Xiao Liu PhD Thesis.pdf - Faculty of Information and Communication ...

path with the longer duration [2], in this thesis, we define the joint distribution of the
parallelism building block as the joint distribution of the path with a lager expected
duration, i.e. if
j l
j
l
∑ µ p ≥ ∑ µ q then Z = ∑ µ p , otherwise Z = ∑ µ q .
p=
i q=
k
p=
i
q=
k
Accordingly, the structure weight for each activity on the path with longer duration
is 1 while on the other path is 0. Therefore, the weighted joint distribution of this
block is
Z
⎧ j ⎛ j j ⎞ j l
⎪ ∑ ⎜
2
X
⎟
p ~ N
≥ ∑
⎪
∑ µ p,
∑ σ p , if
∑ µ p µ q
p= i
=
⎝ p=
i p=
i ⎠ p=
i q=
k
⎨
⎪
l ⎛ l l ⎞
⎪
∑ ⎜
2
X
⎟
q ~ N
∑ µ q,
∑ σ q , otherwise
⎩ q= k ⎝ q=
k q=
k ⎠
.
Figure 5.3 Parallelism Building Block
4) Choice building block. As depicted in Figure 5.4, the choice building block
contains two paths in an exclusive relationship which means that only one path will
be executed at runtime. The probability notation denotes that the probability for the
choice of the upper path is β and hence the choice probability for the lower path is
1 − β . In the real world, β may also follow some probability distribution. However,
similar to the iteration building block, in order to avoid the complex joint
distribution, β is estimated by the mean probability for selecting a specific path, i.e.
the number of times that the path has been selected divided by the total number of
workflow instances. Accordingly, the structure weight for each activity in the choice
building block is the probability of the path it belongs to. The weighted joint
j
l
distribution is Z = ( ∑ X p ) + (1 −β)(
∑ Xq)
p=
i
q=
k
⎛
N⎜
⎝
β ~
j
l
j
l
2 2 2 2 ⎞
( ∑ µ ) + (1 − )( ∑ ), ( ∑ ) + (1 − ) ( ∑ ) ⎟
p β µ q β σ p β σ p
p=
i
q=
k p=
i
q=
k ⎠
β .
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