1. An Optimal Probabilistic Forwarding Protocol in Delay Tolerant Networks ACM MobiHoc ’09 Cong Liu and Jie Wu 2010.02.03 Presenter: Hojin Lee

2. Contents Introduction Preliminary Optimal Stopping Theory Copying decision Conclusion

3. 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(N 1/2 ) Threshold: updated to better on contact

4. Introduction (2/2) Goal: increase delivery ratio with low cost within Time-To-Live Quality – existing metric –Inter-meeting time to destination: I i,d Metric in OPF –Comprehensive: direct, indirect –Dynamic: metric changed with residual time(, remaining hops)

5. Preliminary Hop-count-limited forwarding –Special case of ticket-based forwarding (half- half ticket)

6. Metric Metric: P i,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 –P i,d,0,Tr –Exponential: 1-exp(-Tr/I i,d ) I i,d : mean inter-meeting time between node i and destination

7. Optimal Stopping Theory (1/3) After observing X 1, X 2, … X t, two choices –1) stop (and receive the known reward y t ) –2) continue (and observe X t+1 ) How do we maximize the expected reward? –When do we stop?  optimal stopping rule Probability, statistics – for many ones

8. Optimal Stopping Theory (2/3) A finite horizon –Upper bound T on the number of stages V t (T) : the maximum expected reward starting from stage t –V t (T) =max{y t, E[V t+1 (T) ]} –The above equation implies if y t >E[V t+1 (T) ] then stop, o/w continue How do we know E[V t+1 (T) ]?  backward induction

9. 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[V t (T) ] Example: job selection problem, house selling problem

10. Job selection problem During 10 years Two kinds of job offer with the same probability –Good job: 100/year –Bad job: 44/year 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 –E[X 10 ] = 100 x 0.5 + 44 x 0.5 = 72 At 9 –Bad: 44 x 2 = 88 > 72 (E[X 10 ]) –E[X 9 ] = 144 At 8 –Bad: 44 x 3 = 132 continue –E[X 8 ] = 0.5 x 144 + 0.5 x 300 = 294 At 7 –Bad: 44 x 4 = 176 continue –E[X 7 ] = 0.5 x 294 + 0.5 x 400 = 347 From 9 th year on, accept any jobs Before that, accept only a good job

11. House selling problem Sell a house –Within T days  horizon –Offer on day t: X t y t =X t –X 1, …, X T : i.i.d., uniform distribution over 0 to M

12. Copying decision & P Assumption: time slotted –1-(1-P i,d,K-1,Tr-1 )x(1-P j,d,K-1,Tr-1 )≥P i,d,K,Tr-1 Several other nodes in the same time slot –Choose the node with the highest delivery probability P i,d,K,Tr –The probability the copy will be forwarded in time-slot Tr and then be delivered –Delivery probability P i,d,K,Tr-1 when the message is not forwarded in time- slot Tr

13. 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 E-book Wikipedia Papers Lecture homepages Open courses (MIT, YouTube, …) Formulation! –Optimal stopping theory, … Fundamentally, make your fundamental

14. Resources Backward induction –http://en.wikipedia.org/wiki/Backward_inductionhttp://en.wikipedia.org/wiki/Backward_induction Optimal stopping –http://en.wikipedia.org/wiki/Optimal_stoppinghttp://en.wikipedia.org/wiki/Optimal_stopping Secretary problem –http://en.wikipedia.org/wiki/Optimal_stoppinghttp://en.wikipedia.org/wiki/Optimal_stopping Optimal Stopping and Applications –http://www.math.ucla.edu/~tom/Stopping/Contents.htmlhttp://www.math.ucla.edu/~tom/Stopping/Contents.html Open course –http://academicearth.org/http://academicearth.org/ –http://www.youtube.com/education?b=400http://www.youtube.com/education?b=400 –http://ocw.mit.edu/OcwWeb/web/home/home/index.htmhttp://ocw.mit.edu/OcwWeb/web/home/home/index.htm