Information Technology Reference
In-Depth Information
T
, the set
•
x=
{
y
|
y
F
} is the pre-set of
x
, and
x
•
=
{
y
|
For all
x
∈
P
∪
∈
P
∪
T
∧
(
y, x
)
∈
y
∈
P
∪
T
∧
(
x, y
)
∈
F
} is the post-set of
x
.
p
is marked by
M
iff
M
(
p
)
>
0. A transition
•
t
:
M
(
p
)
t
∈
T
is enabled under
M
, if and only if
∀
p
∈
>
0, denoted as
M
[
t
>
. If
M
[
t
>
holds,
t
may fire, resulting in a new marking
M
′
, denoted as
M
[
t
>
M
′
, such that
∈
•
t
\
t
•
,
M
t
•
\
•
t
, and otherwise
M
M
(
p
)=
M
(
p
).
An initial marking is denoted by
M
0
and
R
(
M
0
) is defined as the set of all reachable
marking set of
′
(
p
)=
M
(
p
)
−
1 if
∀
p
′
(
p
)=
M
(
p
)+1 if
∀
p
∈
′
ʣ
where
∀
M
i
∈
R
(
M
0
) such that
M
0
[
ʴ
>
M
i
.
Definition
3.2.
Let
ʣ
=(
P,
T
;
F, M
0
) be a Petri net. A place
p
∈
P
is
bounded
if there is
an non-negative integer
n
, such that
M
i
(
p
)
≤
n
for all reachable markings
M
i
∈
R
(
M
0
). A
Petri net is
bounded
if all its places are bounded.
Boundedness is a decidable property which needs the construction of the coverabil-
ity (or reachability) graph. If a place is found to be unbounded, a special symbol
ˉ
is
introduced to indicate this fact.
A Petri net which models a workflow process definition is called the workflow net
or WF-net whose definition is briefly reviewed following [9].
Definition
3.3.
A
Petri net
ʣ
=(
P,
T
;
F, M
0
) is a WF-net if: (1) there is one source
P
such that
•
i
=
P
such that
o
•
=
place
i
∈
∅
; (2) there is one sink place
o
∈
∅
; (3) Each
node
x
∈
P
∪
T
is on a path from
i
to
o
; and (4)
∀
p
∈
P
,
M
0
(
p
)=1 if
p=i
, and otherwise
M
0
(
p
)=0.
As any process handled by the WF-net is created if it enters the workflow engine
and is deleted when it is completed, it has an input place and an output place to
represent these two states respectively. In a WF-net, the transition set
T
is used to
represent the tasks in the workflow, and source place and sink place represent its start
and end respectively. For the rest of this paper, a WF-net is formally denoted as
W
=(
P,
T
;
F, M
0
) to be distinguished from traditional Petri net. Moreover, as our WF-
net is specifically used to model workflow processes, we synonymously use the term
transition and task for convenience.
4
Petri Net Model for Cross-Organization Workflow
with Synchronous Interaction Pattern
4.1
CWF-Net with Synchronous Interaction Pattern
When charactering the cross-organizational features of a loosely-coupled workflow,
we often use different kinds of interaction patterns which contain asynchronous and
synchronous ones [4]-[5]. The asynchronous interaction patterns involve the message
interaction pattern and resource interaction pattern while the synchronous interaction
pattern specific refers to task synchronization pattern. In this section, we only focus
on a kind of cross-organization workflow with task synchronization pattern.
