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attributes to evaluate, that is,
c
1
q
. The SFC QoS attributes are the
cumulative effect of the QoS attributes of the nondummy Web service
places that instantiate the modules of the SFC. The dummy Web service
places included in the SFC do not have QoS values. To compute them,
we assume that a QoS attribute for a candidate SFC is a function of the
QoS attribute of all the component services. We explain each attribute's
aggregation function as follows:
1.
Cost
. The cost of an SFC is the sum of the costs of all the Web
services involved. Suppose that the reusability frequency for
Web service denoted by
p
is
p
f
, that is,
p
f
¼
1 if the Web service
is used once, and
p
f
>
1 if the Web service is reused.
p
0
; zÞ
Let
C
ð
be the set of all Web services in an SFC, and
p
0
; zÞÞ
c
1
ð
C
ð
and
c
1
ð
p
; zÞ
denote the cost of the SFC and the
Web
service
p
at
time
z
,
respectively.
Then
we
have
p
0
; zÞÞ ¼
P
p2Cðp
0
;zÞ
c
1
ð
p
f
.
2.
Availability
. The availability of an SFC is given by the product
of the availability of all the Web services involved.
Let
c
1
ð
C
ð
p
; zÞ
p
0
; zÞÞ
c
2
ð
C
ð
c
2
ð
p
; zÞ
denote the results after apply-
ing the logarithm function to the availability of an SFC and the
availability of the Web service
p
at time
z
, respectively. Then
the original product relations among availability values of all
the Web services are converted into addition relations, that
is,
and
p
0
; zÞÞ ¼
P
p2Cðp
0
;zÞ
c
2
ð
p
f
.Thenwetransform
the nonlinear aggregation function into a linear one.
c
2
ð
C
ð
p
; zÞ
5.2.3 Computation of QoS
Suppose that we have
N
c
SFC candidates. The
j
th QoS attribute for the
i
th SFC is denoted as
q
where
q
denotes the
number of QoS parameters under consideration. Different SFC candi-
date corresponds to different solution vector
c
i; j
,1
i
N
c
,1
j
a
i
,1
i
N
c
.
i
, according to the
definition of SFC and the linear aggregation function, for the
j
th QoS
attribute we associate a 1
To compute
c
i; j
by the linear formulation of
a
ð
m
þ
n
Þ
QoS attribute vector
V
j
. Suppose
that
V
j
n
vectors,
respectively. Elements in
V
m
are associated to
m
places, whereas
elements in
V
n
are associated to
n
transitions accordingly. Note that
¼
ð
V
m
V
n
Þ
, where
V
m
and
V
n
are 1
m
and 1
