ANALYSIS OF GSPN MODELS - Modelling with Generalised Stochastic Petri Nets - page 133

p 1

µ 3 T 3 µ 2 T 2 µ 1 T 1

δ p 3 α

p 4 p 2

t 3 t 1

t 4 t 2

γ β

Figure 6.11: A GSPN system with several multiple paths between tangible
markings

These informal considerations are precisely captured by the following for-
mula:

X

u 0 ij e ir P { r → s } d sj = f ij + (6.32)

r:M r ∈V RS

where P { r → s } d sj is the probability that the net moves from vanish-
ing marking M r to tangible marking M j in an arbitrary number of steps,
following a path through vanishing markings only.
In order to provide a general and e cient method for the computation of
the state transition probability matrix U 0 of the REMC, we can observe
that equation (6.32) can be rewritten in matrix notation in the following
form:

U 0 = F + E G D (6.33)

where each entry g rs of matrix G represents the probability of moving from
vanishing marking M r to vanishing marking M s in any number of steps, but
without hitting any intermediate tangible marking. Observing that each
entry c rs of matrix C n represents the probability of moving from vanishing
marking M r to vanishing marking M s in exactly n steps without hitting any
intermediate tangible marking, we can conclude that

∞

X

C n G =

n=0