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There are 2 ways when PDA accepts the input

  1. The PDA has finished reading the input and it is in the final state
  2. 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.

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