You do have schedules? Since they are mainly used within SLA management, it seems like you are just not handling your processes correctly and now need a solution for that. 

Easiest way: create an sla definition (since you aren't using them, it doesn't matter what you put in it. Start time on opened, stop time on closed would be a good start. And then run a 'repair sla' on all your open incidents. It will give you the data you need in a much faster way than creating a script to validate on a schedule that may not even exist to get these records.

Following the SLA definition solution will give you a very fast and up to date, reportable view of the data.