1. A Probabilistic Model for Message Propagation in Two-Dimensional Vehicular Ad-Hoc Networks Yanyan Zhuang, Jianping Pan and Lin Cai University of Victoria, Canada

2. 2 Previous Work Cluster: a connected group of vehicles on a one- dimensional highway, in which messages can be propag ated directly Cluster size: the distance between the first and last vehicle in the same cluster -- [JSAC10ZPLC] to appear Challenges: from 1-d to 2-d

3. 3 Background & Related Work Message propagation in 2-d, infrastructure-less V2V communication Traffic modeling and message propagation 1) Vehicle Traffic Models Assumption: inter-vehicle distances follow an i.i.d. distribution, e.g., exponential distribution

4. 4 Background & Related Work (cont.)‏ 2) Percolation Theory The process of liquid seeping through a porous object: each edge is open with probability p The existence of an infinite connected cluster of open edges: whether p < p c or p ≥ p c Focus: determine the probability that a message is delivered to certain blocks away from the source

5. 5 Background & Related Work (cont.)‏ 3) Message Propagation and Connectivity Network connectivity in 1-d is always limited For 2-d, e.g., city blocks, network connectivity can be guaranteed if the density among nearby nodes is above a certain threshold

6. 6 Contributions Connectivity property of message propagation in two-dimensional VANET scenarios 1) Derive average cluster size in 1-d, with distribution approximation 2) Derive connectivity probability for 2-d ladder 3) Formulate the problem for 2-d lattice Tradeoff between message forwarding schemes w/o geographic constraints: simulation

7. 7 One-Dimensional Message Propagation Cluster size C: the distance between first and last vehicle R: transmission range E[C]: expected cluster size X 1 : distance between the first and second vehicle (RV)‏ If exp. distribution and let therefore,

8. 8 Comparison

9. 9 Cluster Size Characterization Already have: first order Derivation of second order thus

10. 10 Cluster Size Distribution X i : the RV of inter-vehicle distance, given that the i-th and (i+1)-th vehicles are in the same cluster Suppose there are k vehicles in a cluster, the Laplace Transform of the cluster size distribution is

11. 11 Cluster Size Distribution (cont.)‏ Given f C|k, the distribution function of C is f C|k is obtained by taking the inverse-Laplace Transform on f* C|k, and is the probability that there are k vehicles in a cluster Unfortunately, no closed-form by inverse-Laplace Transform: C is the sum of k truncated exponential random variables, and k itself follows a Geometric distribution

12. 12 Cluster Size Distribution (cont.)‏ Gamma Approximation where to ensure: the 1 st and 2 nd order moments of the Gamma approximation are the same as E[C] and E[C 2 ]

13. 13 Gamma Approximation

14. 14 Two-Dimensional Message Propagation Bond probability p: prob. that two adjacent intersections are connected

15. 15 Two-Dimensional Message Propagation If wireless transmissions are heavily shadowed, p can be simplified as Pr{cluster size > d} Otherwise: V 0 is connected to the source

16. 16 Case 1: the cluster originating from V 0 has a size larger than d−d o Case 2: the cluster size originating from V 0 is smaller than d-d 0 ; the last vehicle connected to V 0 is V w, and d e +d w >R

17. 17 Bond Probability p=p 1 +p 2, d=500m

18. 18 Ladder Connectivity Given that each intersection is connected with p, by the principle of inclusion-exclusion (PIE)‏

19. 19 Ladder Connectivity (cont.)‏ For x>1, recursion is needed to derive the probability Generally,

20. 20 R=200m and d=500m

21. 21 Lattice Connectivity Enumerate all the possible paths from (0, 0) to (x, y)‏ by PIE, P(x, y) can be obtained by calculating the probabilities of different combinations of paths and crosschecking their overlapping street segments

22. 22 combinatorial explosion Eg, x = 5, y = 3, # of different paths is # of different combinations of these 56 paths can be, each of which has |x|+|y|=8 segments If store these segments in bitmap, requires 38 bits per path, (x+1)y+x(y+1)=38 unique street segments memory required

23. 23 Simulation Network connectivity (w/o geo-constraints: GF vs. UF)‏

24. 24 Connectivity probability (GF vs. UF, )‏

25. 25 Broadcast Cost (GF vs. UF)‏

26. 26 Conclusion & Further Discussions Network connectivity in 1-d, 2-d ladder, 2-d lattice (simulation)‏ Bond probability: consider packet loss, collisions Vehicle mobility, e.g.,carry-and-forward V2I communications: drive-thru Internet

27. 27 Thanks! Q&A

28. 28 Connectivity probability (GF vs. UF)‏

29. 29

30. 30 References [JSAC10ZPLC] Y. Zhuang, J. Pan, Y. Luo and L. Cai, “Time and Location-Critical Emergency Message Dissemination for Vehicular Ad-Hoc Networks”, to appear in IEEE Journal on Selected Areas in Communications (JSAC) special issue on Vehicular Communications and Networks, 2010. [DGP06CW] L. C. Chen and F. Y. Wu, “Directed percolation in two dimensions: An exact solution”, in Di ff erential Geometry and Physics, Nankai Tracts in Math., Vol. 10, pp. 160-168, 2006.