|Donny Osmond playing Joseph and his coat of many colors.|
Exodus 1:8 speaks of a "new king, who did not know about Joseph" so that sounds like a new dynasty - which could be the Berbers, who came from the southern part of Egypt, the Upper Nile region. Which was approximately 1700 BC, or about 150 years before the biblical account of Moses. [ibid]
The person who was most likely Joseph himself was known to the Egyptians as Imhotep, the "genius" and "architect" who brought Egypt through the great famine. That, according to National Geographic, 1995.
The "Ark Discovery" web site puts Imhotep at about 2600 BC, in the reign of Djoser.[ibid]
2293 BC - Noah leaves the Ark
1750 BC - The birth of Joseph
1711 BC - Jacob enters Egypt
1640 BC - Death of Joseph
1576 BC - Birth of Moses
1496 BC - Exodus from Egypt
So, according to this timeline, the Hebrews were only in Egypt for 135 years . That being said, 135 is quite a long time especially when one considers that even in our own country (the USA) 135 years ago it was 1879. In a time where putting a man on the moon is like "ancient history" to today's youth, in 1879 we didn't have planes, automobiles, public electricity or even indoor plumbing was not gaining in popularity until the mid 19th century - and "outhouses" were still commonplace in 1879.  In 1879 the US population was a bit over 50 million  - in 2014 it is 317 million  - in 135 the US has increased over 6 fold - and that is with modern "birth control" efforts preventing literally millions of US citizens every year. Now, imagine the 12 tribes of Jacob, all living in Egypt for 135 years - and no "birth control!"
Let's use a conservative number and say there were only 5 sons born to each family in the descendants of Jacob (and only Jacob)...
12 sons of Jacob x 5 sons = 60 grandsons
60 grandsons of Jacob x 5 sons = 300 great-grandsons
Now 40 years
300 great-grandsons of Jacob x 5 sons = 1500 great-great-grandsons
Now 80 years
1500 great-great-grandsons x 5 sons = 7500 great-great-great-grandsons
Now 100 years
7500 great-great-great-grandsons x 5 sons = 37500 great-great-great-great-grandsons
Now 120 years
37500 great-great-great-great-grandsons x 5 sons = 187500 great-great-great-great-great-grandsons
Now 140 years
187500 g-g-g-g-g-grandsons x 5 sons = 937500 g-g-g-g-g-g-grandsons
So, in approximately the same length of time from Jacob going to Egypt till the time of the Exodus of Moses, there are, conservatively, about one million Hebrew descendents. Keep in mind, I only went with 5 sons and Jacob had 12 sons! I did not account for the daughters and/or for those Hebrews who intermarried with the Egyptians. Also keep in mind, by the time Jacob was brought into Egypt, his twelve sons were already prolific and he brought with him a small nation of what the Egyptians called "the Shepherd Kings" or "the Hyksos (whose migration to Egypt actually begins at least a couple hundred years before Joseph) . Easily we can see it would have been well over a million, and considering the Hyksos, it would have been several million.
Some believe the Pharoah of Joseph's time (who would not have gone by the name "Pharoah," as was common in Moses' time - and it was Moses who wrote the Joseph account) would have been Sesostris II of Dynasty 12.