ANALYSIS OF GSPN MODELS - Modelling with Generalised Stochastic Petri Nets - page 136

An alternative way of computing the steady-state probability distribution
over the tangible markings is thus that of solving the following system of
linear matrix equations:

8
<

: η 0 Q 0

= 0

(6.40)

η 0 1 T = 1

This result shows that GSPNs, like SPNs, can be analysed by solving proper
associated CTMCs. This obviously implies that the computation of perfor-
mance indices defined over GSPN models can be performed using the reward
method discussed in Section 6.1 without any additional di culty.
The advantage of solving the system by first identifying the REMC is twofold.
First, the time and space complexity of the solution is reduced in most cases,
since the iterative methods used to solve the system of linear equations tend
to converge more slowly when applied with sparse matrices and an improve-
ment is obtained by eleiminating the vanishing states thus obtaining a denser
matrix [ 18, 11] . Second, by decreasing the impact of the size of the set of
vanishing states on the complexity of the solution method, we are allowed a
greater freedom in the explicit specification of the logical conditions of the
original GSPN, making it easier to understand.

6.3.1 Evaluation of the steady-state probability distribution

for the example GSPN system

The GSPN system of Fig. 6.9 can now be analysed to compute the steady-
state probability distribution of the states of the parallel processing system
that it represents.
As was mentioned in Section 6.2.2, the 38 markings of this net can be
grouped within two subsets comprising 20 tangible, and 18 vanishing mark-
ings, respectively. Vanishing markings are tagged with a star in Table 6.2.2.
Using the notation introduced in Section 6.3, we have | RS | = 38, | TRS | =
20, and | V RS | = 18. The transition probability matrix U of the EMC
associated with the GSPN can be subdivided into four blocks whose entries
are computed from the parameters of the model (the W function of the
GSPN system specification). The four submatrices C, D, E, and F of
equation ( 6.28) have dimensions (18 × 18), (18 × 20), (20 × 18), and (20 × 20),
respectively. Because of the relatively simple structure of the model, all these
four blocks are quite sparse (few of the entries are different from zero). Table
6.12 reports the non-zero values of all these matrices, with the understanding
that x i,j (r,s) represents the non-zero component of the X matrix located
in row i and column j, and corresponding to the transition probability from
state r to state s.