# End to End Delay and Energy Consumption in a Two Tier Cluster Hierarchical Wireless Sensor Networks

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## Abstract

**:**

## 1. Introduction

## 2. Related Work

## 3. WSN Scenario

#### Cells, Sensors and Cell Head Selection

## 4. Communication

#### 4.1. Frame Structure

#### 4.2. Slot Assignment for Intra-Cell and Inter-Cell Communications

#### 4.3. The Communication Range

#### 4.4. Inter-Cell Routing and Load Balancing

#### 4.5. Synchronization of the WSN

## 5. The Intra-Cluster Communication

#### 5.1. Traffic Model

#### 5.2. The Contention Process

## 6. The Inter-Cluster Communication

#### 6.1. The Traffic Load and the Stability Conditions

#### 6.2. The Embedded Markov Chain

#### 6.3. On the Input Process in the Inter-Cluster Communication

#### 6.4. On the Output Process in the Inter-Cluster Communication

## 7. End-To-End Delay Analysis

#### 7.1. Sojourn Times in a CH

#### 7.2. Local and Exogenous Traffic

#### 7.3. Delay of a Tagged Data Packet

#### 7.4. Algorithmic Procedure

#### 7.5. Model Validation

## 8. The Energy Consumption Model

## 9. Numerical Results

#### 9.1. Delay Analysis

#### 9.2. Energy Consumption

## 10. Conclusions and Further Work

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

${\langle x,y\rangle}_{h}$ | Hexagonal coordinates for a cluster |

${\langle x,y\rangle}_{p}$ | Polar coordinates for a cluster |

${r}_{\mathrm{max}}$ | Maximum number of rings in the WSN area |

${R}_{C}$ | Minimum range of intra-cell communication |

${R}_{T}$ | Minimum range of inter-cell communication |

${N}_{m}$ | Total number of sensors or motes in the WSN area |

${M}_{c}\left({r}_{\mathrm{max}}\right)$ | Total number of members per cluster |

${N}_{intra}$ | Number of slots per $\mathcal{CONT}$ sub-frame |

${N}_{msC}$ | Number of mini-slots per $\mathcal{CONT}$ slot |

${N}_{inter}$ | Number of slots per $\mathcal{TDMA}$ sub-frame |

${N}_{msT}$ | Number of mini-slots per $\mathcal{TDMA}$ slot |

${N}_{sCF}$ | Number of slots per Combi-Frame |

${N}_{msCF}$ | Number of mini-slots per Combi-Frame |

${T}_{msC}$ | Time duration of the $\mathcal{CONT}$ mini-slot |

${T}_{msT}$ | Time duration of the $\mathcal{TDMA}$ mini-slot |

${T}_{CONT}$ | Time duration of the $\mathcal{CONT}$ sub-frame |

${T}_{TDMA}$ | Time duration of the $\mathcal{TDMA}$ sub-frame |

${T}_{CF}$ | Time duration of the Combi-Frame |

$\lambda $ | Poisson arrival rate of data packets per sensor |

${p}_{act}$ | Probability a sensor becomes active during one mini-slot |

$F\left(z\right)$ | PGF of the input process at a CH |

$L\left(z\right)$ | PGF of the output process after a $\mathcal{CONT}$ slot |

${A}_{i}\left(z\right)$ | PGF of the arrival data packets at CH during slot i |

$D\left(z\right)$ | PGF of the departure process at a CH after a $\mathcal{TDMA}$ slot |

${\rho}_{k}$ | Normalized traffic load at any CH of ring k |

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**Figure 2.**Hexagonal Coordinates ${\langle x,y\rangle}_{h}$, Polar Coordinates ${\langle x,y\rangle}_{p}$, Sectors ${S}_{k}$ and Axes ${A}_{k}$ for $0\le k\le 5$.

**Figure 5.**The role of ${A}_{i}\left(z\right)$ in several nodes located at ${S}_{0}$ when ${r}_{\mathrm{max}}=4$.

**Figure 6.**End to end delay (in mini-slots), from sensor to the sink, for ${r}_{\mathrm{max}}=0$, ${p}_{act}=0.001$ and $({N}_{msC},{N}_{intra};{N}_{msT},{N}_{inter})=(63,1;0,0).$

**Figure 7.**End to end delay (in mini-slots), from sensor to the sink, for ${r}_{\mathrm{max}}=1$, ${p}_{act}=0.001$ and $({N}_{msC},{N}_{intra};{N}_{msT},{N}_{inter})=(10,3;3,7).$

**Figure 8.**End to end delay (in mini-slots), from sensor to the sink, for ${r}_{\mathrm{max}}=2$, ${p}_{act}=0.001$ and $({N}_{msC},{N}_{intra};{N}_{msT},{N}_{inter})=(3,3;1,7).$

**Figure 9.**End to end delay (in mini-slots), from sensor to the sink, for ${r}_{\mathrm{max}}=3$, ${p}_{act}=0.001$ and $({N}_{msC},{N}_{intra};{N}_{msT},{N}_{inter})=(2,3;1,7).$

**Figure 10.**End to end delay (in mini-slots), from sensor to the sink, for ${r}_{\mathrm{max}}=4$, ${p}_{act}=0.001$ and $({N}_{msC},{N}_{intra};{N}_{msT},{N}_{inter})=(2,3;1,7).$

**Figure 11.**End to end delay (in mini-slots), from sensor to the sink, for ${r}_{\mathrm{max}}=5$, ${p}_{act}=0.001$ and $({N}_{msC},{N}_{intra};{N}_{msT},{N}_{inter})=(2,3;1,7).$

**Figure 12.**Energy consumption for several radius of the WSN, with ${N}_{s}=364$ sensors, ${p}_{act}=0.001$ and ${r}_{\mathrm{max}}=5$.

Sector | Condition | Axes | Condition |
---|---|---|---|

${S}_{0}$ | $0<{x}_{p},0<{y}_{p}<{x}_{p}$ | ${A}_{0}$ | $0<{x}_{p},{y}_{p}=0$ |

${S}_{1}$ | $0<{x}_{p},{x}_{p}<{y}_{p}<2{x}_{p}$ | ${A}_{1}$ | $0<{x}_{p},{y}_{p}={x}_{p}$ |

${S}_{2}$ | $0<{x}_{p},2{x}_{p}<{y}_{p}<3{x}_{p}$ | ${A}_{2}$ | $0<{x}_{p},{y}_{p}=2{x}_{p}$ |

${S}_{3}$ | $0<{x}_{p},3{x}_{p}<{y}_{p}<4{x}_{p}$ | ${A}_{3}$ | $0<{x}_{p},{y}_{p}=3{x}_{p}$ |

${S}_{4}$ | $0<{x}_{p},4{x}_{p}<{y}_{p}<5{x}_{p}$ | ${A}_{4}$ | $0<{x}_{p},{y}_{p}=4{x}_{p}$ |

${S}_{5}$ | $0<{x}_{p},5{x}_{p}<{y}_{p}<6{x}_{p}$ | ${A}_{5}$ | $0<{x}_{p},{y}_{p}=5{x}_{p}$ |

${\mathit{i}}_{\mathit{x}},{\mathit{j}}_{\mathit{x}}$ | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | ... |
---|---|---|---|---|---|---|---|---|---|---|---|

0 | 0 | ||||||||||

1 | 1 | 3 | |||||||||

2 | 4 | 7 | 12 | ||||||||

3 | 9 | 13 | 19 | 27 | |||||||

4 | 16 | 21 | 28 | 37 | 48 | ||||||

5 | 25 | 31 | 39 | 49 | 61 | 75 | |||||

6 | 36 | 43 | 52 | 63 | 76 | 91 | 108 | ||||

7 | 49 | 57 | 67 | 79 | 93 | 109 | 127 | 147 | |||

8 | 64 | 73 | 84 | 97 | 112 | 129 | 148 | 169 | 192 | ||

9 | 81 | 91 | 103 | 117 | 133 | 151 | 171 | 193 | 217 | 243 | |

⋮ | ⋮ | ⋮ | ⋮ | ⋮ | ⋮ | ⋮ | ⋮ | ⋮ | ⋮ | ⋮ | ⋱ |

**Table 3.**Cluster Size ${N}_{x}={i}_{x}^{2}+{i}_{x}{j}_{x}+{j}_{x}^{2}$, ($x={N}_{intra},{N}_{inter}$) and Slot Assignment for Transmission for any CH ${\langle x,y\rangle}_{h}$.

${\mathit{i}}_{\mathit{x}}$ | ${\mathit{j}}_{\mathit{x}}$ | ${\mathit{N}}_{\mathit{x}}$ | $\mathit{T}({\mathit{x}}_{\mathit{h}},{\mathit{y}}_{\mathit{h}})$ |
---|---|---|---|

1 | 1 | 3 | $[{x}_{h}+{y}_{h}]\phantom{\rule{0.277778em}{0ex}}mod\phantom{\rule{0.277778em}{0ex}}3$ |

2 | 0 | 4 | $2{x}_{h}\phantom{\rule{0.277778em}{0ex}}mod\phantom{\rule{0.277778em}{0ex}}4+{y}_{h}\phantom{\rule{0.277778em}{0ex}}mod\phantom{\rule{0.277778em}{0ex}}2$ |

2 | 1 | 7 | $[2{x}_{h}+{y}_{h}]\phantom{\rule{0.277778em}{0ex}}mod\phantom{\rule{0.277778em}{0ex}}7$ |

2 | 2 | 12 | $[2{x}_{h}+2{y}_{h}+{x}_{h}\phantom{\rule{0.277778em}{0ex}}mod\phantom{\rule{0.277778em}{0ex}}2]\phantom{\rule{0.277778em}{0ex}}mod\phantom{\rule{0.277778em}{0ex}}12$ |

3 | 0 | 9 | $3{x}_{h}\phantom{\rule{0.277778em}{0ex}}mod\phantom{\rule{0.277778em}{0ex}}9+{y}_{h}\phantom{\rule{0.277778em}{0ex}}mod\phantom{\rule{0.277778em}{0ex}}3$ |

3 | 1 | 13 | $[3{x}_{h}+{y}_{h}]\phantom{\rule{0.277778em}{0ex}}mod\phantom{\rule{0.277778em}{0ex}}13$ |

3 | 2 | 19 | $[3{x}_{h}+2{y}_{h}]\phantom{\rule{0.277778em}{0ex}}mod\phantom{\rule{0.277778em}{0ex}}19$ |

3 | 3 | 27 | $[3{x}_{h}+3{y}_{h}+{x}_{h}\phantom{\rule{0.277778em}{0ex}}mod\phantom{\rule{0.277778em}{0ex}}3]\phantom{\rule{0.277778em}{0ex}}mod\phantom{\rule{0.277778em}{0ex}}27$ |

4 | 0 | 16 | $4{x}_{h}\phantom{\rule{0.277778em}{0ex}}mod\phantom{\rule{0.277778em}{0ex}}16+{y}_{h}\phantom{\rule{0.277778em}{0ex}}mod\phantom{\rule{0.277778em}{0ex}}4$ |

4 | 1 | 21 | $[4{x}_{h}+{y}_{h}]\phantom{\rule{0.277778em}{0ex}}mod\phantom{\rule{0.277778em}{0ex}}21$ |

4 | 2 | 28 | $[4{x}_{h}+2{y}_{h}+{x}_{h}\phantom{\rule{0.277778em}{0ex}}mod\phantom{\rule{0.277778em}{0ex}}2]\phantom{\rule{0.277778em}{0ex}}mod\phantom{\rule{0.277778em}{0ex}}28$ |

4 | 3 | 37 | $[4{x}_{h}+3{y}_{h}]\phantom{\rule{0.277778em}{0ex}}mod\phantom{\rule{0.277778em}{0ex}}37$ |

4 | 4 | 48 | $[4{x}_{h}+4{y}_{h}+{x}_{h}\phantom{\rule{0.277778em}{0ex}}mod\phantom{\rule{0.277778em}{0ex}}4]\phantom{\rule{0.277778em}{0ex}}mod\phantom{\rule{0.277778em}{0ex}}48$ |

5 | 0 | 25 | $5{x}_{h}\phantom{\rule{0.277778em}{0ex}}mod\phantom{\rule{0.277778em}{0ex}}25+{y}_{h}\phantom{\rule{0.277778em}{0ex}}mod\phantom{\rule{0.277778em}{0ex}}5$ |

5 | 1 | 31 | $[5{x}_{h}+{y}_{h}]\phantom{\rule{0.277778em}{0ex}}mod\phantom{\rule{0.277778em}{0ex}}31$ |

5 | 2 | 39 | $[5{x}_{h}+2{y}_{h}]\phantom{\rule{0.277778em}{0ex}}mod\phantom{\rule{0.277778em}{0ex}}39$ |

5 | 3 | 49 | $[5{x}_{h}+3{y}_{h}]\phantom{\rule{0.277778em}{0ex}}mod\phantom{\rule{0.277778em}{0ex}}49$ |

5 | 4 | 61 | $[5{x}_{h}+4{y}_{h}]\phantom{\rule{0.277778em}{0ex}}mod\phantom{\rule{0.277778em}{0ex}}61$ |

5 | 5 | 75 | $[5{x}_{h}+5{y}_{h}+{x}_{h}\phantom{\rule{0.277778em}{0ex}}mod\phantom{\rule{0.277778em}{0ex}}5]\phantom{\rule{0.277778em}{0ex}}mod\phantom{\rule{0.277778em}{0ex}}75$ |

6 | 0 | 36 | $6{x}_{h}\phantom{\rule{0.277778em}{0ex}}mod\phantom{\rule{0.277778em}{0ex}}36+{y}_{h}\phantom{\rule{0.277778em}{0ex}}mod\phantom{\rule{0.277778em}{0ex}}6$ |

6 | 1 | 43 | $[6{x}_{h}+{y}_{h}]\phantom{\rule{0.277778em}{0ex}}mod\phantom{\rule{0.277778em}{0ex}}43$ |

6 | 2 | 52 | $[6{x}_{h}+2{y}_{h}+{x}_{h}\phantom{\rule{0.277778em}{0ex}}mod\phantom{\rule{0.277778em}{0ex}}2]\phantom{\rule{0.277778em}{0ex}}mod\phantom{\rule{0.277778em}{0ex}}52$ |

6 | 3 | 63 | $[6{x}_{h}+3{y}_{h}+{x}_{h}\phantom{\rule{0.277778em}{0ex}}mod\phantom{\rule{0.277778em}{0ex}}3]\phantom{\rule{0.277778em}{0ex}}mod\phantom{\rule{0.277778em}{0ex}}63$ |

6 | 4 | 76 | $[6{x}_{h}+4{y}_{h}+{x}_{h}\phantom{\rule{0.277778em}{0ex}}mod\phantom{\rule{0.277778em}{0ex}}2]\phantom{\rule{0.277778em}{0ex}}mod\phantom{\rule{0.277778em}{0ex}}76$ |

6 | 5 | 91 | $[6{x}_{h}+5{y}_{h}]\phantom{\rule{0.277778em}{0ex}}mod\phantom{\rule{0.277778em}{0ex}}91$ |

6 | 6 | 108 | $[6{x}_{h}+6{y}_{h}+{x}_{h}\phantom{\rule{0.277778em}{0ex}}mod\phantom{\rule{0.277778em}{0ex}}6]\phantom{\rule{0.277778em}{0ex}}mod\phantom{\rule{0.277778em}{0ex}}108$ |

⋮ | ⋮ | ⋮ | ⋮ |

**Table 4.**Frame structure for nodes in ${A}_{0}\cup {S}_{0}\cup {A}_{1}$ and $\mathrm{Ring}\le 4$. $({N}_{Intra},{N}_{Inter})=(4,7)$.

${\mathit{CH}}_{\mathit{h}}$ | $\begin{array}{c}\mathit{Frame}\phantom{\rule{4.pt}{0ex}}\mathit{Structure}:\\ -\mathcal{CONT}-\mathcal{TDMA}-\end{array}$ | $\begin{array}{c}\mathit{Frame}\phantom{\rule{4.pt}{0ex}}\mathit{Structure}:\\ \mathit{With}\phantom{\rule{4.pt}{0ex}}\phantom{\rule{4.pt}{0ex}}\mathit{as}\phantom{\rule{4.pt}{0ex}}\mathit{Starting}\phantom{\rule{4.pt}{0ex}}\mathit{Point}\end{array}$ | $\mathit{CT}$ |
---|---|---|---|

$\langle 4,4\rangle $ | $\mathcal{CS}$$\mathcal{S}$$\mathcal{S}$-$\mathcal{S}$$\mathcal{S}$$\mathcal{S}$$\mathcal{S}$$\mathcal{S}$$\mathcal{T}$$\mathcal{S}$- | $\mathcal{CS}$$\mathcal{S}$$\mathcal{S}$-$\mathcal{S}$$\mathcal{S}$$\mathcal{S}$$\mathcal{S}$$\mathcal{S}$$\mathcal{T}$$\mathcal{S}$- | 9 |

$\langle 4,3\rangle $ | $\mathcal{SC}$$\mathcal{S}$$\mathcal{S}$-$\mathcal{S}$$\mathcal{S}$$\mathcal{S}$$\mathcal{T}$$\mathcal{S}$$\mathcal{S}$$\mathcal{S}$- | $\mathcal{CS}$$\mathcal{S}$-$\mathcal{S}$$\mathcal{S}$$\mathcal{S}$$\mathcal{T}$$\mathcal{S}$$\mathcal{S}$$\mathcal{S}$-$\mathcal{S}$ | 6 |

$\langle 4,2\rangle $ | $\mathcal{CS}$$\mathcal{S}$$\mathcal{S}$-$\mathcal{S}$$\mathcal{T}$$\mathcal{S}$$\mathcal{S}$$\mathcal{S}$$\mathcal{S}$$\mathcal{S}$- | $\mathcal{CS}$$\mathcal{S}$$\mathcal{S}$-$\mathcal{S}$$\mathcal{T}$$\mathcal{S}$$\mathcal{S}$$\mathcal{S}$$\mathcal{S}$$\mathcal{S}$- | 5 |

$\langle 4,1\rangle $ | $\mathcal{SC}$$\mathcal{S}$$\mathcal{S}$-$\mathcal{S}$$\mathcal{S}$$\mathcal{S}$$\mathcal{S}$$\mathcal{S}$$\mathcal{S}$$\mathcal{T}$- | $\mathcal{CS}$$\mathcal{S}$-$\mathcal{S}$$\mathcal{S}$$\mathcal{S}$$\mathcal{S}$$\mathcal{S}$$\mathcal{S}$$\mathcal{T}$-$\mathcal{S}$ | 9 |

$\langle 4,0\rangle $ | $\mathcal{CS}$$\mathcal{S}$$\mathcal{S}$-$\mathcal{S}$$\mathcal{S}$$\mathcal{S}$$\mathcal{S}$$\mathcal{T}$$\mathcal{S}$$\mathcal{S}$- | $\mathcal{CS}$$\mathcal{S}$$\mathcal{S}$-$\mathcal{S}$$\mathcal{S}$$\mathcal{S}$$\mathcal{S}$$\mathcal{T}$$\mathcal{S}$$\mathcal{S}$- | 8 |

$\langle 3,3\rangle $ | $\mathcal{S}$$\mathcal{S}$$\mathcal{SC}$-$\mathcal{S}$$\mathcal{S}$$\mathcal{T}$$\mathcal{R}$$\mathcal{R}$$\mathcal{R}$$\mathcal{S}$- | $\mathcal{C}$-$\mathcal{S}$$\mathcal{S}$$\mathcal{T}$$\mathcal{R}$$\mathcal{R}$$\mathcal{R}$$\mathcal{S}$-$\mathcal{S}$$\mathcal{S}$$\mathcal{S}$ | 3 |

$\langle 3,2\rangle $ | $\mathcal{S}$$\mathcal{SC}$$\mathcal{S}$-$\mathcal{T}$$\mathcal{R}$$\mathcal{S}$$\mathcal{R}$$\mathcal{S}$$\mathcal{S}$$\mathcal{S}$- | $\mathcal{CS}$-$\mathcal{T}$$\mathcal{R}$$\mathcal{S}$$\mathcal{R}$$\mathcal{S}$$\mathcal{S}$$\mathcal{S}$-$\mathcal{S}$$\mathcal{S}$ | 2 |

$\langle 3,1\rangle $ | $\mathcal{S}$$\mathcal{S}$$\mathcal{SC}$-$\mathcal{S}$$\mathcal{R}$$\mathcal{S}$$\mathcal{S}$$\mathcal{S}$$\mathcal{T}$$\mathcal{R}$- | $\mathcal{C}$-$\mathcal{S}$$\mathcal{R}$$\mathcal{S}$$\mathcal{S}$$\mathcal{S}$$\mathcal{T}$$\mathcal{R}$-$\mathcal{S}$$\mathcal{S}$$\mathcal{S}$ | 6 |

$\langle 3,0\rangle $ | $\mathcal{S}$$\mathcal{SC}$$\mathcal{S}$-$\mathcal{S}$$\mathcal{R}$$\mathcal{S}$$\mathcal{T}$$\mathcal{R}$$\mathcal{S}$$\mathcal{R}$- | $\mathcal{CS}$-$\mathcal{S}$$\mathcal{R}$$\mathcal{S}$$\mathcal{T}$$\mathcal{R}$$\mathcal{S}$$\mathcal{R}$-$\mathcal{S}$$\mathcal{S}$ | 5 |

$\langle 2,2\rangle $ | $\mathcal{CS}$$\mathcal{S}$$\mathcal{S}$-$\mathcal{R}$$\mathcal{R}$$\mathcal{R}$$\mathcal{S}$$\mathcal{S}$$\mathcal{S}$$\mathcal{T}$- | $\mathcal{CS}$$\mathcal{S}$$\mathcal{S}$-$\mathcal{R}$$\mathcal{R}$$\mathcal{R}$$\mathcal{S}$$\mathcal{S}$$\mathcal{S}$$\mathcal{T}$- | 10 |

$\langle 2,1\rangle $ | $\mathcal{SC}$$\mathcal{S}$$\mathcal{S}$-$\mathcal{R}$$\mathcal{S}$$\mathcal{S}$$\mathcal{S}$$\mathcal{T}$$\mathcal{R}$$\mathcal{S}$- | $\mathcal{CS}$$\mathcal{S}$-$\mathcal{R}$$\mathcal{S}$$\mathcal{S}$$\mathcal{S}$$\mathcal{T}$$\mathcal{R}$$\mathcal{S}$-$\mathcal{S}$ | 7 |

$\langle 2,0\rangle $ | $\mathcal{CS}$$\mathcal{S}$$\mathcal{S}$-$\mathcal{R}$$\mathcal{S}$$\mathcal{T}$$\mathcal{R}$$\mathcal{S}$$\mathcal{R}$$\mathcal{S}$- | $\mathcal{CS}$$\mathcal{S}$$\mathcal{S}$-$\mathcal{R}$$\mathcal{S}$$\mathcal{T}$$\mathcal{R}$$\mathcal{S}$$\mathcal{R}$$\mathcal{S}$- | 6 |

$\langle 1,1\rangle $ | $\mathcal{S}$$\mathcal{S}$$\mathcal{SC}$-$\mathcal{S}$$\mathcal{S}$$\mathcal{S}$$\mathcal{T}$$\mathcal{R}$$\mathcal{R}$$\mathcal{R}$- | $\mathcal{C}$-$\mathcal{S}$$\mathcal{S}$$\mathcal{S}$$\mathcal{T}$$\mathcal{R}$$\mathcal{R}$$\mathcal{R}$-$\mathcal{S}$$\mathcal{S}$$\mathcal{S}$ | 4 |

$\langle 1,0\rangle $ | $\mathcal{S}$$\mathcal{SC}$$\mathcal{S}$-$\mathcal{S}$$\mathcal{T}$$\mathcal{R}$$\mathcal{S}$$\mathcal{R}$$\mathcal{S}$$\mathcal{R}$- | $\mathcal{CS}$-$\mathcal{S}$$\mathcal{T}$$\mathcal{R}$$\mathcal{S}$$\mathcal{R}$$\mathcal{S}$$\mathcal{R}$-$\mathcal{S}$$\mathcal{S}$ | 3 |

${\mathit{r}}_{\mathit{max}}$ | Ring k $(\#{\mathit{CH}}_{\mathit{k}})$ | ||||||||
---|---|---|---|---|---|---|---|---|---|

0 (1) | 1 (6) | 2 (12) | 3 (18) | 4 (24) | 5 (30) | 6 (36) | 7 (42) | ... | |

0 | 1 (sink) | ||||||||

1 | 1 + 6 | 1 | |||||||

2 | 1 + 18 | 3 | 1 | ||||||

3 | 1 + 36 | 6 | ${}^{5}{/}_{2}$ | 1 | |||||

4 | 1 + 60 | 10 | ${}^{9}{/}_{2}$ | ${}^{7}{/}_{3}$ | 1 | ||||

5 | 1 + 90 | 15 | ${}^{14}{/}_{2}$ | ${}^{12}{/}_{3}$ | ${}^{9}{/}_{4}$ | 1 | |||

6 | 1 + 126 | 21 | ${}^{20}{/}_{2}$ | ${}^{18}{/}_{3}$ | ${}^{15}{/}_{4}$ | ${}^{11}{/}_{5}$ | 1 | ||

7 | 1 + 168 | 28 | ${}^{27}{/}_{2}$ | ${}^{25}{/}_{3}$ | ${}^{22}{/}_{4}$ | ${}^{18}{/}_{5}$ | ${}^{13}{/}_{6}$ | 1 | |

⋮ | ⋮ | ⋮ | ⋮ | ⋮ | ⋮ | ⋮ | ⋮ | ⋮ | ⋱ |

**Table 6.**Frame structure for nodes in ${A}_{0}\cup {S}_{0}\cup {A}_{1}$. $({N}_{Intra},{N}_{Inter})=(3,7)$ and $\mathrm{Ring}\le 4$.

RING | CH | $\begin{array}{c}\mathit{Frame}\phantom{\rule{4.pt}{0ex}}\mathit{Structure}:\\ \mathcal{CONT}-\mathcal{TDMA}-\end{array}$ | $\begin{array}{c}\mathit{Frame}\phantom{\rule{4.pt}{0ex}}\mathit{Structure}:\\ \mathit{With}\phantom{\rule{4.pt}{0ex}}\mathcal{T}\phantom{\rule{4.pt}{0ex}}\mathit{as}\phantom{\rule{4.pt}{0ex}}\mathit{Starting}\phantom{\rule{4.pt}{0ex}}\mathit{Point}\end{array}$ |
---|---|---|---|

4 | $\langle 4,4\rangle $ | $\mathcal{S}$$\mathcal{SC}$-$\mathcal{S}$$\mathcal{S}$$\mathcal{S}$$\mathcal{S}$$\mathcal{S}$$\mathcal{T}$$\mathcal{S}$- | $\mathcal{T}$$\mathcal{S}$-$\mathcal{S}$$\mathcal{SC}$-$\mathcal{S}$$\mathcal{S}$$\mathcal{S}$$\mathcal{S}$$\mathcal{S}$ |

$\langle 4,3\rangle $ | $\mathcal{SC}$$\mathcal{S}$-$\mathcal{S}$$\mathcal{S}$$\mathcal{S}$$\mathcal{T}$$\mathcal{S}$$\mathcal{S}$$\mathcal{S}$- | $\mathcal{T}$$\mathcal{S}$$\mathcal{S}$$\mathcal{S}$-$\mathcal{SC}$$\mathcal{S}$-$\mathcal{S}$$\mathcal{S}$$\mathcal{S}$ | |

$\langle 4,2\rangle $ | $\mathcal{CS}$$\mathcal{S}$-$\mathcal{S}$$\mathcal{T}$$\mathcal{S}$$\mathcal{S}$$\mathcal{S}$$\mathcal{S}$$\mathcal{S}$- | $\mathcal{T}$$\mathcal{S}$$\mathcal{S}$$\mathcal{S}$$\mathcal{S}$$\mathcal{S}$-$\mathcal{CS}$$\mathcal{S}$-$\mathcal{S}$ | |

$\langle 4,1\rangle $ | $\mathcal{S}$$\mathcal{SC}$-$\mathcal{S}$$\mathcal{S}$$\mathcal{S}$$\mathcal{S}$$\mathcal{S}$$\mathcal{S}$$\mathcal{T}$- | $\mathcal{T}$-$\mathcal{S}$$\mathcal{SC}$-$\mathcal{S}$$\mathcal{S}$$\mathcal{S}$$\mathcal{S}$$\mathcal{S}$$\mathcal{S}$ | |

$\langle 4,0\rangle $ | $\mathcal{SC}$$\mathcal{S}$-$\mathcal{S}$$\mathcal{S}$$\mathcal{S}$$\mathcal{S}$$\mathcal{T}$$\mathcal{S}$$\mathcal{S}$- | $\mathcal{T}$$\mathcal{S}$$\mathcal{S}$-$\mathcal{SC}$$\mathcal{S}$-$\mathcal{S}$$\mathcal{S}$$\mathcal{S}$$\mathcal{S}$ | |

3 | $\langle 3,3\rangle $ | $\mathcal{CS}$$\mathcal{S}$-$\mathcal{S}$$\mathcal{S}$$\mathcal{T}$$\mathcal{R}$$\mathcal{R}$$\mathcal{R}$$\mathcal{S}$- | $\mathcal{T}$$\mathcal{R}$$\mathcal{R}$$\mathcal{R}$$\mathcal{S}$-$\mathcal{CS}$$\mathcal{S}$-$\mathcal{S}$$\mathcal{S}$ |

$\langle 3,2\rangle $ | $\mathcal{S}$$\mathcal{SC}$-$\mathcal{T}$$\mathcal{R}$$\mathcal{S}$$\mathcal{R}$$\mathcal{S}$$\mathcal{S}$$\mathcal{S}$- | $\mathcal{T}$$\mathcal{R}$$\mathcal{S}$$\mathcal{R}$$\mathcal{S}$$\mathcal{S}$$\mathcal{S}$-$\mathcal{S}$$\mathcal{SC}$- | |

$\langle 3,1\rangle $ | $\mathcal{SC}$$\mathcal{S}$-$\mathcal{S}$$\mathcal{R}$$\mathcal{S}$$\mathcal{S}$$\mathcal{S}$$\mathcal{T}$$\mathcal{R}$- | $\mathcal{T}$$\mathcal{R}$-$\mathcal{SC}$$\mathcal{S}$-$\mathcal{S}$$\mathcal{R}$$\mathcal{S}$$\mathcal{S}$$\mathcal{S}$ | |

$\langle 3,0\rangle $ | $\mathcal{CS}$$\mathcal{S}$-$\mathcal{S}$$\mathcal{R}$$\mathcal{S}$$\mathcal{T}$$\mathcal{R}$$\mathcal{S}$$\mathcal{R}$- | $\mathcal{T}$$\mathcal{R}$$\mathcal{S}$$\mathcal{R}$$\mathcal{CS}$$\mathcal{S}$-$\mathcal{S}$$\mathcal{R}$$\mathcal{S}$- | |

2 | $\langle 2,2\rangle $ | $\mathcal{SC}$$\mathcal{S}$-$\mathcal{R}$$\mathcal{R}$$\mathcal{R}$$\mathcal{S}$$\mathcal{S}$$\mathcal{S}$$\mathcal{T}$- | $\mathcal{T}$-$\mathcal{SC}$$\mathcal{S}$-$\mathcal{R}$$\mathcal{R}$$\mathcal{R}$$\mathcal{S}$$\mathcal{S}$$\mathcal{S}$ |

$\langle 2,1\rangle $ | $\mathcal{CS}$$\mathcal{S}$-$\mathcal{R}$$\mathcal{S}$$\mathcal{S}$$\mathcal{S}$$\mathcal{T}$$\mathcal{R}$$\mathcal{S}$- | $\mathcal{T}$$\mathcal{R}$$\mathcal{S}$-$\mathcal{CS}$$\mathcal{S}$-$\mathcal{R}$$\mathcal{S}$$\mathcal{S}$$\mathcal{S}$ | |

$\langle 2,0\rangle $ | $\mathcal{S}$$\mathcal{SC}$-$\mathcal{R}$$\mathcal{S}$$\mathcal{T}$$\mathcal{R}$$\mathcal{S}$$\mathcal{R}$$\mathcal{S}$- | $\mathcal{T}$$\mathcal{R}$$\mathcal{S}$$\mathcal{R}$$\mathcal{S}$-$\mathcal{S}$$\mathcal{SC}$-$\mathcal{R}$$\mathcal{S}$ | |

1 | $\langle 1,1\rangle $ | $\mathcal{S}$$\mathcal{SC}$-$\mathcal{S}$$\mathcal{S}$$\mathcal{S}$$\mathcal{T}$$\mathcal{R}$$\mathcal{R}$$\mathcal{R}$- | $\mathcal{T}$$\mathcal{R}$$\mathcal{R}$$\mathcal{R}$-$\mathcal{S}$$\mathcal{SC}$-$\mathcal{S}$$\mathcal{S}$$\mathcal{S}$ |

$\langle 1,0\rangle $ | $\mathcal{SC}$$\mathcal{S}$-$\mathcal{S}$$\mathcal{T}$$\mathcal{R}$$\mathcal{S}$$\mathcal{R}$$\mathcal{S}$$\mathcal{R}$- | $\mathcal{T}$$\mathcal{R}$$\mathcal{S}$$\mathcal{R}$$\mathcal{S}$$\mathcal{R}$-$\mathcal{SC}$$\mathcal{S}$-$\mathcal{S}$ |

**Table 7.**Parameters, ${N}_{m}=364;{p}_{act}=0.001;{({N}_{intra},{N}_{inter})}^{*}=(3,7)$; ${N}_{msCF}={N}_{msC}.{N}_{intra}+{N}_{msT}.{N}_{inter}.$ *If ${r}_{\mathrm{max}}=0,{N}_{intra}=1,{N}_{inter}=0.$

${\mathit{r}}_{\mathit{max}}$ | ${\mathit{M}}_{\mathit{c}}\left({\mathit{r}}_{\mathit{max}}\right)$ | ${\mathit{N}}_{\mathit{m}}^{{}^{\prime}}\left({\mathit{r}}_{\mathit{max}}\right)$ | $({\mathit{N}}_{\mathit{msC}},{\mathit{N}}_{\mathit{msT}})$ | ${\mathit{N}}_{\mathit{msCF}}$ | ${\mathit{\rho}}_{1}$ | ${\mathit{\rho}}_{2}$ | ${\mathit{\rho}}_{3}$ | ${\mathit{\rho}}_{4}$ | ${\mathit{\rho}}_{5}$ |
---|---|---|---|---|---|---|---|---|---|

0 | 363 | 364 | (63, 0) | 63 | - | - | - | - | - |

1 | 51 | 364 | (10, 3) | 51 | 0.8254 | - | - | - | - |

2 | 18 | 361 | (3, 1) | 16 | 0.8552 | 0.2850 | - | - | - |

3 | 9 | 370 | (2, 1) | 13 | 0.6967 | 0.2903 | 0.1161 | - | - |

4 | 5 | 366 | (2, 1) | 13 | 0.6456 | 0.2905 | 0.1506 | 0.0645 | - |

5 | 3 | 364 | (2, 1) | 13 | 0.5813 | 0.2712 | 0.1550 | 0.0871 | 0.0387 |

© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

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**MDPI and ACS Style**

Casares-Giner, V.; Inés Navas, T.; Smith Flórez, D.; Vargas Hernández, T.R.
End to End Delay and Energy Consumption in a Two Tier Cluster Hierarchical Wireless Sensor Networks. *Information* **2019**, *10*, 135.
https://doi.org/10.3390/info10040135

**AMA Style**

Casares-Giner V, Inés Navas T, Smith Flórez D, Vargas Hernández TR.
End to End Delay and Energy Consumption in a Two Tier Cluster Hierarchical Wireless Sensor Networks. *Information*. 2019; 10(4):135.
https://doi.org/10.3390/info10040135

**Chicago/Turabian Style**

Casares-Giner, Vicente, Tatiana Inés Navas, Dolly Smith Flórez, and Tito Raúl Vargas Hernández.
2019. "End to End Delay and Energy Consumption in a Two Tier Cluster Hierarchical Wireless Sensor Networks" *Information* 10, no. 4: 135.
https://doi.org/10.3390/info10040135