A Probabilistic Model for Message Propagation in Two-Dimensional Vehicular Ad-Hoc Networks

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

2. Previous Work • Cluster: a connected group of vehicles on a one-dimensional highway, in which messages can be  propagated 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. 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. 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 < pc or p ≥ pc • Focus: determine the probability that a message is delivered to certain blocks away from the source

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. 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. One-Dimensional Message Propagation • Cluster size C: the distance between first and last vehicle • R: transmission range E[C]: expected cluster size • X1: distance between the first and second vehicle (RV)‏ • If exp. distribution • and let • therefore,

8. Comparison

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

10. Cluster Size Distribution • Xi: 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. Cluster Size Distribution (cont.)‏ • Given fC|k, the distribution function of C is • fC|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. Cluster Size Distribution (cont.)‏ • Gamma Approximation • where to ensure: the 1st and 2nd order moments of the Gamma approximation are the same as E[C] and E[C2]

13. Gamma Approximation

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

15. Two-Dimensional Message Propagation • If wireless transmissions are heavily shadowed, p can be simplified as Pr{cluster size > d} • Otherwise: • V0 is connected • to the source

16. Case 1: the cluster originating from V0 has a size larger than d−do • Case 2: the cluster size originating from V0 is smaller than d-d0; the last vehicle connected to V0 is Vw, and de+dw>R

17. Bond Probability • p=p1+p2, d=500m

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

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

20. R=200m and d=500m

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

24. Connectivity probability (GF vs. UF, )‏

25. Broadcast Cost (GF vs. UF)‏

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. Thanks! • Q&A

28. Connectivity probability (GF vs. UF)‏

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 Differential Geometry and Physics, Nankai Tracts in Math., Vol. 10, pp. 160-168, 2006.