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                     (Eq. 172)

Mathinline
body ${{F}_{median}}({{f}_{VLR}},{{h}_{VLR}},{{h}_{VLT}},{{R}_{\max }},env)+{{F}_{slowfading}}(X%)={{P}_{VLT}}+{{g}_{VLT}}+{{g}_{VLR}}-sen{{s}_{VLR}}$

where the path loss is defined by a median loss plus an additional term representing the distribution

                                                                                                                    (Eq. 173)

where:

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The distribution of  the path loss  can be expressed in a general way by the following equation: Image Removed

Unit
body $Q(\mu +a,{{R}_{\max }})=y$

                                                                                        

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where Q  is the cumulative distribution for R_max and the resulting mean path loss (symbol to be inserted)

Mathinline
body $\mu $

 and an additional path loss due

Mathinline
body $a$

 due to availability or coverage  

Mathinline
body $y$

. The availability y of the system is linked to the coverage loss through the simple relation y = 1 – x. Assuming that slow fading can be approximated by log-normal distribution, i.e. median  mean, the relation  can be introduced where  b stands for a multiple of the standard deviation (symbol to be inserted)

Mathinline
body $\sigma $

.

Here are some examples for illustration:

• for 68 % coverage, b = 1;
• for 95 % coverage, (symbol to be inserted)

  Mathinline
  body $b$

   = 1.96;
• for 95.5 % coverage,  (symbol to be inserted)

  Mathinline
  body $b$

  = 2;
• for 99 % coverage, (symbol to be inserted)

  Mathinline
  body $b$

   = 2.58;
• for 99.9 % coverage, (symbol to be inserted)

  Mathinline
  body $b$

   =3.29.
  
  

The exact values can be easily determined by using the inverse Gaussian function.

Mathinline
body $v({{R}_{\max }})={{P}_{VLT}}+{{g}_{VLT}}+{{g}_{VLR}}-sen{{s}_{VLR}}-{{F}_{median}}({{f}_{VLR}},{{h}_{VLR}},{{h}_{VLT}},{{R}_{\max }},env)-b\sigma $

                      (Eq. 174)

The determination of the zero of function v, is made through a recursive method such as regula-falsi used in logarithmic scale which should yield a better precision. The solution of such a method provides the following equation:

           

Mathinline
body ${{\tilde{R}}_{\max }}={{R}_{\max 0}}-\frac{{{R}_{\max 0}}-{{R}_{\max 1}}}{v({{R}_{\max 0}})-v({{R}_{\max 1}})}v({{R}_{\max 0}})$

                        (Eq. 175)

In this case, formulas given for  need to be inverted. 

Then the equation:

Image Removed                (Eq. 165)

The determination of the zero of function v, is made through a recursive method such as regula-falsi used in logarithmic scale which should yield a better precision. The solution of such a method provides:

   Image Removed                                 (Eq. 166)

In this case, formulas given for (EQ to be inserted - only half is copied with math tool)  need to be inverted.

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Note 1: The inverse of the normalised Gaussian cumulative distribution is implemented through a piece approximation.

 

Note 2: Ro is to be set to 1 m (0.001 km) 

Note 3: If after running a simulation it appears that the resulting coverage radius is equal or very close to the minimum distance or the maximum distance used in the calculation of the coverage radius, it is likely that there is a mistake in the values provided. This can be solved by reducing the minimum distance or increasing the minimum distance used in the calculation, so that the algorithm can calculate the corresponding coverage radius.

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