ANALYSIS OF GSPN MODELS - Modelling with Generalised Stochastic Petri Nets - page 138

Using the method outlined by equation ( 6.3) we obtain the transition prob-
ability matrix of the REMC reported in Table 6.13. Solving the system of
linear equations ( 6.35) , we obtain the steady state probabilities for all the
states of the REMC.
Supposing that the transition weights assume the following values:

λ = 0.2, µ 1 = 2.0, µ 2 = 1.0, θ = 0.1, ν = 5.0, α = 0.99, and β = 0.01

we obtain, for example:

ψ 0 0 = 0.10911, ψ 0 2 = 0.11965, and ψ 0 37 = 0.00002

Assuming marking 1 as a reference, we compute the visits to any other state
using equation (6.36) . Focusing our attention on markings M 0 , M 2 , and

M 37 , we obtain:

v 0,0 = 1.0, v 2,0 = 1.09666, and v 37,0 = 0.00020.

Knowing the mean sojourn time for the markings of the GSPN:

SJ 0 = 2.5, SJ 2 = 0.3125, and SJ 37 = 10.0

we can compute the mean cycle time with respect to reference marking M 0 :

CY (M 0 ) = 4.31108

and consequently the steady-state probability of each marking. For the three
markings arbitrarily chosen before we have:

η 0 = 0.57990, η 2 = 0.07949, and η 37 = 0.00047

6.3.2 Performance analysis of the example GSPN

The performance of the parallel processing system of Fig. 6.9 can be conve-
niently studied by evaluating the probability of having at least one process
waiting for synchronization.

For this purpose, we can use the following

reward function:

8
< (M(p 5 ) ≥ 1) ∨ (M(p 6 ) ≥ 1)

: 1
0

r(M) = (6.41)

otherwise

With this definition the desired probability is computed to have the value
R = 0.23814.
Similarly, the processing power of this system (i.e., the average number of
processors doing useful work accessing local data only) can be obtained using
the following reward function:

r(M) = M(p 1 ) (6.42)

which provides R = 1.50555.