ANALYSIS OF GSPN MODELS - Modelling with Generalised Stochastic Petri Nets - page 128

set of a GSPN system is a subset of the reachability set of the associated
PN system, because the priority rules introduced by immediate transitions
maayy not allow some states to be reached. The reachability set of an SPN
system is, instead, the same as for the associated PN system. Moreover,
as was pointed out in Chapter 5, the reachability set of the GSPN sys-
tem can be divided in two disjoint subsets, one of which comprises tangible
markings, i.e., markings that enable timed transitions only, while the other
one comprises markings that enable immediate transitions (vanishing mark-
ings); in our case the set of tangible markings comprises 20 elements while
the number of vanishing markings is 18.
Once the GSPN system is defined, some structural properties may be com-
puted to perform a validation of the model. First, P and T semi − flows can
be computed to check whether the net is structurally bounded and whether
it may have home-states.
In the case of the GSPN system of this example, two P semi-flows are
identified (PS 1 = p 1 + p 2 + p 3 + p 6 + p 7 + p 8 + p 9 and PS 2 = p 1 + p 2 +
p 4 + p 5 + p 7 + p 8 + p 9 ) that completely cover the PN system that is thus
structurally bounded. Similarly, two T semi-flows are obtained (TS 1 =
T ndata + t start + T par1 + T par2 + t syn + t OK + T I/O and TS 2 = t start +
T par1 + T par2 + t syn + t KO + T check ) that again cover the whole net, making
the presence of home-states possible. Other structural results that may be
computed are the ECSs of this model that, as we already said before, are
{ t start } , { t syn } , and { t OK , t KO } .
These results ensure that the net is suitable for a numerical evaluation yield-
ing the steady-state probabilities of all its markings.

6.3 Numerical Solution of GSPN Systems

The stochastic process associated with k-bounded GSPN systems with M 0
as their home state can be classified as a finite state space, stationary (ho-
mogeneous), irreducible, and continuous-time semi-Markov process.
In Appendix A we show that semi-Markov processes can be analysed iden-
tifying an embedded (discrete-time) Markov chain that describes the transi-
tions from state to state of the process. In the case of GSPNs, the embedded
Markov chain (EMC) can be recognized disregarding the concept of time and
focusing the attention on the set of states of the semi-Markov process. The
specifications of a GSPN system are su cient for the computation of the
transition probabilities of such a chain.
Let RS, TRS, and V RS indicate the state space (the reachability set), the
set of tangible states (or markings) and the set of vanishing markings of the
stochastic process, respectively. The following relations hold among these

sets:

[ \

V RS = ∅ .

RS = TRS V RS, TRS