# Advancements in the Statistical Study, Modeling, and Simulation of Microwave-Links in Cellular Backhaul Networks

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## Abstract

**:**

## 1. Introduction

- Spatial density of the CMLs;
- Orientations of the CMLs;
- Lengths of the CMLs.

## 2. Spatial Distribution

^{2}) [14]. It is recommended to maintain a minimum of such region size. Accordingly, in this paper we examined regions that are indeed 10 × 10 (km

^{2}).

## 3. Length and Orientation

## 4. Modeling the Relationship between Cellular Microwave-Links’ (CMLs) Length and Density

#### 4.1. Approach to Modeling

#### 4.2. Methodology

^{2}), and it is noted as an explicit function of $l$, the CML’s mean length (km). ${a}_{1}$ and ${a}_{2}$ are constant coefficients to be optimized via non-linear regression. They are chosen to be those that minimize the mean squared error (MSE). For example, this is the optimization process for the third model:

## 5. Simulating CMLs for Computational Experiments

- It allows for a controlled study. Simulated CMLs allow one to account for every attribute they possess.
- It strengthens the integrity of the results. When deducing an outcome of an experiment, since the simulation of the CMLs is controlled, one can determine a clear set of assumptions under which the outcome holds.
- It provides statistical robustness of the results. When simulating CMLs, the amount of CMLs is not limited, thus allowing one to utilize as many CMLs as necessary for the statistical experiment.
- It introduces a new experimental feature, sensitivity analysis. The computer simulation allows one to tweak the CMLs’ parameters and evaluate their effect.

#### 5.1. Modeling a Single CML as a Computational Structure

^{2}) area), each array element represents a pixel, and thus the size of the array is directly derived from the chosen resolution. A pixel that the CML crosses has a positive value equal to the length of the overlap that the CML has with that pixel. A pixel that the CML does not cross is zeroed. Here we use the terms “pixel” and “array element” interchangeably. Figure 6 portrays how an array models a CML.

#### 5.2. Key Simulation Factors

^{2}), a set of 1000 CMLs is generated. The experiment of interest requires a density of 2 CMLs per km

^{2}, thus 200 CMLs are randomly selected, and the fact that there is a “sufficiently high” number of CMLs allows for many Monte-Carlo iterations with different subsets of CMLs. Here “sufficiently high” refers to a number so high that it allows for the maximal number of CMLs to be randomly selected out of the set, multiple times. The contrary would be to limit the set to having “just enough” CMLs, thus allowing only one manner for selecting the maximal number of CMLs by simply selecting the entire set. The latter would not allow for repetitions of the experiment in a Monte-Carlo setting.

#### 5.3. The CMLs Simulation Algorithm

- Region Area:This is a physical parameter, e.g., 10 × 10 (km
^{2}). - Spatial Resolution—N:N is the number of pixels the area is being partitioned to.
- Environment Type—Mean CML Length:As established in Section 3, simulating a different environment corresponds to choosing a different mean CML length. Values typically range from 1.5 to 10 (km).
- The Number of CMLs to Generate n
_{set}:Given the region area and the maximum spatial density desired for the experiment, let n_{set}be sufficiently high so to allow for multiple random draws of the maximal number of CMLs.

- Generate an Array of n
_{set}Objects, Each is a CML Length Value:Each length is drawn randomly from an exponential distribution with the above pre-set mean length. - Assign Each Object an Orientation:Each of the generated objects is assigned an angle drawn uniformly from $\left[0,\pi \right)$.
- Assign Each Object a Position:Each object’s beginning point is drawn uniformly in a square of dimensions $\sqrt{N}\times \sqrt{N}$. Then, the end point is defined by drawing a line based on the length and orientation of the object.Note that in this step, each CML object is defined by “continuous”, i.e. not discrete, measures.
- Calculate Quantized Pixels Values:In order to suit the discrete model, the CML is being represented as an array.For each CML object, partition the $\sqrt{N}\times \sqrt{N}$ area to $\sqrt{N}\times \sqrt{N}$ squares, each represents a pixel.Set 0 to a pixel that does not have the CML pass through it. For a pixel that the CML does pass through, assign a positive value equal to the physical length of the CML’s overlap with the pixel’s region (i.e., overlapping with the square). See Figure 6 for a graphical description.

## 6. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A

**Figure A1.**Israel’s surroundings. The yellow lines show the distribution of Cellcom’s CMLs [13].

**Table A1.**The urban regions analyzed [13].

Geographic Boundary | Tel Aviv | Jerusalem | Haifa |
---|---|---|---|

Min. latitude coordinate | 32.013 | 31.74 | 32.765 |

Max. latitude coordinate | 32.096 | 31.81 | 32.825 |

Min. longitude coordinate | 34.776 | 35.175 | 34.985 |

Max. longitude coordinate | 34.8739 | 35.235 | 35.075 |

**Table A2.**The suburban regions analyzed [13].

Geographic Boundary | Hasharon | Caesarea | Nazareth |
---|---|---|---|

Min. latitude coordinate | 32.15 | 32.41 | 32.615 |

Max. latitude coordinate | 32.3 | 32.52 | 32.732 |

Min. longitude coordinate | 34.83 | 34.91 | 35.224 |

Max. longitude coordinate | 34.98 | 35.04 | 35.374 |

**Table A3.**The rural regions analyzed [13].

Geographic Boundary | Top North Area | Kseifa Area |
---|---|---|

Min. latitude coordinate | 32.7 | 31.4 |

Max. latitude coordinate | 33.09 | 31.008 |

Min. longitude coordinate | 35.15 | 35.26 |

Max. longitude coordinate | 35.82 | 34.905 |

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**Figure 1.**A single cellular microwave–link (CML) [6].

**Figure 2.**Typical cellular microwave-link (CML) topologies. All four are such that the number of CMLs is approximately identical to the number of BS [6].

**Figure 3.**The distribution of CML angles. (

**a**) all Israel, (

**b**) top of northern Israel as rural, (

**c**) Hasharon as suburban, (

**d**) Tel Aviv as urban [6].

**Figure 4.**Lengths distributions for various environments. (

**a**) all Israel, (

**b**) top of northern Israel as rural, (

**c**) Hasharon as suburban, (

**d**) Tel Aviv as urban [6].

**Figure 5.**The empirical relationship between a CML’s density and mean length in a given region. The black curve was derived through non-linear regression [13].

**Figure 6.**A discrete model of a CML. (

**a**)—a single CML of length 4.8 pixels, (

**b**)—its discrete representation.

**Figure 7.**Portraying the main attributes of the synthetic CMLs. (

**a**) spatial distribution, (

**b**) lengths distribution, (

**c**) angles distributions [6].

**Table 1.**Statistical base station (BS) densities [10]. BS densities depend on population densities. Since cellular microwave-links (CMLs) topologies are such that the number of CMLs is nearly identical to the number of BS, this table also reflects spatial densities of CMLs.

Region | Area (km^{2}) | BS Amount | BS Density (1/km^{2}) |
---|---|---|---|

Most Dense City | 60 × 40 | 6251 | 2.604 |

Second Densest City | 30 × 50 | 1911 | 1.274 |

Third Densest City | 40 × 40 | 977 | 0.611 |

Rural | 200 × 200 | 12,691 | 0.317 |

Region | Area (km^{2}) | CMLs Amount | CMLs Density (1/km^{2}) | CMLs Mean Length (km) | |
---|---|---|---|---|---|

- | All of Israel | 22,770 | 3624 | 0.16 | 3.54 |

Urban | Tel-Aviv | 85.18 | 264 | 3.1 | 1.48 |

Jerusalem | 44.16 | 141 | 3.2 | 1.26 | |

Haifa | 56.14 | 159 | 2.8 | 1.58 | |

Sub-urban | Hasharon | 235.54 | 124 | 0.53 | 2.5 |

Caesarea Area | 149.27 | 77 | 0.52 | 2.3 | |

Nazareth Area | 182.78 | 89 | 0.49 | 2.4 | |

Rural | Top North Israel | 2718.74 | 278 | 0.1 | 4.7 |

Kseifa Area | 1474.73 | 69 | 0.05 | 8.07 |

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**MDPI and ACS Style**

Gazit, L.; Messer, H.
Advancements in the Statistical Study, Modeling, and Simulation of Microwave-Links in Cellular Backhaul Networks. *Environments* **2018**, *5*, 75.
https://doi.org/10.3390/environments5070075

**AMA Style**

Gazit L, Messer H.
Advancements in the Statistical Study, Modeling, and Simulation of Microwave-Links in Cellular Backhaul Networks. *Environments*. 2018; 5(7):75.
https://doi.org/10.3390/environments5070075

**Chicago/Turabian Style**

Gazit, Lior, and Hagit Messer.
2018. "Advancements in the Statistical Study, Modeling, and Simulation of Microwave-Links in Cellular Backhaul Networks" *Environments* 5, no. 7: 75.
https://doi.org/10.3390/environments5070075