$P_s(SINR_{threshhold}) = exp((-SINR_{threshhold}(r_{rx})^{\alpha}\rho^2)/P)$
$P_s(SINR_{threshhold})$ is reception probability according to SINR threshold. $\alpha$ is loss exponent and $\rho^2$ is noise variance in AWGN. $r_{rx}$ is the distance between the receiver and the transmitter. $P$ is the transmission power. I think the unit of power is W. But I don't know what the unit of SINR is. dB? or W? For example, when SINR threshhold is 8dB, if unit is dB, $SINR_{threshhold}$ is 8. But if the unit is W, it will be $10^8$. How can I find it?
Thank you
SNR stands for signal-to-noise-ratio. If you discuss SINR instead, then it reads (not 100% about the details) signal-to-interference-and-noise-ratio.
Anyway, both signal power and noise power (and also the power of the interfering signal from other users) is measured in Watts. But, this implies that their ratio has units Watts/Watts, in other words, a plain number. The relevant scale is logarithmic so this is more often than not coverted into decibels. So if the power of the (useful) signal is $X$ (Watts) and interference+noise combine to power $Y$, then $$ \text{SINR}=10\cdot\log_{10}\frac{X}{Y}\,\text{dB}. $$ Because it is in decibels, we multiply the logarithm of the ratio by ten.
Remember that the energy of a harmonic oscillator (like a radio wave) is proportional to the square of the amplitude. Squaring doubles the logarithm, so if you give $X$ and $Y$ in terms of the amplitudes as opposed to Watts (may happen naturally, if you do a simulation), then you need to multiply by two.
But, A Big But. Looking at the formula you gave it looks like it requires the actual SINR. NOT THE LOG-SCALED VERSION ABOVE. So the unit will be Watts/Watts, a plain number. Not the log-scale version measured in decibels.