ANALYSIS OF GSPN MODELS - Modelling with Generalised Stochastic Petri Nets - page 117

2
4

3
5

− 3
0
10
0
0
0
5
0

1
− 102
0
0
0
0
0
5

0
100
− 12
0
0
0
0
0

0
0
2
− 10
100
0
0
0

0
2
0
0
− 200
1
0
0

2
0
0
10
0
− 101
0
0

0
0
0
0
0
100
− 6
0

0
0
0
0
100
0
1
− 5

(η 0 , η 1 , η 2 , η 3 , η 4 , η 5 , η 6 , η 7 ) = 0

and

7

X

η i = 1

i=0

By solving this linear system we obtain:

η 0 = 0.61471, η 1 = 0.00842, η 2 = 0.07014, η 3 = 0.01556

η 4 = 0.00015, η 5 = 0.01371, η 6 = 0.22854, η 7 = 0.04876
As an example, we can now compute the average number of tokens in place
p idle from which we can easily compute the utilization of the shared memory.
Since one token is present in place p idle in markings M 0 , M 1 , M 4 , and M 5 ,
we have:

E[M(p idle )] = η 0 + η 1 + η 4 + η 5 = 0.6370

and the utilization of the shared memory is:
U[shared memory] = 1.0 − E[M(p idle )] = 0.3630

The throughput of transition T req1 , that represents the actual rate of access
to the shared memory from the first processor, can be obtained using equa-
tion ( 6.12) and observing that T req1 is enabled only in M 0 , M 5 , and M 6 .
The following expression allows the desired quantity to be obtained:

f req1 = (η 0 + η 5 + η 6 ) λ 1 = 0.8570

6.2 The Stochastic Process Associated with a GSPN

In Chapter 5 we described the qualitative effect that immediate transitions
have on the behaviour of a GSPN system. GSPNs adopt the same firing
policy of SPNs; when several transitions are enabled in the same marking,
the probabilistic choice of the transition to fire next depends on parameters
that are associated with these same transitions and that are not functions
of time. The general expression for the probability that a given (timed or
immediate) transition t k , enabled in marking M i , fires is:
P { t k | M i } = w k
q i

(6.15)