Graphics Programs Reference
In-Depth Information
µ
1
T
1
, that happens with probability
(µ
1
+µ
2
)
, and that leads to the desired
(target) marking M
j
in one step only. The second corresponds to selecting
transition T
2
to fire first, followed by transition t
1
. The first of these two
events happens with probability
µ
2
(µ
1
+µ
2
)
, and the second with probability
α
(α+β)
. The total probability of this second path from M
i
to M
j
amounts to
µ
2
(µ
1
+µ
2
)
α
(α+β)
. Notice that firing transition T
2
followed by transition t
2
would
lead to a different marking (in this case the initial one). Firing transition T
2
leads the system into an intermediate (vanishing) marking M
r
. The total
probability of moving from marking M
i
to marking M
j
is thus in this case:
µ
1
(µ
1
+ µ
2
)
µ
2
(µ
1
+ µ
2
)
α
(α + β)
u
0
ij
=
+
(6.30)
In general, upon the exit from a tangible marking, the system may “walk”
through several vanishing markings before ending in a tangible one. To
make the example more interesting, and to illustrate more complex cases,
marking M
2
= p
2
following four different paths. The first corresponds again
to firing transition T
1
and thus to a direct move from the initial marking
to the target one.
µ
1
(µ
1
+µ
2
+µ
3
)
. The second
path corresponds to firing transition T
2
followed by t
1
(total probability
µ
2
(µ
1
+µ
2
+µ
3
)
This happens with probability
α
(α+β)
); the third path corresponds to firing transition T
3
followed
by transition t
4
(total probability
µ
3
(µ
1
+µ
2
+µ
3
)
γ
(γ+δ)
). Finally, the last path
corresponds to firing transition T
3
followed by transition t
3
and then by
transition t
1
which happens with probability
µ
3
(µ
1
+µ
2
+µ
3
)
δ
(γ+δ)
α
(α+β)
.
In this case the total probability of moving from marking M
i
to marking M
j
becomes:
µ
1
(µ
1
+ µ
2
+ µ
3
)
µ
2
(µ
1
+ µ
2
+ µ
3
)
α
(α + β)
u
0
ij
=
+
+
µ
3
(µ
1
+ µ
2
+ µ
3
)
γ
(γ + δ)
µ
3
(µ
1
+ µ
2
+ µ
3
)
α
(α + β)
δ
(γ + δ)
+
(6.31)
In general, recalling the structure of the U matrix, a direct move from
marking M
i
to marking M
j
corresponds to a non-zero entry in block F
(f
ij
6
= 0), while a path from M
i
to M
j
via two intermediate vanishing
markings corresponds to the existence of
1. a non-zero entry in block E corresponding to a move from M
i
to a
generic intermediate marking M
r
;
2. a non-zero entry in block C from this generic state M
r
to another
arbitrary vanishing marking M
s
;
3. a corresponding non-zero entry in block D from M
s
to M
j
.
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