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these places and transitions are in a disassembly Petri net to be defined
in the next section. The values of the elements in
V
j
are determined in
the following algorithm:
Algorithm 5.1. (Association)
Step 1
. If the
j
th QoS attribute is cost, we associate a positive large
enough number
L
þ
to nonleaf Web service places (to be updated
during a configuration process). If the
j
th QoS attribute is benefit,
we associate a negative large enough number
L
.
Step 2
. Every leaf Web service place
p
is associated with QoS attribute
c
j
ð
; zÞ
. Note that a leaf Web service place corresponds to a known
service whose QoS values are given already.
Step 3
p
. For every nonleaf and nondummy Web service place
p
,we
associate
p
.
c
j
ð
p
; zÞ
to all of its output transitions
t
2
Step 4
. For every nonleaf and dummy Web service place
p
,we
associate a number 0 to all of its output transitions
t
p
.
In the following theorem, we need the knowledge of disassembly
Petri nets to be discussed next. Note that
2
i
is (
M
i
x
i
)
T
where
M
i
is
the marking of the net corresponding to the
i
-th candidate SFC and
x
i
is the firing vector that leads initial marking
M
0
to
M
i
.
a
Theorem5.1.
The jth QoS attribute for the i-th candidate SFC, that
is,
c
i; j
, can be computed as V
j
a
i
.
T
,
V
j
a
Proof:
Since
x
i
.For
the Web service
s
denoted by the nonleaf Web service place, the
reusability frequency of the Web service in the configuration is equal
to the firing number of the transitions
t
a ¼
ð
M
i
x
i
Þ
is equal to
V
m
M
i
þ
V
n
P
s
.Thenthe
QoS attribute of a nonleaf Web service place can be aggregated through
V
n
2
p
where
p
2
x
i
.
For the Web service denoted by the leaf Web service place
p
,the
reusability frequency of the Web service in the configuration is equal
to the number of tokens in marking
M
i
ð
. Then the QoS attribute of
leaf Web service places can be aggregated through
V
m
p
Þ
M
i
. Dummy
Web services chosen in a configuration do not affect the final result.
Since the aggregation functions can be treated linearly, the conclu-
sion holds.
