There are 2 ways when PDA accepts the input
- The PDA has finished reading the input and it is in the final state
- The PDA has finished reading the input and its stack is empty.
in your pda if we input "abbb"
$\left ( q_{0},abbb, z_{0} \right )$
$\vdash$ $\left ( q_{0},bbb, az_{0} \right )$
$\vdash$ $\left ( q_{1},bb, z_{0} \right )$
$\vdash$ $\left ( q_{2},b, a \right )$
we still got "b" unread and $abbb$ is a valid string.