I always thought the more realistic answer was that the window to detect an emerging civilization is on the order of a couple decades. In that time they go from simple easily detectable modulation schemes that are broadcast at very high power to modulation schemes that are nearly impossible to differentiate from noise at much higher efficiency/lower power levels. Much like we have, in the 1950's it was all high power TV/Radio signals which were radiating into space on fairly narrow bands. Now days we have really wide band signals using really advanced modulation techniques frequently sitting at or below the noise floor (GPS!) and frequently very short range (5G!).

Then using your Drake equation variables calculate how many are within those couple decades at any given point that you can see.