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

from a p+ 1 to a p + m
, the probability time redundancy of subsequent activities
is PTR ( U ( SW ),( a p + 1,
a p + m )) which is equal
to u ( SW ) −[
R(
a1,
a p ) + M ( a p+ 1,
a p+
m)
+ θ ( a p+
m+
1,
an
)] . Here, M ( a p + 1,
a p + m ) is
equal to
p+
m
n
∑(µ k ) and θ ( a p+
m+
1,
an
) is equal to ∑ ( µ k + λθ σ k ) .
k = p+
1
k = p+
m+
1
The probability time deficit is for measuring the occurred time deficit at the
current checkpoint. The probability time redundancy is for measuring the expected
time redundancy (i.e. the time redundancy between the mean completion time and
the temporal constraints) of the subsequent activities at the current checkpoint. For
example, at checkpoint a p , if the temporal constraint for activity a p+ 1 to a p + m is
equal to
µ θ
+ λ σ , then the value of λ θ σ is regarded as the expected time
redundancy which can be used to compensate for the occurred time deficit. Based on
the probability time deficit and the probability time redundancy, the probability of
self-recovery is defined as follows.
Definition 7.3: (Probability of Self-Recovery).
For activity point a p which is covered by U (SW ) , given the probability time
deficit (denoted as PTD ( a p )
) and the probability time redundancy (denoted
as PTR ( a p ) ), the probability of self-recovery, i.e. the probability that PTD ( a p ) can
be compensated for by PTR ( a p ) is defined as:
−x
2
1 T
P(
T ) = ∫ e 2
2π
−∞
, where
T
=
PTR(
a
p
) − PTD(
a
PTD(
a
p
)
p
)
If without any prior knowledge, it is difficult to decide which probability
distribution model that T fits. Therefore, in this chapter, we assume that T follows a
standard normal distribution 6 , i.e. N (0,1)
with the expected value of 0 and the
standard deviation of 1. It is obvious that the larger the difference between
PTR ( a p ) and PTD a )
( p
, the higher the probability for self-recovery. For example,
according to Definition 7.3, if PTR ( a p ) is equal to PTD ( a p ) , i.e. T is equal to 0,
6 Note that we have tested our strategy with other distribution models such as exponential, uniform
and a mixture of them. The results are similar.
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