I am wondering how one could simulate a repeated measurement design with a continous treatment in each group without crossover of treatments?

Imagine you want to test a specific behavioural intervention that increases waste seperation at the workplace (e.g. better labels). You test your treatment against a placebo treatment. You only have a fixed number of subjects (e.g. one recycling bins at each floot), but you assume that treatment effects increase over time each week and you potentially have an unlimited amount of time and can measure every week. How could one set up a simulation to find out the amount of time needed to have a power of at least 0.8?

I assume one knows the following:

```
N = 20
b = .5 #correlation between each week
c = .1 #change over time in control group
d = .002 #decrease of effectiveness of treatment each week
ATE_time = tau + .3 - t*d #effect of treatment per week, decreasing by d for each week t
```