ANALYSIS OF GSPN MODELS - Modelling with Generalised Stochastic Petri Nets - page 130

The transition probability matrix U of the EMC can be obtained from the
specification of the model using the following expression:

P
T k ∈E j (M i ) w k
q i

u ij =

(6.27)

Assuming for the sake of simplicity, that all enabled transitions produce a
different change of marking upon their firing, the sum contained in equation
( 6.27) reduces to a single element. In this way, except for the diagonal
elements of matrix U, all the other transition probabilities of the EMC can
be computed using equation ( 6.15) independently of whether the transition
to be considered is timed or immediate, according to the discussion contained
at the end of Section 6.2.
By ordering the markings so that the vanishing ones correspond to the first
entries of the matrix and the tangible ones to the last, the transition prob-
ability matrix U can be decomposed in the following manner:

2
4 C

3
5

2
4

3
5

D 0 0

U = A + B = + (6.28)

0 0 E F

The elements of matrix A correspond to changes of markings induced by the
firing of immediate transitions; in particular, those of submatrix C are the
probabilities of moving from vanishing to vanishing markings, while those of
D correspond to transitions from vanishing to tangible markings. Similarly,
the elements of matrix B correspond to changes of markings caused by
the firing of timed transitions: E accounts for the probabilities of moving
from tangible to vanishing markings, while F comprises the probabilities of
remaining within tangible markings.
The solution of the system of linear matrix equations
8
<

: ψ = ψ U
ψ 1 T

(6.29)

= 1

in which ψ is a row vector representing the steady-state probability dis-
tribution of the EMC, can be interpreted in terms of numbers of (state-)
transitions performed by the EMC. Indeed, 1/ψ i is the mean recurrence
time for state s i (marking M i ) measured in number of transition firings
(see Appendix A). The steady-state probability distribution of the stochas-
tic process associated with the GSPN system is then obtained by weighting
each entry ψ i with the sojourn time of its corresponding marking SJ i and
by normalizing the whole distribution (see Appendix A) .
The solution method outlined so far, is computationally acceptable when-
ever the size of the set of vanishing markings is small (compared with the