PETRI NETS AND THEIR PROPERTIES - Modelling with Generalised Stochastic Petri Nets - page 40

As a consequence, if in a PN model all places are covered by P-semiflow s 4 ,
then for any reachable marking in any PN system obtained by instantiation
of the parametric initial marking, the maximum number of tokens in any
place is finite (since the initial marking is finite) and the net is said to be
structurally bounded.
All P-semiflows of a PN can be obtained as linear combinations of the P-
semiflows that are elements of a minimal set PS. See [ 40, 47, 6, 7] for

P-semiflows computation algorithms.
For example, the set PS for the readers & writers PN model in Fig. 2.1
contains two P-semiflows:

• PS 1 = [1, 1, 1, 1, 0, 1, 1]

• PS 2 = [0, 0, 0, 0, 1, 0, 1]

We can also write the P-semiflows as bags of places as follows:

• PS 1 = p 1 + p 2 + p 3 + p 4 + p 6 + p 7

• PS 2 = p 5 + p 7

It is possible to visualize the P-semiflows above as (possibly cyclic) paths
in the representation of the Petri net, as shown in Fig. 2.13, where we have
drawn with dotted lines the part of the Petri net that is not related to the
P-semiflow. With the usual parametric initial marking MP that assigns K
tokens to p 1 , and one token to p 5 , the two P-semiflows define the following
P-invariants:

• M(p 1 ) + M(p 2 ) + M(p 3 ) + M(p 4 ) + M(p 6 ) + M(p 7 ) = K

• M(p 5 ) + M(p 7 ) = 1

As a consequence, places p 5 and p 7 are 1-bounded, while all other places are
K-bounded. Moreover, since the marking is an integer vector, it can never
happen that places p 5 and p 7 are marked together.
Observe that any linear combination of the two P-semiflows PS 1 and PS 2 is
also a P-semiflow: for example PS 1 + PS 2 = [1, 1, 1, 1, 1, 1, 2] is a solution
of (2.14) .

T-semiflows and T-invariant relations — As observed in Equation
( 2.12) , if V σ is a firing count vector of a transition sequence σ, then

M 0 = M + [CV σ ] T (2.18)

4 A place p is covered by P-semiflows if there is at least one vector of Y with a non null
entry for p