Lifetime vs. Age
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Lifetime - E-Commerce - Lecture Slides, Slides of Fundamentals of E-Commerce
Students of Computer Science, study E-Commerce as an auxiliary subject. these are the key points discussed in these Lecture Slides of E-Commerce : Lifetime, Documents, Probability, Expected Fraction, Probability, Change Rate, Empirical Distributions, Essential Property, Exponential Distribution, Growth Model
Typology: Slides
2012/2013
1 / 11
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Lifetime vs. Age
Aging of Documents
•^
S(t)
is the probability that a document last changed at time zero will remain unmodified at time
t
•^
G(t)
is the expected fraction of documents that are older that
t
•^
Probability that a document will be modified beforean additional time
h^
has passed is the conditional
probability
P( t < T
≤^ t + h | T > t )
•^
The change rate -^ where
f(t)
is the lifetime pdf
)( )(
) |
( 1 lim )(
0
t S
tf
t T h t T t
Ph
t^
h^
=
≤ <
=^
→
λ
Empirical Distributions
Growth with Time
•^
Essential property that the Web is growing with time– not captured by the model
-^
Most documents are young
-^
Suppose exponential distribution for growth^ – if documents were never edited, their age is the time sincetheir creation – trivial growth model yields an exponential age distribution
-^
Realistic model should take to account both Webgrowth and document refreshing^ – use hybrid model
Improved Model
-^
Document changes are controlled by anunderlying Poisson process^ –
probability of observing a change is independent ofprevious changes
-^
Mean lifetimes are Weibull distributed
-^
Change rates and timespans are independent
-^
The resulting lifetime distribution^ where w(1/
λ) is an estimate of the mean lifetime
−
=^
0
λ^
λ^
d
w
e
t
f^
t )/ (^1) ( ˆ^ λ w
Estimated Mean Lifetime Distribution
Refresh Interval
-^
The probability that for a particular time
t^
the
document is unmodified in [0,t-
β] is
-^
The probability that a collection of documentsis^
β-current is
^
∈ −−
[, )( , t te βλ
∈ ∈
− −
) , 0 (
, 1
) , [
) , (
β^ β
β α
t
I
t
e^
t
dt I e t
I t w
t I
∞ ∫
− −
^
−
=^
0
)/ ( / 1 )(
β
Example
-^
Brewington and Cybenko (2000)^ –
reindexing period of about 18 days – assuming a Web size of 800 million pages – to guarantee that 95% of repository was currentup to one week ago