Does Process deadline triggers escalation?
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No. Process deadline does not trigger anything. Modern Appian apps typically consist of multiple process models working together. A deadline on process level is rarely used. At least in my experience.
Why is the deadline present in the process model properties tab? I tried multiple ways to complete my active task by the deadline but it is not working. Could you give me an example of using a deadline?
As I already tried to explain, deadlines on process and task level to not do anything on their own. They can be used in process reports to display to the user. In case you want to skip a task, add an exception to that task and configure the timer in conjunction with the process and/or task deadline.