A Probabilistic Model for Message Propagation in Two-Dimensional

Vehicular Ad-Hoc Networks

Yanyan Zhuang, Jianping Pan and Lin Cai

University of Victoria, Canada

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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

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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

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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

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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

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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

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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,

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Comparison

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Cluster Size Characterization

Already have: first order

Derivation of second order

thus

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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

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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

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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]

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Gamma Approximation

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Two-Dimensional Message Propagation

Bond probability p: prob. that two adjacent intersections are connected

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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

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Case 1: the cluster originating from V0 has a size larger than d−d

o

Case 2: the cluster size originating from V0 is smaller than d-d

0;

the last vehicle connected to V0 is V

w, and d

e+d

w>R

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Bond Probability

p=p1+p

2, d=500m

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Ladder Connectivity

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

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Ladder Connectivity (cont.)

For x>1, recursion is needed to derive the probability

Generally,

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R=200m and d=500m

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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

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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

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SimulationNetwork connectivity (w/o geo-constraints: GF vs.

UF)

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Connectivity probability (GF vs. UF, )

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Broadcast Cost (GF vs. UF)

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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

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Thanks!

Q&A

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Connectivity probability (GF vs. UF)

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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 erential ffGeometry and Physics, Nankai Tracts in Math., Vol. 10, pp. 160-168, 2006.