Information Technology Reference
In-Depth Information
For example, given
CP
1
to
CP
3
defined in the previous page. Since
t
1
and
t
2
are enabled in
CP
1
,
t
3
is enabled in
CP
2
, and no transition can
be enabled in
CP
3
, only the set of places of
CP
3
is a candidate SFC.
Definition5.10.
In an SC-net, a subgraph that corresponds to a Solution
of the State-shift Equation (SSE) consists of the transitions correspond-
ing to nonzero elements of
x
, their input and output places, and places
corresponding to nonzero elements of
M
.
For example, for the SC-net in Figure 5.3, a solution of the state-shift
equation can be
T
a
1
¼ð
0
;
0
:
2
;
0
:
8
;
0
;
0
;
0
;
0
;
0
;
0
;
0
;
0
:
2
;
0
:
8
;
0
;
0
;
0
;
0
Þ
T
,thentwo
or
a
2
¼ð
0
;
0
:
1
;
0
;
0
:
9
;
0
:
9
;
0
;
0
;
0
;
0
;
0
;
1
;
0
;
0
:
9
;
0
;
0
;
0
Þ
p
0
;
SSEs are SSE
1
¼fð
P
;
T
Þj
P
¼f
p
1
;
p
2
g;
T
¼f
t
1
;
t
2
gg
and SSE
2
¼
p
0
;
fð
P
;
T
Þj
P
¼f
p
1
;
p
3
;
p
4
g;
T
¼f
t
1
;
t
3
gg
, respectively.
If an SSE corresponds to a CP, and the final marking corresponds to a
realizable configuration state, then the solution of the state-shift equa-
tion is a
realizable solution
.
Definition 5.11.
corresponding to nonzero
elements of the solution are linearly independent of one another, the
solution is a basis solution.
If the column vectors of
E
-
A
½
Definition5.12.
If all elements of a solution are nonnegative integer, this
solution is called nonnegative integer solution.
Theorem 5.3. [127]
Column vectors corresponding to the nonzero
elements of the realizable solution in E
-
A
½
are linearly independent.
Theorem 5.4.
A solution is a nonnegative integer solution if and only
if it is a realizable solution.
Proof:
From Definition 5.9, it is obvious that if a solution is a realizable
one, it is a nonnegative integer one.
Necessity: Consider the Web service place
p
0
. Following the
equation
M
0
ð
p
0
Þþ
P
ð
p
0
Þ
l
p
0
Þ¼
p
0
Þ¼
M
ð
¼
1,
we
have
M
ð
1
and
P
ð
0 and
P
ð
p
0
Þ
l
p
0
Þ
l
p
0
Þ¼
p
0
Þ¼
¼
0or
M
ð
¼
1. When
M
ð
1 and
P
ð
p
0
l
Þ
¼
0, the other elements of the solution have to be 0. This means
that the corresponding SSE consists only of place
p
0
. In this case, the
configuration process is in the initial state and this solution is obviously a
