OK, making a REALLY SIMPLE model, I allowed that we started with 2 people, and that the population doubled each 30 years. This is an underestimation - most humans produce more than 2 kids, and usually before they are 30 years old. However I had to allow for die-off of the older folk, so an average of double per cycle seemed OK.
Results: (population, years from zero)
<table border="1"><tr><td>2</td><td> 60</td></tr><tr><td>4</td><td> 90</td></tr><tr><td>8</td><td> 120</td></tr><tr><td>16</td><td> 150</td></tr><tr><td>32</td><td> 180</td></tr><tr><td>64</td><td> 210</td></tr><tr><td>128</td><td> 240</td></tr><tr><td>256</td><td> 270</td></tr><tr><td>512</td><td> 300</td></tr><tr><td>1,024</td><td> 330</td></tr><tr><td>2,048</td><td> 360</td></tr><tr><td>4,096</td><td> 390</td></tr><tr><td>8,192</td><td> 420</td></tr><tr><td>16,384</td><td> 450</td></tr><tr><td>32,768</td><td> 480</td></tr><tr><td>65,536</td><td> 510</td></tr><tr><td>131,072 </td><td>540</td></tr><tr><td>262,144</td><td> 570</td></tr><tr><td>524,288</td><td> 600</td></tr><tr><td>1,048,576</td><td> 630</td></tr><tr><td>2,097,152</td><td> 660</td></tr><tr><td>4,194,304</td><td> 690</td></tr><tr><td>8,388,608</td><td> 720</td></tr><tr><td>16,777,216</td><td> 750</td></tr><tr><td>33,554,432</td><td> 780</td></tr><tr><td>67,108,864</td><td> 810</td></tr><tr><td>134,217,728</td><td> 840</td></tr><tr><td>268,435,456</td><td> 870</td></tr><tr><td>536,870,912</td><td> 900</td></tr><tr><td>1,073,741,824</td><td> 930</td></tr><tr><td>2,147,483,648</td><td> 960</td></tr><tr><td>4,294,967,296</td><td> 990</td></tr><tr><td>8,589,934,592</td><td> 1,020</td></tr><tr><td>17,179,869,184</td><td> 1,050</td></tr></table>
This suggests that the current population of about 8.5 billion people could be reached in under 1000 years. It also suggests that 200 years ago, when the USA was born, the population of the Earth was just under 70 million people. And that King Richard the Lionheart probably sat on the throne of England with a only few dozen people to cheer him (so who did he crusade against, then?).
Do you think I should adjust my model parameters at all?
[Edit: numbers formatted - I hope]