It is generally hard find closed-form expressions. It always the same math involved:

Say that the SINR is S/(I+sigma^2), where S and I are random due to fading. Then the outage probability when communicating at the rate x is:

Pr{ log2(1+S/(I+sigma^2)) < x } = Pr{ S/(I+sigma^2) < 2^x - 1 }.

If you know the distribution of S/(I+sigma^2), then you can compute this directly.

If you only know the PDFs of S and I individually, then you need to rewrite it as

Pr{ S/(I+sigma^2) < 2^x - 1 } = Pr{ S - I (2^x - 1) < sigma^2 (2^x - 1) }.

Now "sigma^2 (2^x - 1)" is a constant, but you need to compute the PDF of "S - I (2^x - 1)", which can be done via integration. The joint PDF of the sum of random variables is obtained by convolution.

At the end of the day, the chance that you get a nice PDF of "S - I (2^x - 1)" so that you then compute the probability is closed form by integration over the PDF from 0 to sigma^2 (2^x - 1) is not so big.

Emil Björnson thanks for the detailed answer. Isn't possible to directly consider the expression of the SINR for a specific value of a threshold x, so that Poutage=Pr{ S/(I+sigma^2) < x } ?

You can look up the note Capacity Limits of MIMO Systems - by Andrea Goldsmith, Syed Ali Jafar, Nihar Jindal, and Sriram Vishwanath available at https://web.stanford.edu/class/archive/ee/ee359/ee359.1062/cup_mimo.pdf

I would highly recommend you follow what Dr. Emil Björnson said. So the outage probability can be calculated based on a certain parameter which so-called SINR. Therefore, when the SINR goes below a threshold ((TH): which can be determined based on whether the UE that achieves such value of SINR can or can not send or receive to/from the base station that connects to) so now we can count this UE is in an outage. To calculate the outage probability of the whole system takes the CDF of the SINR values of all the UEs.

Pr(outage)= Pr(SINR

In my view, you should have the distribution of SINR which is dependent on the channel distribution. Then, due to outage = P(SINR