I checked my own models and i'm using absolute path names, so probably i was wrong, this was just an idea 🙂
But when you just build and upload a small modell and then try different kinds of path-names you'll figure it out! If you want you can change the input file to a model parameter, then you can enter the path names as a string value before running the model - perhaps this is easier to find out the correct path for you.
And if this doesn't work, you should check the read/write permissions again. And if it doesn't work after these steps, you can write me an email and i'll have a look at it (but i'm just a non-professional and not working for esri).