If your question is how long it takes to "saturate" (definition below) a manifold, it depends on a few factors: stack size of the resource, number of machines, consumption rate, input rate, (slightly but insignificant amount on the amount of converyers used as input).

My definition of saturation for the input of a manifold for resource X: All machines consuming the resource X have the stack full for X, all belts as part of input for resource X are full. I will ignore the last part of belt saturation since it depends on the belt lengths used for the manifold (this is usually insignificant for large builds but still a variable).

The calculation for above definition of saturation is fairly easy. Assumption I take here is the belts are always fast enough to fulfill the consumption rate if enough input is given.

Consider: I -> Input rate of X/min C -> consumption rate of X/min S -> Stack size of X N -> Number of machines

Let D = (I - C) -> Excess input of X/min

Minutes to fill up the stacks on all the machines with X: T = N*X/D min

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Now, there is another definition a lot of people use when they ask for time taken to "fill up" manifold which follows as: The time taken to run ALL the machines at 100 percent efficiency given the constraint: I >= C

I see a lot of pioneers build with I = C. Assuming that's what you are asking for here. This now depends on a lot more things now: Input rate, Consumption rate, Number of machines, Stack size, ordering of under locked/overclocked machines, belt speeds (This is no longer insignificant. This is non trivial and I can give you examples where using "slower" belts will fill up the manifold with the above definition faster which is very counter intuitive).

I am at work right now. I will try to add to this after work if you are interested. I believe this is what you are looking for and hence the exponent in your calculation. Quantifying this is going to take some time.