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Not 100% sure if this is what you're looking for, but I've got something similar setup...
What I've done was created a new email template (Setup/Email/Templates) that triggers an automatic email to the client when a tech changes the status to 'resolved'.
The email to the client has a voting button in this case asking them if the issue was fixed or not.
If Yes, then the ticket is automatically closed.
If No, then the ticket is reopened and the tech is notified.
To achieve something like based off of client response times (which is what I think you're after), would require at least an email template and a new action rule and probably a new status type.
Action Rule Info
* on all Ticket updates
* only if criteria did not match before ticket was updated
(matching ALL) Latest Update Trigger Email From a Client
(matching ANY) Date Updated wihtin the last ? business days
Change Status No Response
To really make sure you're covered, you'd create another action rule based off the above, but choose Client Interface instead of Email From a Client.
Then in turn, create an email template to fire off a notice of impending closure to the client if the ticket status is No Response.
(you'd reserve the No Response status for just this scenario)
It looks good on paper but you might have to tweak it a bit