Its failing because Server is trying to overwrite the output which exists, but it cant (you know this).
The issue here is you should be writing your output to a unique location, not your working directory where you authored your model. Server has ways to write to a unique location (arcgisjobs directory)
I'd read this help topic: http://help.arcgis.com/en/arcgisdesktop/10.0/help/index.html#/Creating_models_for_geoprocessing_serv...
In particular, you want to look at this section: Intermediate and output data
You'll see here that Server can handle the output and put it in a location where no output exists (thus no chance for collision), by using the %scratchworkspace% inline variable.
In short, simply changing your output path from the c:\student\test___buffer.shp you have to something like %scratchworkspace%\buffer.shp, republishing or restarting the service, should make it work.