# Rec. Itu-r bo. 1696

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## 1 Exact system availability

The following algorithm describes one possible approach to implementing the exact solution methodology described in § 2.3.2 to determine the overall system availability Ps.

Note that the propagation models in Recommendation ITU R P.618-8 for cloud, rain and scintillation fading are only valid over a combined range of exceedance (pu or pd) of 0.01% to 5%, the lower bound being imposed by the scintillation fading model. In the following procedure, this range is extended down to 0.001% by assuming that the scintillation fading at 0.01% is maintained for lower percentages.

Step 1: Set pu  pd  0.001% and calculate the associated C/(N + I)u and C/(N + I)d respectively using equations (1a), (1b), (2), (3), (4a) and (4b). Denote these as Yu  C/(N + I)u and Yd  C/(N + I)d. These represent the minimum C/(N + I) of interest in each link.

Step 2: Set pu  pd  5% and calculate the associated C/(N + I)u and C/(N + I)d respectively using the same equations as in Step 1. Denote these as Xu  C/(N + I)u and Xd  C/(N + I)d. These represent the maximum C/(N + I) of interest in each link.

Step 3: Set X  max(XuXd) and Y  min(Yu Yd).

Step 4: Define the number of points, M, in the required uplink and downlink PDFs. M should be chosen such that the required resolution on the final PDF is achieved. As a guideline, M > round[(X Y)/0.1] should be sufficient where round(x) is the next integer greater than x.

Step 5: Define M equally spaced values in the interval [10X/10 – dw, 10Y/10] and denote them as w(n) where for M – j + 1 we get w(M – j + 1)  10Y/10 – ( j – 1)*dw; dw  (10Y/10 − 10–/10)/ (M   2) and j  1, …, M. The array w(n) defines the values of over which the uplink and downlink PDFs will be defined.

Step 6: For j  1 to M

if w( j) < 10Xu/10

set Puj)  1;

else, if wj) > 10Yu/10

set Puj)  0;

else,

calculate Apu(pu) required to achieve C/(N + I)u = −10 log wj);

calculate pu associated with this Apu using Recommendation ITU R P.618 8;

set Puj)  pu/100;

end.

End-for-loop.

At the end of this step, we have the array Pu ( j) defining the CDF for the values of interest (i.e. wj)).

Step 7: Repeat step (5) to find Pd ( j) given C/(N + I)d, Xd and Yd. At the end of this step, we have the array Pd ( j) defining the CDF for the values of interest (i.e. wj)).

Step 8: Denote the PDF of as fu () and of as fd () defined by:  Step 9: Define k  m + j – 1, then z(k)  w(m) + wj) for m, j  1, …, M − 1 and thus  1, 2, …, 2*M − 3.

Step 10: Apply the convolution of the individual PDFs as follows: Note that if n is not in the interval [1, M – 1], then fu(n)  0 and fd(n)  0.

Step 11: The PDF of the overall aggregate C/(N + I) is then given by:

Prob(C/(N + I) 10 log z(k)) f (z(k))

Step 12: The system availability Ps which is the probability of the overall aggregate C/(N + I) being greater than a threshold (Z) is given by: where L is such that –10 log z(L)  Z and –10 log z(L + 1) < Z.

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