An Optimal Probabilistic
Forwarding Protocol in Delay
Tolerant Networks
ACM MobiHoc ’09
Cong Liu and Jie Wu
2010.02.03
Presenter: Hojin Lee
Contents
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•
•
•
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Introduction
Preliminary
Optimal Stopping Theory
Copying decision
Conclusion
Introduction (1/2)
• Multi-copy forwarding
• Blind:
– Epidemic routing – O(N)
– spray-and-wait – O(C)
• Statistics: probabilistic forwarding
– Quality: forwarded to node j only if j’s quality >
i’s – O(N) worst cast
– Delegation: forwarded to node j only j’s
quality > i’s threshold – O(N1/2)
• Threshold: updated to better on contact
Introduction (2/2)
• Goal: increase delivery ratio with low cost
within Time-To-Live
• Quality – existing metric
– Inter-meeting time to destination: Ii,d
• Metric in OPF
– Comprehensive: direct, indirect
– Dynamic: metric changed with residual time(,
remaining hops)
Preliminary
• Hop-count-limited forwarding
– Special case of ticket-based forwarding (halfhalf ticket)
Metric
• Metric: Pi,d,K,Tr
– Node i to destination
– Remaining hop-count K
– Residual time: Tr
• Assumption
– Full routing information known (mean inter-meeting times
between every pairs of nodes
– (amortized, or learning)
• Direct forwarding probability with residual time Tr
– Pi,d,0,Tr
– Exponential: 1-exp(-Tr/Ii,d)
• Ii,d: mean inter-meeting time between node i and destination
Optimal Stopping Theory (1/3)
• After observing X1, X2, … Xt, two choices
– 1) stop (and receive the known reward yt)
– 2) continue (and observe Xt+1)
• How do we maximize the expected
reward?
– When do we stop?
 optimal stopping rule
• Probability, statistics – for many ones
Optimal Stopping Theory (2/3)
• A finite horizon
– Upper bound T on the number of stages
• Vt(T): the maximum expected reward starting
from stage t
– Vt(T)=max{yt, E[Vt+1(T)]}
– The above equation implies if yt>E[Vt+1(T)] then
stop, o/w continue
• How do we know E[Vt+1(T)]?
 backward induction
Optimal Stopping Theory (3/3)
• Backward induction
– Find the optimal rule back to the initial
stage (stage 0)
• From T to 0
• The optimal rule: E[Vt(T)]
• Example: job selection problem, house
selling problem
Job selection problem
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During 10 years
Two kinds of job offer with the same probability
–
–
Good job: 100/year
Bad job: 44/year
–
E[X10] = 100 x 0.5 + 44 x 0.5 = 72
–
–
Bad: 44 x 2 = 88 > 72 (E[X10])
E[X9] = 144
–
–
Bad: 44 x 3 = 132 < 144 (E[X9]) -> continue
E[X8] = 0.5 x 144 + 0.5 x 300 = 294
–
–
Bad: 44 x 4 = 176 < 294 (E[X8]) -> continue
E[X7] = 0.5 x 294 + 0.5 x 400 = 347
Once accept a job, you will remain in that job for the rest of 10 years
You have two choices, accept a job (stop), or continue
At 10
At 9
At 8
At 7
From 9th year on, accept any jobs
Before that, accept only a good job
House selling problem
• Sell a house
– Within T days  horizon
– Offer on day t: Xt
• yt=Xt
– X1, …, XT: i.i.d., uniform distribution over 0 to
M
M
(T )


E VT   E  yT   E  X T  
E Vt
(T )



 
  E  max yt , E Vt (T1 )     max x, E Vt (T1 )  dF  x 

 0
E Vt(T1 ) 


x
   ( T )  xd

E Vt 1
0
M


M

2
M
E V
(T )
t 1

x M  E V
 d

M
2M
2
(T )
t 1
2
Copying decision & P
• Assumption: time slotted
– 1-(1-Pi,d,K-1,Tr-1)x(1-Pj,d,K-1,Tr-1)≥Pi,d,K,Tr-1
• Several other nodes in the same time slot
– Choose the node with the highest delivery probability
•
• Pi,d,K,Tr
– The probability the copy will be forwarded in time-slot Tr and then
be delivered
– Delivery probability Pi,d,K,Tr-1 when the message is not forwarded in
time-slot Tr
Conclusion
• New area of math
– Not complex and complicated, just unknown to us
• It just needs time and effort
– But, interesting, useful, and applicable
– Abundant resources
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E-book
Wikipedia
Papers
Lecture homepages
Open courses (MIT, YouTube, …)
• Formulation!
– Optimal stopping theory, …
• Fundamentally, make your fundamental
Resources
• Backward induction
– http://en.wikipedia.org/wiki/Backward_induction
• Optimal stopping
– http://en.wikipedia.org/wiki/Optimal_stopping
• Secretary problem
– http://en.wikipedia.org/wiki/Optimal_stopping
• Optimal Stopping and Applications
– http://www.math.ucla.edu/~tom/Stopping/Contents.html
• Open course
– http://academicearth.org/
– http://www.youtube.com/education?b=400
– http://ocw.mit.edu/OcwWeb/web/home/home/index.htm