Petri nets boundedness
For the given Petri Net, the number of tokens for P_1,P_2  and P_3  is represented by m_1,m_2  and m_3  respectively. Based on the definition of boundedness of a Petri Net by Popova-Zeugmann (2013), the given Petri Net is bounded because m_1 ≤ 2,m_2 ≤ 2 and m_3 ≤ 2. In other words, there is a natural number (2 in this case) such that the number of tokens in P_1,P_2  and P_3  never exceeds the number.

I am also providing an example of an unbounded Petri Net and an example of a bounded Petri Net. Regardless of the initial marking, the Petri Net in Figure 1 is not bounded (or it is unbounded) because m_0 → ∞ when T_0  keeps firing.

For the given initial marking, the Petri Net in Figure 2 is bounded because m_0 ≤ 1 and m_1 ≤ 2. In general, the Petri Net in Figure 2 is bounded: let the initial value of m_0 = q_0  and the initial value of m_1 = q_1; m_0 ≤ q_0  and m_1 ≤ 2q_0+q_1



References

Popova-Zeugmann, L. (2013). Chapter 2 The Classic Petri Net [Electronic version]. Time and Petri Nets. Berlin; Heidelerg: Springer-Verlag.