osTicket doesn't really have the concept of escalation in it.
You could setup three departments.
Tech Support - Tier 1
Tech Support - Tier 2
Tech Support - Tier 3
If you have multiple staff
Say
Janus Doh (t1)
Jane Doe (t2)
Luke Skysaunter (t3)
Each one of them has access to one of the Departments (as denoted by tier 1 = t1, tier 2 = t2, etc).
A ticket comes in and is opened in Tech Support - Tier 1.
Janus does some basic troubleshooting, and realize he does not have access to complete task.
He updates the ticket and Transfers to Tech Support - Tier 2.
Jane sees the ticket in her queue and starts working on it. She does have the access to complete the task, but something goes wrong with the fix. So she updates the ticket and transfers it to Tech Support - Tier 3.
Luke Skysaunter recently back from his trip to the Dago Bar for a late lunch see the ticket and does the work to fix it. Updates the ticket and closes it.
It's not an automated process, but it does have an easily defined process of escalation.