 𝑘 0,1, … , 𝑘 − 1 𝑘 ≤ 𝑚 𝑝𝑖 𝐼𝑇𝑖 𝑘 𝐼𝑇𝑖 𝑘 𝑘 𝑝𝑖 TSHS𝑗 𝑘 𝑝𝑗 TSHS𝑗 TSHS𝑖 𝐼𝑇𝑖 TSHS𝑖 𝑝𝑗 𝐼𝑇𝑖 TSHS𝑖 TSHS𝑖 TSHS𝑗 𝑝𝑗 𝑝𝑖 𝑝𝑖 𝐼𝑇𝑖 𝑡 𝑡 ≡ 𝑔 𝑘 𝑔

4.3

Performance Analysis

4.3.1 Coverage Performance 𝑡 (𝑘 ≤ 𝑡) 𝑗 ( 𝑗 < 𝑘) 1 𝑘𝑡 −1 𝑖 𝑗 −1 𝑖=0 𝑘 𝑗 𝑗 𝑖 𝑗 − 𝑖 𝑡 𝑘𝑡 _𝑘 𝑡 𝑗 −1 𝑖 𝑗 −1 𝑖=0 𝑗 𝑖 𝑗 − 𝑖 𝑡 𝑘 𝑗 𝑗 𝑘 𝑃 𝑡, 𝑘, 𝑗 = 1 𝑘𝑡 −1 𝑖 𝑗 −1 𝑖=0 𝑘 𝑗 𝑗 𝑖 𝑗 − 𝑖 𝑡 (1) 𝑃 𝑡, 𝑘, 𝑗 min ⁡{𝑘,𝑡} 𝑗 =1 = 1 𝑆 𝑛, 𝑗 = 1 𝑗! −1 𝑖 𝑗 −1 𝑖=0 𝑗 𝑖 𝑗 − 𝑖 𝑛 ,

𝑃 𝑡, 𝑘, 𝑗 = _𝑘𝑗 !_𝑡 𝑘_𝑗 𝑆(𝑡, 𝑗) 𝑆(𝑛 + 1, 𝑗) = 𝑆(𝑛, 𝑗 − 1) + 𝑗𝑆(𝑛, 𝑗) (2) 𝑚 𝑘 (𝑘 ≤ 𝑚) 𝑝 𝑡(𝑡 ≥ 𝑚) 𝑃_{𝐶𝐶𝑀𝑆} = 1 𝑖𝑓 𝑡₂ ≥ 𝑘; 1 𝑘𝑡1 (−1) 𝑖 𝑗 −1 𝑖=0 min {𝑘,𝑡1} 𝑗 =(𝑘−𝑡2) 𝑘 𝑗 𝑗 𝑖 𝑗 − 𝑖 𝑡1 𝑖𝑓 𝑡2 < 𝑘; 𝑡₁+ 𝑡₂ = 𝑡 𝑡₁ 𝑡 𝑝 𝑡 𝑡 𝑡 𝑡 𝑡1 𝑡₂ ≥ 𝑘 𝑘 𝑡2 {0, 1, … , 𝑘 − 1} 𝑘 𝑘 𝑡 𝑡2 < 𝑘 𝑡1 𝑗 1 𝑘𝑡 −1 𝑖 𝑗 −1 𝑖=0 𝑘 𝑗 𝑗 𝑖 𝑗 − 𝑖 𝑡 𝑃 𝑡₁, 𝑘, 𝑗 = 1 𝑘𝑡1 −1 𝑖 𝑗 −1 𝑖=0 𝑘 𝑗 𝑗 𝑖 𝑗 − 𝑖 𝑡1 𝑃 𝑡1, 𝑘, 𝑗 min {𝑘,𝑡1} 𝑗 =(𝑘−𝑡2) 𝑡2

{0, 1, … , 𝑘 − 1} 𝑡 − 𝑐𝑙𝑖𝑞𝑢𝑒 (𝑡 ≥ 𝑘) 𝑃𝑅𝑆𝐺𝐶 𝑘 𝑘 (𝑘 ≤ 𝑚) {0, 1, … , 𝑘 − 1} 𝑃_{𝑅𝐶𝐶𝑀𝑆} − 𝑃_{𝑅𝑆𝐺𝐶} 𝑃(𝑡,𝑘,𝑗)𝑘−1 𝑗 =1 𝑖𝑓 𝑡2 ≥ 𝑘; 1 𝑘𝑡−𝑖( 𝑘 − 𝑖 − 1 𝑘 − 𝑖 − 1 ! 𝑘 𝑘 − 𝑖 − 1 𝑆(𝑡 − 𝑖 − 1, 𝑘 − 𝑖 − 1)) 𝑡2−1 𝑖=0 𝑖𝑓 𝑡₂ < 𝑘; 𝑃𝐶𝐶𝑀𝑆 = 1 𝑖𝑓 𝑡₂ ≥ 𝑘; 1 𝑘𝑡1 (−1) 𝑖 𝑗 −1 𝑖=0 min {𝑘,𝑡1} 𝑗 =(𝑘−𝑡2) 𝑘 𝑗 𝑗 𝑖 𝑗 − 𝑖 𝑡1 𝑖𝑓 𝑡2 < 𝑘; 𝑃𝑅𝑆𝐺𝐶 = 1 𝑘𝑡 −1 𝑖 𝑘−1 𝑖=0 𝑘 𝑖 𝑘 − 𝑖 𝑡 𝑡₂ ≥ 𝑘 𝑃_{𝑅𝑆𝐺𝐶} < 𝑃_{𝐶𝐶𝑀𝑆} 𝑃_{𝐶𝐶𝑀𝑆} = 1 𝑃_{𝑅𝑆𝐺𝐶} < 1 𝑡₂ < 𝑘 𝑃𝑅𝑆𝐺𝐶 = 1 − 𝑃(𝑡, 𝑘, 𝑗) 𝑘−1 𝑗 =1

𝑃_{𝐶𝐶𝑀𝑆} = 1 − 𝑃(𝑡 − 𝑡₂, 𝑘, 𝑗) 𝑘−𝑡2−1 𝑗 =1 𝑓 𝑥 = 𝑃(𝑡 − 𝑥, 𝑘, 𝑗) 𝑘−𝑥−1 𝑗 =1 = 𝑗! 𝑘𝑡−𝑥 𝑘−𝑥−1 𝑗 =1 𝑘 𝑗 𝑆(𝑡 − 𝑥, 𝑗) 𝑃_{𝑅𝑆𝐺𝐶} = 1 − 𝑓(0) 𝑃_{𝐶𝐶𝑀𝑆} = 1 − 𝑓(𝑡₂) 𝑓(0) − 𝑓(𝑡₂) 𝑏 𝑘𝑡−𝑏_{𝑓 𝑏 = 𝑗!} 𝑘 𝑗 𝑘−𝑏−1 𝑗 =1 𝑆 𝑡 − 𝑏, 𝑗 = 𝑘 + 𝑗! 𝑘 𝑗 𝑘−𝑏−1 𝑗 =2 𝑆 𝑡 − 𝑏, 𝑗 = 𝑘 + 𝑗! 𝑘 𝑗 𝑘−𝑏−1 𝑗 =2 (𝑆 𝑡 − 𝑏 − 1, 𝑗 − 1 + 𝑗𝑆(𝑡 − 𝑏 − 1)) = 𝑘 + 𝑗! 𝑘 𝑗 𝑘−𝑏−1 𝑗 =2 𝑆 𝑡 − 𝑏 − 1, 𝑗 − 1 + 𝑗! 𝑘 𝑗 𝑘−𝑏−1 𝑗 =2 𝑗𝑆(𝑡 − 𝑏 − 1, 𝑗) = 𝑘 + (𝑗 + 1)! 𝑘 𝑗 + 1 𝑘−𝑏−2 𝑗 =2 𝑆 𝑡 − 𝑏 − 1, 𝑗 + 𝑗! 𝑘 𝑗 𝑘−𝑏−1 𝑗 =2 𝑗𝑆(𝑡 − 𝑏 − 1, 𝑗) = (𝑗 + 1)! 𝑘 𝑗 + 1 𝑘−𝑏−2 𝑗 =2 𝑆 𝑡 − 𝑏 − 1, 𝑗 + 𝑗! 𝑘 𝑗 𝑘−𝑏−1 𝑗 =1 𝑗𝑆(𝑡 − 𝑏 − 1, 𝑗) = 𝑗! (𝑘 − 𝑗) 𝑘 𝑗 𝑘−𝑏−2 𝑗 =2 𝑆 𝑡 − 𝑏 − 1, 𝑗 + 𝑗! 𝑘 𝑗 𝑘−𝑏−1 𝑗 =1 𝑗𝑆(𝑡 − 𝑏 − 1, 𝑗) = 𝑗! 𝑘 𝑘 𝑗 𝑘−𝑏−2 𝑗 =2 𝑆 𝑡 − 𝑏 − 1, 𝑗 + 𝑗! 𝑘 𝑗 𝑘−𝑏−1 𝑗 =𝑘−𝑏−1 𝑗𝑆(𝑡 − 𝑏 − 1, 𝑗) = 𝑗! 𝑘 𝑘 𝑗 𝑘−𝑏−2 𝑗=2 𝑆 𝑡 − 𝑏 − 1, 𝑗 + 𝑡 − 𝑏 − 1 𝑘 − 𝑏 − 1 ! 𝑘 𝑘 − 𝑏 − 1 𝑆(𝑡 − 𝑏 − 1, 𝑘 − 𝑏 − 1)

= 𝑘𝑡−𝑏𝑓 𝑏 + 1 + 𝑘 − 𝑏 − 1 (𝑘 − 𝑏 − 1!) 𝑘 𝑘 − 𝑏 − 1 𝑆(𝑡 − 𝑏 − 1, 𝑘 − 𝑏 − 1) 𝑓 𝑏 − 𝑓 𝑏 + 1 = 1 𝑘𝑡−𝑏( 𝑘 − 𝑏 − 1 𝑘 − 𝑏 − 1 !) 𝑘 𝑘 − 𝑏 − 1 𝑆(𝑡 − 𝑏 − 1, 𝑘 − 𝑏 − 1) 𝑃𝐶𝐶𝑀𝑆 − 𝑃𝑅𝑆𝐺𝐶 = 𝑓 0 − 𝑓(𝑡2) = (𝑓 𝑖 − 𝑓(𝑖 + 1)) 𝑡2−1 𝑖=0 = 1 𝑘𝑡−𝑖 𝑡2−1 𝑖=0 ( 𝑘 − 𝑖 − 1 (𝑘 − 𝑖 − 1!) 𝑘 𝑘 − 𝑖 − 1 𝑆(𝑡 − 𝑖 − 1, 𝑘 − 𝑖 − 1)) 𝑝 𝑡 𝑘 𝑃_{𝐶𝐶𝑀𝑆} − 𝑃_{𝑅𝑆𝐺𝐶} 𝑡₂ 𝑃_{𝐶𝐶𝑀𝑆} 𝑃_{𝑅𝑆𝐺𝐶}

4.3.2 Detection Performance 𝑙 𝑅𝑐𝑑 𝑙 = 1 𝑙 𝑃 𝑡 𝑑𝑡 𝑙 0 𝑃(𝑡) 𝑙 𝑙 𝑡 𝑠𝑡 𝑅_𝑐𝑑 𝑙 𝑅𝑐𝑑 = 1 𝑙 𝑙 0 × 1 − 1 − 1 𝑗 min 𝑘,𝑠𝑡 𝑗 =1 × 𝑃(𝑠𝑡, 𝑘, 𝑗) 𝑠𝑡 𝑑𝑡 , 𝑃 𝑠_𝑡, 𝑘, 𝑗 = _𝑘1_𝑠𝑡 (−1)𝑖 𝑘 𝑗 𝑗𝑖 (𝑗 − 𝑖)𝑠𝑡 𝑗 −1 𝑖=0 𝑠𝑡 𝑠_𝑡 𝑠𝑡 1 𝑗 min 𝑘,𝑠𝑡 𝑗 × 𝑃 𝑠𝑡, 𝑘, 𝑗 , 1 − 1 𝑗 min 𝑘,𝑠𝑡 𝑗 × 𝑃 𝑠𝑡, 𝑘, 𝑗 , 𝑠_𝑡 𝑔 1 − 1 𝑗 min 𝑘,𝑠𝑡 𝑗 × 𝑃 𝑠_𝑡, 𝑘, 𝑗 𝑠𝑡 ,

𝑠𝑡 1 − 1 − 1 𝑗 min 𝑘,𝑠𝑡 𝑗 × 𝑃 𝑠𝑡, 𝑘, 𝑗 𝑠𝑡 . 𝑔 𝑠_𝑡 𝑔 1 − 1 − min 𝑘,𝑠𝑡 1_𝑗 𝑗 × 𝑃 𝑠𝑡, 𝑘, 𝑗 𝑠𝑡 𝑔 − 𝑡𝑕 1 − 1 − 1 𝑗 min 𝑘,𝑠𝑡 𝑗 × 𝑃 𝑠𝑡, 𝑘, 𝑗 𝑠𝑡 . 𝑃_𝑐𝑑 𝑙 𝑅𝑐𝑑 = 1 𝑙 𝑙 0 × 1 − 1 − 1 𝑗 min 𝑘,𝑠𝑡 𝑗 =1 × 𝑃(𝑠𝑡, 𝑘, 𝑗) 𝑠𝑡 𝑑𝑡 𝑙 𝑡 𝑠𝑡 𝑅_𝑐𝑑𝐶𝐶𝑀𝑆 _𝑙 𝑅_𝑐𝑑𝑅𝑆𝐺𝐶 𝑅_𝑐𝑑𝐶𝐶𝑀𝑆 _{≥ 𝑅} 𝑐𝑑𝑅𝑆𝐺𝐶 𝑅_𝑐𝑑𝑅𝑆𝐺𝐶 𝑅_𝑐𝑑𝑅𝑆𝐺𝐶 = 1 𝑙 𝑙 0 × 1 − 1 −1 𝑘 𝑠𝑡 𝑑𝑡 . 1 𝑗 min 𝑘,𝑠𝑡 𝑗 =1 × 𝑃 𝑠_𝑡, 𝑘, 𝑗 ≥ 1 𝑘 . 1 𝑗 min 𝑘,𝑠𝑡 𝑗 =1 × 𝑃 𝑠𝑡, 𝑘, 𝑗 ≥ 1 min 𝑘, 𝑠𝑡 × 𝑃 𝑠𝑡, 𝑘, 𝑗 min 𝑘,𝑠𝑡 𝑗 =1

≥ 1 𝑘 × 𝑃 𝑠𝑡, 𝑘, 𝑗 min 𝑘,𝑠𝑡 𝑗 =1 = 1 𝑘 𝑃 𝑠_𝑡, 𝑘, 𝑗 min 𝑘,𝑠𝑡 𝑗 =1 = 1 𝑙 𝑠_𝑡 𝑙 𝑃𝑐𝑠= 1 𝑖𝑓 𝑙 ≥ 𝑘 × 𝑇; 1 − 𝑙 𝑇− 𝑙 𝑇 × 1 − 1 − 𝑙 𝑇 × 1 𝑗 min 𝑘,𝑠𝑡 𝑗 =1 × 𝑃 𝑠𝑡, 𝑘, 𝑗 𝑠_𝑡 + 𝑙 𝑇− 𝑙 𝑇 × 1 − 1 − 𝑙 𝑇 + 1 × 1 𝑗 min 𝑘,𝑠𝑡 𝑗 =1 × 𝑃 𝑠𝑡, 𝑘, 𝑗 𝑠_𝑡 𝑂𝑡𝑕𝑒𝑟𝑤𝑖𝑠𝑒. 𝑠_𝑡 𝑠_𝑡 𝑙 ≥ 𝑘 × 𝑇 𝑃𝑐𝑠 = 1 𝑙 < 𝑘 × 𝑇 𝑙 𝑇 𝑙 𝑇 + 1 𝑙 𝑇 1 − (_𝑇𝑙 − _𝑇𝑙 ) _𝑇𝑙 + 1 𝑙 𝑇− 𝑙 𝑇 𝑎 𝑠𝑡 𝑎 𝑔 min 𝑘,𝑠𝑡 1_𝑗 𝑗 =1 × 𝑃(𝑠𝑡, 𝑘, 𝑗) 𝑙 𝑇 𝑙 𝑇 + 1

𝑙 𝑇 × 1 𝑗 min 𝑘,𝑠𝑡 𝑗 =1 × 𝑃 𝑠_𝑡, 𝑘, 𝑗 𝑙 𝑇 + 1 × 1 𝑗 min 𝑘,𝑠𝑡 𝑗 =1 × 𝑃 𝑠_𝑡, 𝑘, 𝑗 𝑙 𝑇 𝑙 𝑇 + 1 1 − 𝑙 𝑇 × 1 𝑗 min 𝑘,𝑠𝑡 𝑗 =1 × 𝑃 𝑠_𝑡, 𝑘, 𝑗 1 − 𝑙 𝑇 + 1 × 1 𝑗 min 𝑘,𝑠𝑡 𝑗 =1 × 𝑃 𝑠𝑡, 𝑘, 𝑗 𝑠𝑡 𝑙 𝑇 𝑙 𝑇 + 1 1 − 𝑙 𝑇 × 1 𝑗 min 𝑘,𝑠𝑡 𝑗 =1 × 𝑃 𝑠_𝑡, 𝑘, 𝑗 𝑠𝑡 1 − 𝑙 𝑇 + 1 × 1 𝑗 min 𝑘,𝑠𝑡 𝑗 =1 × 𝑃 𝑠_𝑡, 𝑘, 𝑗 𝑠𝑡 𝑠𝑡 𝑙 𝑇 𝑙 𝑇 + 1 1 − 1 − 𝑙 𝑇 × 1 𝑗 min 𝑘,𝑠𝑡 𝑗 =1 × 𝑃 𝑠_𝑡, 𝑘, 𝑗 𝑠𝑡 1 − 1 − 𝑙 𝑇 + 1 × 1 𝑗 min 𝑘,𝑠𝑡 𝑗 =1 × 𝑃 𝑠_𝑡, 𝑘, 𝑗 𝑠𝑡

𝑙 𝑇 1 − ( 𝑙 𝑇− 𝑙 𝑇 ) 𝑙 𝑇 + 1 𝑙 𝑇− 𝑙 𝑇 𝑠𝑡 𝑃𝑐𝑠 𝑃𝑟𝑠 𝑙 𝑃𝑐𝑠 ≥ 𝑃𝑟𝑠 𝑃𝑟𝑠 = 1 − ( 𝑙 𝑇− 𝑙 𝑇 ) × 1 − 1 − 𝑙 𝑇 × 1 𝑘 𝑠𝑡 + 𝑙 𝑇− 𝑙 𝑇 × 1 − [1 − 𝑙 𝑇 + 1 × 1 𝑘]𝑠𝑡 1 𝑗 min 𝑘,𝑠𝑡 𝑗 =1 × 𝑃 𝑠𝑡, 𝑘, 𝑗 ≥_𝑘1 𝑃𝑐𝑠 ≥ 𝑃𝑟𝑠

𝑁_𝑚𝑎𝑥

𝐻𝑚𝑎𝑥 𝑁

𝑛

log 𝑁 𝑁_𝑚𝑎𝑥 ∙ log 𝑁 + 𝑁_𝑚𝑎𝑥2∙ log 𝑁 + log 𝐻_𝑚𝑎𝑥 + log 𝑘

[𝑘 ∙ log 𝑘] 𝑘 ≤ 𝑚 𝑚 ≤ 𝑁_𝑚𝑎𝑥 < 𝑁 𝑂(𝑁_𝑚𝑎𝑥2∙ log 𝑁) 𝑁_𝑚𝑎𝑥 𝑁

4.4

Simulation Results

250𝑚 × 250𝑚 𝑟𝑠 𝑟𝑐 2𝑟𝑠

4.4.1 Group Coverage Maintenance Performance 𝑁 𝑘 𝑔 (1 ≤ 𝑔 ≤ 𝑘) 𝑅𝐺𝐶𝑀 𝑔 𝑅_𝐺𝐶𝑀 𝑔 ₌ 𝐴𝑟𝑒𝑎(𝑔) 𝐴𝑟𝑒𝑎[𝑁 𝑘 ] 𝐴𝑟𝑒𝑎(𝑔) 𝑔 𝐴𝑟𝑒𝑎 _{𝑁 𝑘} 𝑁/𝑘 𝑅_{𝐴𝐺𝐶𝑀} 𝑅𝐴𝐺𝐶𝑀 = 𝑅_𝐺𝐶𝑀𝑖 𝑘 𝑖=1 𝑘 250𝑚 × 250𝑚

𝑘 = 2

Chapter 5

Second Proposal: An Efficient

Node Scheduling Scheme Based

on Combinatorial Assignment

Code

5.1

Combinatorial Assignment Code

≥ 𝔅 𝔅 𝑥_𝑖1 𝑥_𝑖2 𝑥_𝑖𝑘 ⊂ ∈ 𝐵_𝑖 𝑥_𝑖1 𝑥_𝑖2 𝑥_𝑖𝑘 |𝐵_𝑖| 𝐵_𝑖 𝑘 |𝐵_𝑖| 𝑖=1 ⟺ ∀ ≤ ≤ 𝔅 𝔅 𝔅 𝑘 × 𝑚 𝑎_𝑖𝑗 = 1: if 𝑥_{0: if 𝑥}𝑗 ∈ 𝐵𝑗 𝑗 ∉ 𝐵𝑗 𝑘 × 𝑚 𝐴 = (𝑎𝑖𝑗) 𝑥₁ 𝑥2 … 𝑥𝑚 𝐵1 𝐵_…2 𝐵𝑘 𝑎₁₁ 𝑎₁₂ … 𝑎1𝑚 𝑎₂₁ 𝑎₂₂ … 𝑎2𝑚 … 𝑎_𝑘1 𝑎…_𝑘2 …… … 𝑎_𝑘𝑚

𝑘 × 𝑚 (𝑚, 𝑁, 𝑘) 𝑚 − 𝑘 𝑘 × 𝑚 (𝑚, 𝑁_𝑚𝑖𝑛, 𝑘) ⟺ 𝑘 × 𝑘 𝑚 − 𝑘 + 1 𝑚 − 𝑘 + 1 𝑘(𝑚 − 𝑘 + 1) = 𝑘𝑚 − 𝑘(𝑘 − 1) 𝑘𝑚 − 𝑘(𝑘 − 1) 𝑁_𝑚𝑖𝑛 = 𝑘𝑚 − 𝑘(𝑘 − 1) 𝑚 − 𝑘 + 1 𝑘 × 𝑘 𝑘 × 𝑘 (𝑚, 𝑁_𝑚𝑖𝑛, 𝑘) − (𝑚, 𝑘) 𝑘 × 𝑘 𝑚 − 𝑘 + 1 𝑘(𝑚 − 𝑘 + 1) = 𝑘𝑚 − 𝑘(𝑘 − 1) 𝑁_𝑚𝑖𝑛 = km − k(k − 1). 𝑘 × 𝑘

(𝑚, 𝑁_𝑚𝑖𝑛, 𝑘) 𝑚 = 7 𝑘 = 4 1 1 1 1 0 0 0 0 1 1 1 1 0 0 0 0 1 1 1 1 0 0 0 0 1 1 1 1 1 1 1 1 0 0 0 1 1 1 0 1 0 0 1 1 1 0 0 1 0 1 1 1 0 0 0 1 𝑘 × 𝑚 𝐴 = (𝑎_𝑖𝑗) (𝑚, 𝑁_𝑚𝑖𝑛, 𝑘) ⟺ 𝑘 × 𝑘

5.2

CAC-based Scheduling Scheme

5.2.1 Network Model and Introduction to the Scheme

𝑟_𝑠 𝑟_𝑐 2𝑟𝑠 00 … 001 𝑛 11 … 111 𝑛 00 … 000 𝑛 00 … 001 𝑛 11 … 111 𝑛 00 … 000 𝑛

5.2.2 Distributed Cluster Approach

5.2.3 CAC-based Node Scheduling Scheme for node in cluster

∈

≤

≤ ≤ ≤ ≤

(𝑚, 𝑁, 𝑘) (𝑚, 𝑁, 𝑘) ≤ 𝑘 × 𝑘 𝑝1 𝑝2 … 𝑝𝑚−𝑘 𝑝𝑚−𝑘+1 𝑝𝑚−𝑘+2 … 𝑝𝑚 −1 𝑝𝑚 𝐴 = 𝑎_𝑖𝑗 = 𝑔𝑟𝑜𝑢𝑝(0) 𝑔𝑟𝑜𝑢𝑝(1) … 𝑔𝑟𝑜𝑢𝑝(𝑘 − 1) 1 1 … 1 1 0 … 0 0 1 1 … 1 0 1 … 0 0 … 1 …1 …… …1 0 … … 0 … … … … 0 1 (1) ≥ 𝑝1 𝑝2 … 𝑝𝑎−𝑐 𝑝𝑎−𝑐+1 𝑝𝑎−𝑐+2 … 𝑝𝑚 −1 𝑝𝑚 𝐴 = 𝑎_𝑖𝑗 = 𝑔𝑟𝑜𝑢𝑝(0) 𝑔𝑟𝑜𝑢𝑝(1) … 𝑔𝑟𝑜𝑢𝑝(𝑘 − 1) 1 1 … 1 1 0 … 0 0 1 1 … 1 0 1 … 0 0 … 1 … 1 … 1 … 1 … … … … … 1 … 1 … 0_… 0 … 0 … 0 … ……_… … … …0 0 1 … 0 (2)

∪

5.2.4 ID selection scheme for nodes in the whole sensor network

5.2.5

5.2.6 CAC based node scheduling scheme

≡

5.3

Performance analysis

𝑚_{𝑘 𝑘!} 𝑘𝑚

1 if 𝑚1 ≥ 𝑘; (𝑘−𝑚1) 𝑚−𝑚1 𝑘−𝑚1 (𝑘−𝑚1)! 𝑘(𝑘−𝑚1)𝑚 −𝑚 1 : otherwise. ≥ 𝑘−𝑚 1 𝑘 𝑚−𝑚1 𝑘−𝑚1 (𝑘−𝑚1)! (𝑘−𝑚1)𝑚 −𝑚 1 1: if 𝑚₁ ≥ 𝑘; (𝑘−𝑚1) 𝑚 −𝑚_𝑘−𝑚11 (𝑘−𝑚1)! 𝑘(𝑘−𝑚1)𝑚 −𝑚 1 : otherwise 𝑚_{𝑘 𝑘!} 𝑘𝑚 ≥ 𝑓 𝑥 = (𝑘−𝑥) 𝑚 −𝑥_{𝑘(𝑘−𝑥)}𝑘−𝑥 (𝑘−𝑥)!_{𝑚 −𝑥}

𝑓 𝑥 = 𝑘 − 𝑥 𝑚 − 𝑥 𝑘 − 𝑥 𝑘 − 𝑥 ! 𝑘 𝑘 − 𝑥 𝑚−𝑥 = 1 𝑚 − 𝑘 !∗ 𝑘 − 𝑥 𝑘 ∗ (𝑚 − 𝑥)! (𝑘 − 𝑥)𝑚−𝑥 = 1 𝑚 − 𝑘 !∗ 𝑘 − 𝑥 𝑘 ∗ { 𝑚 − 𝑥 𝑘 − 𝑥 ∗ 𝑚 − 𝑥 − 1 𝑘 − 𝑥 ∗ … ∗ 1 𝑘 − 𝑥} = 1 𝑚 − 𝑘 !∗ { 𝑘 − 𝑥 𝑘 ∗ 𝑚 − 𝑥 𝑘 − 𝑥} ∗ { 𝑚 − 𝑥 − 1 𝑘 − 𝑥 ∗ … ∗ 1 𝑘 − 𝑥} = 1 𝑚 − 𝑘 !∗ 𝑚 − 𝑥 𝑘 ∗ (𝑚 − 𝑥 − 1)! (𝑘 − 𝑥)𝑚−𝑥−1 = 1 𝑘 𝑚 − 𝑘 !∗ (𝑚 − 𝑥)(𝑚 − 𝑥 − 1)! (𝑘 − 𝑥)𝑚−𝑥−1 . 1 𝑚 −𝑘 !∗ 𝑘−𝑥−1 𝑘 ∗ (𝑚−𝑥−1)! (𝑘−𝑥−1)𝑚 −𝑥−1 1 𝑘 𝑚 −𝑘 !∗ (𝑘−𝑥−1)(𝑚−𝑥−1)! (𝑘−𝑥−1)𝑚 −𝑥−1 α β (1 +1_β)α _{> 1 +}α β > 1 β+ α β 𝛼 = 𝑚 − 𝑥 − 1 𝛽 = 𝑘 − 𝑥 − 1 (1 + 1 𝑘 − 𝑥 − 1)𝑚−𝑥−1> 1 𝑘 − 𝑥 − 1+ 𝑚 − 𝑥 − 1 𝑘 − 𝑥 − 1 = 𝑚 − 𝑥 𝑘 − 𝑥 − 1⟹ ( 𝑘 − 𝑥 𝑘 − 𝑥 − 1)𝑚−𝑥−1 > 𝑚 − 𝑥 𝑘 − 𝑥 − 1⟹ 𝑘 − 𝑥 − 1 (𝑘 − 𝑥 − 1)𝑚−𝑥−1 > 𝑚 − 𝑥 (𝑘 − 𝑥)𝑚−𝑥−1⟹ (𝑘 − 𝑥 − 1)(𝑚 − 𝑥 − 1)! (𝑘 − 𝑥 − 1)𝑚−𝑥−1 > (𝑚 − 𝑥)(𝑚 − 𝑥 − 1)! (𝑘 − 𝑥)𝑚−𝑥−1 ⟹ 1 𝑘 𝑚 − 𝑘 !∗ (𝑘 − 𝑥 − 1)(𝑚 − 𝑥 − 1)! (𝑘 − 𝑥 − 1)𝑚−𝑥−1 > 1 𝑘 𝑚 − 𝑘 !∗ (𝑚 − 𝑥)(𝑚 − 𝑥 − 1)! (𝑘 − 𝑥)𝑚−𝑥−1 .

𝑓(𝑥 + 1) > 𝑓(𝑥) 𝑓(𝑥) 𝑚_{𝑘 𝑘!} 𝑘𝑚 53 3!₃5 𝑚_{𝑘 𝑘!} 𝑘𝑚 43 3!₃4 (𝑘−𝑚1) 𝑚−𝑚_𝑘−𝑚11 (𝑘−𝑚1)! 𝑘(𝑘−𝑚1)𝑚 −𝑚 1 1∗ 21 (1)! 3∗(1)2

Chapter 6

Conclusions and Future Work