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

SW consists of n workflow activities with a ~ N(
µ , σ ) , the fine-grained upper
bound temporal constraint for activity a i is U ( a i ) and can be obtained with the
following formula:
⎛ ⎛ n n ⎞ n ⎞
⎜ ⎜
2 2
u( a = + × − ∑ − ∑ ⎟ ∑ ⎟
i ) µ i λσ i 1 w
⎜ ⎜ iσi
wi
σ i σ
⎟ i Formula 5.2
⎟
⎝ ⎝i=
1 i=
1 ⎠ i=
1 ⎠
Here,
i
µ i and σ i are obtained directly from the mean value and standard
deviation of activity a i and λ denotes the same probability with the coarse-grained
temporal constraint. Based on Formula 5.2, we can claim that with our setting
strategy, the sum of weighted fine-grained temporal constraints is approximately the
same to their overall coarse-grained temporal constraint. Here, we present a
theoretical proof to verify our claim.
Proof: Assume that the distribution model for the duration of activity a i is
2
N(
µ i,
σi
) , hence with Formula 5.1, the coarse-grained constraint is set to be of
n
n
u( SW ) = µ sw + λσ sw where µ sw = ∑ w i µ i and σ sw = ∑ w 2 i σi
2 . As defined in
i=
1
i=
1
Formula 5.2, the sum of weighted fine-grained constraints
n
n ⎛ ⎛
⎞
is ∑ ∑
⎜
⎛ n n ⎞ n ⎞
⎜ ⎜
2 2
w ⎟ ⎟⎟
iu( ai
) = wi
µ + × − −
⎜ i λσi
1 ∑ w ∑ ∑
⎜ ⎜ iσi
wi
σi
σ
⎟ i . Evidently,
⎟⎟
i= 1 i= 1 ⎝ ⎝ ⎝i=
1 i=
1 ⎠ i=
1 ⎠⎠
n
n
n
2 2
since w i and σ i are all positive values, ∑ w i σi
≥ ∑ wi
σi
holds and ∑σi
is
i=
1 i=
1
i=
1
normally big for a large size scientific workflow SW , hence the right hand side of
the equation can be extended and what we get is w i ( + λσ × ( 1−
A)
)
n
∑
i=
1
i
2
i
µ i i where A
equals
⎛ n n ⎞ n
⎜
2 2
∑w − ∑ ⎟ ∑
⎜ i σ i wi
σi
σ
⎟ i . Therefore, it can be expressed as
⎝i=
1 i=
1 ⎠ i=
1
n
n n
∑ wiu( ai
) = ∑ wi
µ i + λ ∑ wiσ
i − ∆t1
(Equation Ⅰ) where ∆ t n
1 = ∑
1w i A . Meanwhile,
i=
1 i=
1 i=
1
i=
since
n
n
n
n n n
2 2
2 2
∑ w i σ i ≥ ∑ wi
σi
, thus ∑ wi
µ i + λ ∑ wi
σi
≤ ∑ wi
µ i + λ ∑ wiσ
i .
i=
1 i=
1
i=
1 i=
1 i=
1 i=
1
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