How to write stories for internal projects

Question

I'm working on a project which runs as a process, on a regular basis. The results of the projects daily run are used by other projects. This brings me to question how our team writes stories.

The normal format, As a [role] I want [goal/desire] so that [reason] does not seem to apply, but at the same time I'm doubting that.

This is what we do now: The system shall execute after process ABC has finished successfully.

Should we be using the other projects (that utilize the data from our process) as the role portion of the stories? What we do now just seems too much like requirements..