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(Xu, Xd) 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 [10–X/10 – dw, 10–Y/10] and denote them as w(n) where for n M – j + 1 we get w(M – j + 1) 10–Y/10 – ( j – 1)*dw; dw (10–Y/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) < 10–Xu/10
set Pu( j) 1;
else, if w( j) > 10–Yu/10
set Pu( j) 0;
calculate Apu(pu) required to achieve C/(N + I)u = −10 log w( j);
calculate pu associated with this Apu using Recommendation ITU R P.618 8;
set Pu( j) pu/100;
At the end of this step, we have the array Pu ( j) defining the CDF for the values of interest (i.e. w( j)).
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. w( j)).
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) + w( j) for m, j 1, …, M − 1 and thus k 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.