Download On the limiting performance of broadcast algorithms over

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)
Drawback:
• Needed topologic
information
= Silent node
= Transmitting node
Linear nodes deployment modeled as
an inhomogeneous Poisson arrivals
Broadcast
source
s0
s2
s1
s3 s4
s5
s6 s7
x
{
} = MCDS
s8
x
x=0
Aim: mathematical characterization of the MCDS-broadcast
propagation dynamic with inhomogeneous density of nodes
Theorem
Hypothesis
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:
• Ideal channel
• Deterministic transmission radius (R)
Notations
k 1
 f1 ( x1 )   ( x1  R)

xk
Pk 1

l ( xk  R  xk 1 )
f
(x
)

λ(x

R)
e
f k 1 ( xk 1 )dxk 1 k  2,3...
k
 k k

P
k
xk  R

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
l (x ) = nodes density function
where Pk can, in turn, be recursively derived as
 P1  1

xk

 l ( xk  R  xk 1 )
f k 1 ( xk 1 )dxk 1dxk
 Pk (xk )  Pk 1  λ(xk  R)  e
xk  R

Performance metrics
Ck(x) = Connection probability of x in k hops
NkC = Mean number of nodes reached in k hop
k 1
k  2,3...
Results: Connection probability, Propagation statistics, …
Inhomogeneous (general) Case
Homogeneous Case
Example: nodes reached at each hop
Connection
Probability
-- = analytical
x = simulated
0.8
R=1
0.8
Asymp. value*
0.7
0.7
Ck(xa)
0.6
k=1
0.9
Number
of hops
k=
0.5
k=6
0.4
k=11
- = analytical
x = simulated
k=16
30
25
k=1
0.5
20
0.4
Number of hops for
broadcast "completion"
k=6
15
0.2
0.2
k=11
k=21
10
k=16
0.1
0.1
0
0
5
10
Asymptotic mean number
of reached nodes
0.3
0.3
0
35
0.6
Ck(xa)
0.9
40
1
NC
k
1
15
20
25
30
0
2
4
6
Distance (x a)
8
10
12
14
16
18
20
0
Distance (x a)
l(xa)
10
15
20
25
30
35
40
variable
node density
4
2
0
5
Hop
6
* O. Dousse,et. al. “Connectivity in ad-hoc and
hybrid networks” Proc. IEEE Infocom02
5
0
2
4
6
8
10
12
14
16
18
20
This work was supported by MIUR within the framework of the
”PRIMO” project FIRB RBNE018RFY (http://primo.ismb.it/firb/index.jsp).