# Multipath Routing Scheme for Optimum Data Transmission in Dense Internet of Things

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## Abstract

**:**

## 1. Introduction

- Developing a multipath routing scheme for dense IoT networks.
- Developing a route selection model that selects a group of routes from available routes to transmit data at higher speeds.
- Optimizing the proposed route selection model to obtain the optimum routes that achieve the maximum transmission speed.
- Performance evaluation of the proposed route selection scheme for dense IoT networks.

## 2. Multipath Routing Model

_{i}, where b

_{i}is defined in Equation (2).

_{i}is the delivery time (s). Data transferred over a group of routes are processed in parallel. Simultaneous transmission over several routes is impossible in the case under consideration; however, this applies only to the route’s first and last links. When transmitting over several routes, it is necessary to distribute the data transmitted. The distribution is built on the basis that each route carries a fraction of the data.

_{i}is the fraction of data carried by the ith route. The delivery time for the allocated fraction of data on each route used is a random variable with a distribution function F

_{i}(X). Then, there is a set of distribution functions of the delivery time of available routes (F), defined as follows.

_{wi}is the delivery time for the ith route of the selected route group, and W is the set of selectable routes. The distribution function of the delivery time T is G

_{w}(x) and is defined as follows.

## 3. Route Selection Method

_{i}, and selecting routes with certain distribution functions F

_{i}(t).

_{1}, x

_{2}, and x

_{3}. The considered mathematical expectations of these random variables are introduced as follows.

_{1}and x

_{3}).

_{1,2}(x), g

_{1,3}(x), and g

_{2,3}(x). According to Equation (5), the probability density is defined for the resulting solutions as follows.

_{i,j}(x)) is defined as follows.

_{1}and x

_{2}since the lowest m

_{i,j}in Figure 4 is m

_{1,2}(m

_{1,2}= 60.0, 5.0). For other choices, the values of mathematical expectation, according to Equation (6), are m

_{1,3}= 60.0, 9.0, and m

_{2,3}= 76.0, 7.0.

## 4. Optimizing the Proposed Route Selection Model

_{i}), which depends on the proportion of the data transmitted (β

_{i}), according to Equation (7). Thus, it is necessary to solve the problem of choosing β

_{i}that meets the problem’s solution, which significantly complicates the solution.

_{i}= β = 1/k, i = 1,…, k)). In the second stage, the problem of selecting values of distribution coefficients βi is solved. Algorithms 1 and 2 provide the pseudo-code of the proposed two stages.

Algorithm 1 First stage of the optimum route selection | ||||

Input data: | ||||

R: Set of known routes. | ||||

n: Number of known routes. | ||||

W: Set-group of selected routes. | ||||

K: Number of selected routes. | ||||

Step 1. | Input initial data R, n, W, k and initialize variables i = j = s = 1, W = φ, T = φ. | |||

Step 2. | Choose a route r_{i.} | |||

[This step selects a route from the set R to further evaluate its effect on the target function r_{i} ∈ R, r_{i} ∉ W.] | ||||

Step 3. | Calculate the value of the target function (Solve Equation (16)), Then: | |||

$T=T\cup {\overline{t}}_{w.g}$ | ||||

[This step calculates the next value of the target function and includes it in the set of values of the target function T. (At each iteration i (iteration of Phase I) the value of the target function is calculated for g selected routes, and g varies in Phase II from 1 to k.)] | ||||

Step 4. | Check the end of Phase I. | |||

If (i < n − g) | ||||

i = i + 1 | ||||

go to Step 2. | ||||

Else | ||||

perform the next step (go to step 5) | ||||

End if | ||||

[Phase I runs until all available routes from the set R that are not included in the set of selected routes W are checked] | ||||

Step 5. | Include the route in a group of selected routes. | |||

Calculate r: | ||||

$r=arg\underset{s}{\mathrm{min}}\left\{T,s=1,2,\cdots ,n-g\right\}$ | ||||

Then: | ||||

$W=W\cup r$ | ||||

[At this step, the set of target function values T is examined, and the minimum value is selected, and the route for which the corresponding value of the target function was obtained is included in the group of routes.] | ||||

Step 6. | Ending Phase II: | |||

If (g < k) | ||||

g = g + 1, | ||||

i = 1, | ||||

go to step 2 | ||||

Else | ||||

Perform the next step (go to step 7). | ||||

End if | ||||

Step 7. | Output the set of selected routes W | |||

End |

Algorithm 2 Second stage of the optimum route selection | |||||

Input data: | |||||

W: Set (group) of selected routes. | |||||

β: Set of traffic distribution coefficients. | |||||

b: Set of data transmission rate values for the selected channels. | |||||

K: Number of chosen routes. | |||||

Step 1. | Input initial data W, k and initialize variables i = j = s = 1, β = 1/k, T = φ. | ||||

Step 2. | Choosing the β_{i} coefficient | ||||

[This step selects the coefficient β_{i} from the set β to further evaluate its effect on the target function, β_{i} ∈ β.] | |||||

Step 3. | Calculation of the value of the target function for the changed traffic distribution coefficients β_{i}. | ||||

In this step, the value of β_{i} is changed by Δβ, and all other values β_{j} are corrected through multiplication by η_{j} coefficients so that the condition is: | |||||

$\sum _{j=1}^{k}}{\beta}_{j}{\eta}_{j}+{\beta}_{i}=1\forall k\in \mathbb{R},kn,j\ne i$ | |||||

Then, calculate the objective function as follows. | |||||

${\overline{t}}_{i}=\mathrm{min}\left\{E\left(\mathit{max}\left\{\frac{{{\beta}_{1}}^{*}}{{b}_{1}},\frac{{{\beta}_{2}}^{*}}{{b}_{2}},\cdots ,\frac{{{\beta}_{i}}^{*}}{{b}_{i}},\cdots ,\frac{{{\beta}_{k}}^{*}}{{b}_{k}}\right\}\right),E\left(\mathit{max}\left\{\frac{{{\beta}_{1}}^{**}}{{b}_{1}},\frac{{{\beta}_{2}}^{**}}{{b}_{2}},\cdots ,\frac{{{\beta}_{i}}^{**}}{{b}_{i}},\cdots ,\frac{{{\beta}_{k}}^{**}}{{b}_{k}}\right\}\right)\right\},$ | |||||

Where: | |||||

${{\beta}_{i}}^{*}=\left\{\right)separators="|">\begin{array}{c}{\beta}_{i}+\u2206\beta if{\beta}_{i}+\u2206\beta 1\\ 1otherwise\end{array}$ | |||||

${{\beta}_{i}}^{**}=\left\{\right)separators="|">\begin{array}{c}{\beta}_{i}-\u2206\beta if{\beta}_{i}-\u2206\beta 0\\ 1otherwise\end{array}$ | |||||

${{\beta}_{j}}^{*}={\beta}_{j}\frac{1-{\beta}_{i}-\u2206\beta}{1-{\beta}_{i}},j=1,2,\cdots ,k,j\ne i$ | |||||

${{\beta}_{j}}^{**}={\beta}_{j}\frac{1-{\beta}_{i}+\u2206\beta}{1-{\beta}_{i}},j=1,2,\cdots ,k,j\ne i$ | |||||

[This step calculates the value of the target function and includes it in the set of values of the target function T. (At each iteration i (iteration of phase I) the value of the target function is calculated for k routes.)] | |||||

Step 4. | Check the end of Phase I. | ||||

If (i < k) | |||||

i = i + 1 | |||||

go to step 2. | |||||

Else | |||||

perform the next step (go to step 5) | |||||

End if | |||||

[Phase I is executed until all traffic distribution coefficients β have been checked.] | |||||

Step 5. | Changing traffic distribution ratios. | ||||

$\beta ={\beta}^{*}=arg\underset{s}{\mathrm{min}}\left\{T,s=1,2,\cdots ,k\right\}$ | |||||

[In this step, the set of values of the target function T is examined and the minimum value is selected, the traffic distribution coefficients corresponding to this solution are chosen.] | |||||

Step 6. | Ending Phase II: | ||||

If the resulting solution gave the value of the target function t_{0}, less than the value obtained at the previous iteration t_{0} < T_{0} | |||||

i = 1, | |||||

go to step 2 | |||||

Else | |||||

Perform the next step (go to step 7). | |||||

End if | |||||

Step 7. | Output the set of traffic distribution coefficients β. | ||||

End |

#### 4.1. First Stage

_{i}. The use of dynamic programming is also proposed to solve this problem [25].

#### 4.2. Second Stage

## 5. Numerical Evaluation and Analysis

## 6. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 6.**Simulation results using the proposed method (red curve) and traditional method (blue curve).

**Table 1.**Simulation parameters [13].

Parameter | Value |
---|---|

Number of network nodes | 32 |

MAC protocol | IEEE 802.11 |

Packet size | 128 bytes |

Number of available routes | 12 |

Initial energy of nodes | 4 J |

Energy dissipated per bit | 50 nJ/bit |

Traditional routing protocol | OLSR protocol |

The Proposed Model | The Common Traditional Routing Schemes |
---|---|

${\alpha}_{T}=k.O\left(Logk\right)$ | ${\alpha}_{T}=n.O\left(Logn\right)$ |

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## Share and Cite

**MDPI and ACS Style**

Ateya, A.A.; Bushelenkov, S.; Muthanna, A.; Paramonov, A.; Koucheryavy, A.; Allaoua Chelloug, S.; Abd El-Latif, A.A.
Multipath Routing Scheme for Optimum Data Transmission in Dense Internet of Things. *Mathematics* **2023**, *11*, 4168.
https://doi.org/10.3390/math11194168

**AMA Style**

Ateya AA, Bushelenkov S, Muthanna A, Paramonov A, Koucheryavy A, Allaoua Chelloug S, Abd El-Latif AA.
Multipath Routing Scheme for Optimum Data Transmission in Dense Internet of Things. *Mathematics*. 2023; 11(19):4168.
https://doi.org/10.3390/math11194168

**Chicago/Turabian Style**

Ateya, Abdelhamied A., Sergey Bushelenkov, Ammar Muthanna, Alexander Paramonov, Andrey Koucheryavy, Samia Allaoua Chelloug, and Ahmed A. Abd El-Latif.
2023. "Multipath Routing Scheme for Optimum Data Transmission in Dense Internet of Things" *Mathematics* 11, no. 19: 4168.
https://doi.org/10.3390/math11194168