This is all handled within the Service Level Agreements and associated areas. Effectively each Service has it's hours of operations and each Service would typically be handled by a separate department. There's a wealth of information on how to set these up in the admin help file:
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In addition to the comments made by Jonathan.Schmidt if you are running a server with multiple supporting teams or running without SLA and just using the escalation model then you can create exceptions in the Escalation Engine for the Supporting teams. When the Incident (or SR) is assigned to a new team the escalation clocks will then change to run from the new team clocks as shown below for a simple priority model. Here the defaults are all to the Weekly HOP (default) but if the ticket is a P1 and assigned to the Server and Support team then it will move to their HOP. This can cater for a lots of teams having different hours.
As Mention previously is is simpler if you can do it by service but sometimes even using services you may find a need to override in this manner. To setup just click the level you wish to create an exception at and click Set Exceptions.
Now you can pick any drop down field to create an exception for:
You can now configure the escalation level etc. for that new branch or just use the default levels from above but they will fire based on your new HOP.