Sorry its taken a long time to reply to this, but Ive been rather busy with work.
Im not looking at producing the occasional 1 off Settler here, but churning them out one after the other.
The ability to grow to and maintain the population is only half of the ability to produce Settlers. Building a Granary will allow us to replace our population in half the time, giving 4 turns to gain/replace the population for a Settler compared to 8 turns without. Produce Settlers quicker than this and our population will not grow quickly enough, and we will need to produce other units/improvements in between Settlers to maintain the population. Produce them more slowly and our population will continue to grow resulting in unhappiness problems.
The other half of the equation is therefore the ability to produce enough shields to keep producing the Settlers at that optimum rate. This is not changed by building a Granary, but by the tiles in the city radius, improvements to those tiles and the city population.
In order to produce lots of Settlers we need both high growth and enough production. Some cities may have high growth, but not achieve the required production levels resulting in the population growing out of control and unhappiness problems here we may be better off producing Workers as we can produce 3 Workers for the shields of 1 Settler. Other cities may have high production, but low growth, here we can produce Settlers far faster than the city can cope with and it would be better to produce military, improvements and Wonders with only the occasional Settler.
So looking at Deux Rivières. Here we have both the potential of growth and production.
We need 30 shields to build a Settler so to achieve our maximum sustainable rate thats an average of 3.75 shields/turn without a granary and 7.5 shields/turn with.
We need to keep the labourer on the wheat to maintain the growth, so we can currently produce, with no further improvement:
Code:
Pop Labourers on produces
1 Wheat +5 food, 1 shield
2 Wheat and Bonus grass +5 food, 3 shields
3 Wheat and 2 Bonus grass +5 food, 5 shields
4 Wheat, 2 Bonus grass, and Forest +4 food, 7 shields
4 Wheat, 2 Bonus grass, Irrigated Flood Plain +6 food, 5 shields
5 Wheat, 2 Bonus grass, 2 Forest +3 food, 9 shields
5 Wheat, 2 Bonus grass, Forest, Irrigated FP +5 food, 7 shields
The unhappiness at populations of 3 and 4 can be dealt with using Military Police. 5 would currently require a bit of slider. The slider would no longer be required once the dyes (or wines) are linked
So without a granary all we need (simply) is:
Code:
pop 2 for 4 turns x 3 shields = 12 shields
+pop 3 for 4 turns x 5 shields = 20 shields
After 8 Turns Total = 32 shields enough for the Settler
This will produce our Settler in the same time it takes the city to replenish the population so our city will balance at 2 or 3 in size, growing to 4 as the Settler is produced. If the city grows larger, the extra production would produce another Settler before the population was large enough and our population would drop lower, lowering our production, so the next Settler couldnt be built as quickly allowing the population to rise again.
So what about with a Granary? Well to produce 7.5 shields/turn over 4 turns we obviously need a higher population, but at the moment, two of our high producing tiles the forests do not produce enough food to maintain our 5 excess. This has the effect of slightly slowing the growth rate and slightly extending the time between Settlers.
So with a granary what we get is
Code:
pop 4 for 1 turn x 5 shields = 5 shields (+6 food)
+pop 4 for 1 turn x 7 shields = 7 shields (+4 food)
+pop 5 for 2 turns x 9 shields = 18 shields (+3 food x 2)
After 4 Turns Total = 30 shields enough for the Settler
But this does not generate enough food (4 short) to grow at the same time as the Settler is produced, so the population drops to 3. We therefore need to look at the next Settler as well:
Code:
pop 3 for 1 turn x 5 shields = 5 shields (+5 food)
+pop 4 for 1 turn x 5 shields = 5 shields (+6 food)
+pop 4 for 1 turn x 7 shields = 7 shields (+4 food)
+pop 5 for 2 turns x 7 shields = 14 shields (+5 food x 2)
After 5 Turns Total = 31 shields enough for the Settler
This time the Settler is produced in 5 turns and the population grows to 6 just as the Settler is produced, and we are back where we started at a population of 4.
So at the moment with a Granary we cant quite get a Settler every 4 turns, but we can get 2 Settlers in 9 turns. This compares to 2 in 16 turns without.
The increase in production of Settlers is obvious, but what about the time it takes to build a Granary? How many Settlers could we have produced in that time?
Well if we take our current starting population of 2 and use the time it takes to grow to the required size to build the Granary, then it can be built in 12 turns:
Code:
pop 2 for 4 turn x 3 shields = 12 shields (+5 food)
+pop 3 for 4 turn x 5 shields = 20 shields (+5 food)
+pop 4 for 4 turn x 7 shields = 28 shields (+4 food)
After 12 Turns Total = 60 shields enough for a Granary
The population is still at 4 and will not grow to 5 for another turn. We can therefore embark on our 4,5,4,5 turn Settler building cycle.
So our potential Settler output is as follows
Code:
No. Settlers Turns to Produce: No Granary With Building Granary
1 8 16 (12+4)
2 16 21
3 24 25
4 32 30
5 40 34
10 80 57
So in terms of founding our cities taking the time to build the granary will have the following effect:
Code:
1st New City founded [COLOR=red]8 turns later[/COLOR]
2nd New City [COLOR=red]5 turns later[/COLOR]
3rd New City [COLOR=red]1 turn later[/COLOR]
4th New City [COLOR=green]2 turns earlier[/COLOR]
5th New City [COLOR=green]6 turns earlier[/COLOR]
10th New City [COLOR=green]23 turns earlier[/COLOR]
This is without looking at rushing the Granary or improving any further tiles.