ANALYSIS OF GSPN MODELS - Modelling with Generalised Stochastic Petri Nets - page 134

Table 6.11: Specification of the transitions of the SPN of Fig. 6.11

transition rate semantics

T 1 µ 1 single-server

T 2 µ 2 single-server

transition weight priority ECS

T 3 µ 3 single-server

t 1 α 1 1

t 2 β 1 1

t 3 δ 1 1

t 4 γ 1 1

In the computation of C n , two possibilities may arise. The first corresponds
to the situation in which there are no loops among vanishing markings. This
means that for any vanishing marking M r ∈ V RS there is a value n 0r such
that any sequence of transition firings of length n ≥ n 0r starting from such
marking must reach a tangible marking M j ∈ TRS. In this case

: ∀ n ≥ n 0 C n ∃ n 0 = 0

and

∞ n 0

X X

C k C k

G = =

k=0 k=0

The second corresponds to the situation in which there are possibilities of
loops among vanishing markings, so that there is a possibility for the GSPN
to remain “trapped” within a set of vanishing markings. In this case the
irreducibility property of the semi-Markov process associated with the GSPN
system ensures that the following results hold [ 71] :

n→∞ C n

lim = 0

so that

∞

X

C k = [I − C] −1 . G =

k=0

We can thus write (see [5, 4] for details):

P
n 0

8
<

k=0 C k

no loops among vanishing states

D

H =

:

[I − C] −1 D

loops among vanishing states

from which we can conclude that an explicit expression for the desired total
transition probability among any two tangible markings is:

X

u 0 ij e ir h rj ∀ i,j ∈ TRS = f ij +

r∈V RS