After some research and some help from other google-apps-script developers, the best solution is to use a combination of Gmail filtering system in addition to a time-driven-trigger.
So basically for a normal Gmail account there is a 1 hour/day computing time as mentioned in the documentation See reference here.
So what I did is set up a filter that adds a Label and a star to the incoming emails that need to be processed.
In my script I add the Labels in an array, I loop over the array of labels so that I process only the desired emails and not the whole inbox.
Once processed, the script removes the star from the processed email.
This way you don't lose your precious compute time, and you don't reach the daily limit.
I then set a time driven trigger that runs every 10 minutes.
You can also set up the time driven trigger to send you a daily "Summary of failures " so that you can see what went wrong with your script and fix what has to be fixed.