PETRI NETS AND THEIR PROPERTIES - Modelling with Generalised Stochastic Petri Nets - page 42

Obviously, if [CV σ ] T = 0, we obtain that M 0 = M and we can observe
that the firing sequence σ brings the PN back to the same marking M. The
vectors X, that are integer solutions of the matrix equation

C · X = 0 (2.19)

are called T-semiflows of the net. This matrix equation is equivalent to the
set of linear equations

∀ p ∈ P : C(p,.) · X = 0

(2.20)

In general, the invariant relation (called transition invariant or T-invariant)
produced by a T-semiflow is the following:

X

∀ p ∈ P : C(p,t) · X(t) = 0 (2.21)

t∈T

This invariant relation states that, by firing from marking M any transi-
tion sequence σ whose transition count vector is a T-semiflow, the marking
obtained at the end of the transition sequence is equal to the starting one,
provided that σ can actually be fired from marking M (M[σ i M).
Note again that the T-semiflows computation is independent of any notion
of (parametric) marking, so that T-semiflows are identical for all PN models
and systems with the same structure.
Observe the intrinsic difference between P- and T-semiflows. The fact that
all places in a Petri net are covered by P-semiflows is a su cient condition
for boundedness, whereas the existence of T-semiflows is only a necessary
condition for a PN model to be able to return to a starting state, because
there is no guarantee that a transition sequence with transition count vector
equal to the T-semiflow can actually be fired.
Like P-semiflows, all T-semiflows can be obtained as linear combinations of
the elements of a minimal set TS.
The reader can easily verify that the set TS for our readers & writers model
is:

• TS 1 = [1, 1, 0, 1, 0, 1, 0]

• TS 2 = [1, 0, 1, 0, 1, 0, 1]

We can also write the T-semiflows as bags of transitions as follows:

• TS 1 = t 1 + t 2 + t 4 + t 6

• TS 2 = t 1 + t 3 + t 5 + t 7

These two T-semiflows correspond to a whole cycle through the system made
respectively by a reader and by a writer. In this case, both TS 1 and TS 2
correspond to transition sequences that are fireable from the initial marking.