Spotfire Statistics Services startup issues

My company does monthly patch deployment that requires servers to be restarted. We're having issues bringing up Spotfire Statistics Services after restarting the worker and manager servers. We have no indication that anything is wrong until a user tries to work with a Spotfire analysis in the web player and we see errors like this:

Could not execute function call.

The remote server returned an error: (500) Internal Server Error

We ultimately have to log on to the Manager and Worker servers and restart the Stat Services service on each, until eventually we're back to life. The most reliable method seems to be actually stopping the service on each server, waiting a minute, then starting the service on the Manager node, waiting another minute, and then starting the service on the Worker node.

Is there any configurations we can review or modify to ensure Stat Services comes back up successfully after an outage?