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

Definition 6.2: (Minimum Probability Time Redundancy).
Let U 1 , U 2 ,.., U N be N upper bound constraints and all of them cover a p .
Then, at
a p
, the minimum probability time redundancy is defined as the minimum
of all probability time redundancies of U 1 , U 2 ,..., U N and is represented as
MPTR ( a p ) = Min{ PTR( Us , a p ) s = 1,2,...,
N}
.
The purpose of defining minimum probability time redundancy is to detect the
earliest possible temporal violations. Based on Definition 6.2, Theorem 6.1 is
presented to locate the exact temporal constraint which has the temporal consistency
state below the θ % bottom line.
Theorem 6.1. At workflow activity point
a p , if ( a p ) > ( a p ) + MPTR(
a p−1)
R θ ,
then at least one of the temporal constraints is violated and it is exactly the one
whose time redundancy at a p−1
is MPTR ( a p− 1)
.
Proof: Suppose that U ( a k , a l ) is an upper bound constraint whose probability is
above the threshold before execution of
a p ( k < p < l ) and it is the one with
MPTR ( a p− 1) . Then, according to Definition 6.1, at a p−1
, we
have u( ak
, al
) > R(
ak
, a p−1)
+ θ ( a p,
al
) and here MPTR ( a p− 1)
= u( ak
, al
) − R(
ak
, a p− 1)
−θ
( a p,
al
) . Now, assume that at activity a p , we have
R a ) θ ( a ) + MPTR(
a )
which means
( p > p
p−1
R( ap
p k l k p 1 p l
) > θ(
a ) + u(
a , a ) − R(
a , a − ) −θ(
a , a ) , and that is
u( ak
, al
) < R(
a p ) + R(
ak
, a p−1)
+θ ( a p,
al
) −θ
( a p ) where the right hand side
equals R( ak
, a p ) +θ ( a p−1,
al
)
. Since
( a , a ) + θ ( a − , a ) < R(
a , a ) + θ ( a 1,
a ) , we have
R k p p 1 l k p p+
l
u ( ak
, al
) < R(
ak
, a p ) + θ ( a p+
1,
al
) and this results in a probability of temporal
consistency which is lower than that of θ % where
u ( ak
, al
) = R(
ak
, a p ) + θ ( a p+
1,
al
) . Therefore, a potential temporal violation is
detected and it is exactly the one whose time redundancy at a p−1
is MPTR ( a p− 1)
.
Thus, the theorem holds.
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