I'm trying to derive the language it represents. However, I'm kind of new to those topics. What happens if input b is gathered once or more than one time at state q without a in the stack? It does not have any transition, does it fail or since it is stuck is it accepted? example: abb
For many DFAs and PDAs, there is a computation path that will lead to a permanent reject state (and once computation reaches that reject state, any further transitions only return to that one state). For DFAs and PDAs that have such a state, these states (and all the transitions that lead to that state) are not shown in a diagram or are even left out of the transition description of such machines, with the understanding that there exists an implicit reject state.
For machines with an implicit reject state, any computational step that is encountered that is not explicitly described in the machine's definition is understood to just lead to the implicit reject state, never to be able to return back to acceptance. It can be thought of as a "crash" state or a "dead" state.