Permutation or Combination on number of states in Petri Net

Question

I have got a Petri Net below,

Petri Net Sample

The circles are places and squares are transitions. There are 6 tokens in each place as shown and this is the initial markings. The total number of reachable states is 7 to the power of 6 = 117,649.

I suspect the calculation is based on permutation or combination but I couldn't figure out why is it 7 to the power of 6. Can someone enlighten me please?

Thank you in advance, Lobbie

Answer 1

Given the following type of Petri Net, based on Lobbie’s example (2016):

1. Every transition has one input and one output.
2. Every place is an input of one and only one transition or an output of one and only one transition.
3. The initial marking of every input place is k (in Lobbie's example k=6).
4. The initial marking of every output place is 0.
5. There are s transitions (in Lobbie's example s=6).

If the number of reachable markings (n) include the initial marking then it is equal to the formula: (k+1) raised to the power of s:

n = (k+1) ^ s

The following two examples illustrate the computation for the number of reachable markings using the formula.

Example 1

If k=1 and s=2 then the number of reachable markings is (1+1) raised to the power of 2 which is 4.

Example 2

If k=1 and s=3 then the number of reachable markings is (1+1) raised to the power of 3 which is 8.

Example 3

If k=2 and s=2 then the number of reachable markings is (2+1) raised to the power of 2 which is 9.