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R i represents the reachability relations between any two operations
in W i , and by aggregating them we can examine the reachability
relations among operations in a global, cross-workflow perspective,
that is, in S 0 .
Given the workflow set W
¼f
W 1 ;
W 2 ; ...;
W n g
and the operation set
O
¼f
o 1 ;
o 2 ; ...;
o m g
, we now combine
f
R 1 ;
R 2 ; ...;
R n g
into an aggre-
gated relation R
¼½
r ij mm :
r ij ¼ P k f
, that is, r ij equals the number of workflows in
which o i can reach o j , and r ii ¼
R kij ¼
1
g
0.
We then calculate the n th power of R :
R n
r i hi mm and r ij ¼ P
n
r n 1
ik r kj
Therefore, r ij equals the number of times for which o i can reach o j
by traversing n workflows, that is, transferring n
¼
k
¼
1
1 times. Now we
know for each given operation pair, the existence of a directed chain
that crosses a certain number of workflows and connects these two
operations.
Step 3: Calculate a Transfer Matrix.
At the beginning of Section 8.3.4, we emphasize that we want a service
chain crossing the least workflows , the way passengers want an
itinerary crossing the least routes (i.e., with the least transfers). To
achieve that goal, we need to introduce another transfer matrix T .
Given m operations and n workflows:
T
r ij >
¼½
t ij mm :
t ij ¼f
min k
2f
1
;
2
; ...;
n
g
s
:
t
:
0
g
if such k exists,
and 0 otherwise.
T reveals the least transfer distance between two given operations.
By definition, t ij
1 is the number of transfers through which o i can
reach o j .If t ij ¼
0, there is no path between them crossing the available
workflows.
Step 4: Calculate Transfer Paths.
If t ij
2, at least k - 1 transfer is needed for any path from o i
to o j . The least transfer path spans over k workflows that can be
denoted
¼
k
f
w 1
;
w 2
; ...;
w k
g
. Then
this
path
can
be
denoted
w 1 o q 1 !
w 2
w k 1 o q k 1 !
w k
o i !
!
o j
in which two adjacent operations
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