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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
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−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
17
Bond Probability
p=p1+p
2, 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,
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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
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
SimulationNetwork 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)
29
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 erential ffGeometry and Physics, Nankai Tracts in Math., Vol. 10, pp. 160-168, 2006.