Beylerbeyi:
If only 10% of the population were Greek, then Anatolia was pretty much unpopulated before the turks arived (or did they just die of the black death or something?). The fact that there are not that many greeks today is because of cultural/religios conversion over 400 years or more. Why stay christian, if you have to give your oldest son to the ottoman army and pay more taxes.
Anatolia was not unpopulated, but it was not densely populated either when the Turks conquered it. The thing is the Turks conquered it in 1070's and the game starts in the 1440s. That's 370 years after the conquest which is plenty of time to convert and absorb the local population so that the ratio of Greeks is about 10% around 1500. As I wrote before the number of Greeks in the region increased in the 19th and 20th centuries. It is wrong to assume that since they were 20% in 1910 they were more in 1810.
We also know these ratios because Ottomans kept detailed records of taxable assets which included the number of households. Studies of these census records were published by historians such as Barkan and Uzuncarsili decades ago, and form the material basis of Ottoman history studies.
You are wrong about "giving your oldest son" thing, but I won't discuss it here.
Here's what I wrote about the Devshirme system before (I edited a bit):
OE actually ruled the Balkans for 200 years with less problems than Anatolia. Balkan problems started with nationalism (still causing problems there), centuries after Devshirme practice was abolished. The Devshirme numbers were too small to make a major impact anyway. However Balkan romantic nationalism selectively resurrected tales of people hating the Devshirme: Turks raided villages and kidnapped kid, with moms crying etc. It is bullshit. In fact for each peasant who cried there was another who actually wanted to give their child away (some even bribed the officers to take the child), because the kids would have great jobs (anything from janissary to grand vizier) and most would remember their family. And the whole selection was organised by the local Orthodox church.
Also this:
Devshirme practice was introduced in the 1390s and effectively lasted until about 1590. The devshirme collected from the whole Balkans numbered at most around 3000 in one go, usually it was in the hundreds. Some peasants even bribed the collectors to give their children away. The children were mostly teenagers, so most remembered their families and helped them if they had the chance later on. It was of course not done to provide education for the Balkan peasants, it was done to produce an effective ruling class loyal to the Sultan and replace the Turkish nobility who could challenge the House of Osman.