Zanella Andrea – Pierobon Gianfranco – Merlin Simone
Dept. of Information Engineering, University of Padova,
{zanella,pierobon,merlo}@dei.unipd.it
Ad hoc linear networks
Optimum Broadcast strategy
•Sensor networks
•Car Networks
Limiting performance:
• Minimum latency
• Minimum traffic
• Maximum reliability
• minimized redundancy
• preserved connectivity
MCDS
(Only nodes in a connected
set of minimum cardinality
rebroadcast packets)
= Silent node
= Transmitting node
Linear nodes deployment modeled as
an inhomogeneous Poisson arrivals
Broadcast
source
s0
s2
s1
s3 s4
s5
s6 s7
Drawback:
• Needed topologic
information
x
{
} = MCDS
s8
x
x=0
Aim: mathematical characterization of the MCDS-broadcast
propagation dynamic with inhomogeneous density of nodes
Notations
Hypothesis
wk = distance reached by the k-th rebroadcast
Pk = probability of the existence of the k-th rebroadcast
fk (x ) = probability density function of wk, given that wk exists
 (x ) = nodes density function
• Ideal channel
• Deterministic transmission radius (R)
Theorem
The dynamic of the MCDS-broadcast propagation along the network is
statistically determined by the family of functions fk(x), which can be
recursively obtained as follows:
Example with a variable node density
k 1
 f1 ( x1 )   ( x1  R)

xk
Pk 1

 ( xk  R  xk 1 )
f k 1 ( xk 1 )dxk 1 k  2,3...
 f k (xk )  P λ(xk  R)  e
k
xk  R

(xa )
6
4
2
0
where Pk can, in turn, be recursively derived as
0
5
10
15
20
Connection probability
as a function of
distance and
number of hops
20
p.d.f. of the
front position
weighted with the
probability
of its existence
1
P1  1

xk

  ( xk  R  xk 1 )
f k 1 ( xk 1 )dxk 1dxk
Pk (xk )  Pk 1  λ(xk  R)  e
xk  R

k 1
- = analytical
x = simulated
R=1
Ck(xa )
k=1
0.5
k=6
k=11
k  2,3...
0
0
5
1
15
fk(xa ) * Pk
k=6
0.5
Homogeneous Case
k=16
10
k=1
Connection
Probability
variable
node density
k=11
0
0
5
k=16
10
Distance (xa )
15
20
1
Asymptotic value*
-- = analytical
x = simulated
0.9
R=1
0.8
40
35
Asymptotic mean number
of reached nodes
0.7
Number of reached
nodes as a function
of number of hops
30
0.6
k=1
Ck(xa )
k=
Number of hops
0.5
k=6
0.4
k=11
NC
k
25
20
0.3
needed hops for
broadcast "completion"
k=16
0.2
15
k=21
0.1
10
0
0
5
10
15
20
25
30
Distance (xa )
5
0
5
10
15
20
25
30
35
40
Hop
* O. Dousse,et. al. “Connectivity in ad-hoc and hybrid networks” Proc. IEEE Infocom02
This work was supported by MIUR within the framework of the
”PRIMO” project FIRB RBNE018RFY (http://primo.ismb.it/firb/index.jsp).