0 Patrick Thiran - Ne X tworking’03 - Chania, Crete, Greece, June 23-25,2003 The First...

1- NeXtworking’03 - Chania, Crete, Greece, June 23-25,2003The First COST-IST(EU)-NSF(USA) Workshop on EXCHANGES & TRENDS IN NETWORKING

Patrick Thiran

Connectivity in sensor networks

[email protected]://lcawww.epfl.ch

Patrick Thiran (joint work with Olivier Dousse, François Baccelli and Martin Hasler)

LCA-ISC-I&C EPFL

LCA


Patrick Thiran

Ad-hoc <-> sensor <-> hybrid <-> cellular

Connectivity is an essential issue in wireless ad hoc networks (many to many, global) sensor networks (many to one, global) hybrid (multi-hop cellular) networks (many to many, local)

Position of nodes is a (often homogeneous Poisson) spatial point process, with intensity .

Position of base stations at nodes of a honey-comb grid, all connected to each other by a wired network.


Patrick Thiran

The Boolean model with circular grains

rji xx

Fixed radio range r Nodes i and j, at positions xi and xj are directly connected iff

References:

• E. N. Gilbert, « Random plane networks », SIAM Journal, 1961.

• R. Meester and R. Roy, « Continuum percolation », Cambridge University Press, 1996.

• O. Dousse, P. Thiran, M. Hasler, « Connectivity in ad hoc and hybrid networks », Infocom 2002.


Patrick Thiran

Connectivity in packet radio networks Finite domain: is the network fully connected ?

Kleinrock & Silvester (1978) « Optimum transmission radii in packet radio networks or why six is a magic number » : r2 = 5.89 is a good value for throughput

Philips, Panwar, Tantawi (1989) « Connectivity properties of a packet radio network » : r2 must grow logarithmically with the area of the domain

Gupta & Kumar (1998) « Critical power for asymptotic connectivity in wireless networks » : for , r2 = log where .

Shakkottai, Srikant, Shroff (2003) « Unreliable sensor grids: Coverage, Connectivity and Diameter»: r2 log pwhere pis the node failure prob.

Infinite domain: is there an infinite connected component (percolation theory): Continuum percolation: there exist a finite (r2)c, below which all connected

components are a.s. bounded and above which there is a unique infinite connected component (Gilbert (1961), Hall (1985), Zuev & Siderenko (1985), Menshikov (1986), Meester & Roy (1990, 1994))

1.64 < (r2)c < 17.9 (Gilbert (1961)) 2.195 < (r2)c < 10.526 (Philips, Panwar, Tantawi (1989)) (r2)c 4.5 (numerical value, Quintanilla, Torquato, Ziff (2000))


Patrick Thiran

λr2

(r2)

Full or partial connectivity ? Long range connectivity appears much before full connectivity

because of a phase transition mechanism (percolation)


Patrick Thiran

Ad hoc or sensor network ?

Ad hoc network : multiple transmissions, many to many. Connectivity metric = probability that an arbitrary pair of nodes is connected to the rest of the network Pc

Sensor network : many to one (the base station collecting data). Connectivity metric = probability that one arbitrary node is connected to the base station


Patrick Thiran

Ad hoc or sensor network ?

Ad hoc network : connectivity

Sensor network : connectivity (probability that an arbitrary node is connected to the base

station)and coverage (probability that an arbitrary point is covered by a node).

λr2

Pconnectivity

(λr2)c

Pcoverage =

1-exp(-r2)


Patrick Thiran

Network on a line

Let Pc(x) be the probability that two nodes distant of x space units are connected, given r and

In 1-dim: Full connectivity Full coverage In 1=dim:

Pc(x) exp(-(x-r)e-r) decreases exponentially fast with x

irx

i

rr

irx

i

r

c i

rixee

i

irxexP

1/

0

/

0 !

))1((

!

)()(

x


Patrick Thiran

Network on a plane Percolation theory: Let (r, ) be the probability that an

arbitrary node belongs to an infinite cluster (percolation probability). Then there is (r2)c such that (r, ) = 0 if r2 < (r2)c (“sub-critical”)

(r, ) > 0 if r2 > (r2)c (“super-critical”)

λr2

Pc(x)

(λr2)csuper-critical (r slightly > rc) sub-critical (r slightly < rc)


Patrick Thiran

Bottlenecks are unavoidable

Let P be the number of alternate paths between any pair of nodes A and B.

Thm: min(NA, NB

) P max(NA0, NB

0)

A

CA(p)

p

C

)(sup)(lim

)(inf)(lim

}bydintersecteedges{#min)(

00

0

0

)(

pNpNN

pNpNN

CpN

Ap

Ap

A

Ap

Ap

A

pCCA

A

0/)( AA NpN

p p

0/)( AA NpNsupercriticalsubcritical


Patrick Thiran

Beyond the Boolean model with circular grains: irregularity helps

Percolation occurs sooner for elongated shapes (Penrose (1993), Booth, Bruck, Cook, Franceschetti (2003))

Possible advantage of directional antennas Uni-directional links


Patrick Thiran

Beyond the Boolean model: the physical (STIRG) model

Signal to Noise Ratio at Node j receiving from Node i is

– P = Emitting power – L(d) = Attenuation function at distance d (e.g., L(d) = d )– N0 = Background thermal noise– γ = degree of orthogonality of the code (γ = 1 for a narrowband system, 0 γ

< 1 for a CDMA system)

Nodes i and j are directly connected iff

Reference:O. Dousse, F. Baccelli, P. Thiran, « Impact of Interferences on Connectivity in ad hoc networks », Infocom 2003.

jikki

ji xxPLN

xxPLSNR ki

,0

},min{ ijji SNRSNR


Patrick Thiran

Interferences can destroy connectivity…

()

= 0

> 0

= 0 > 0


Patrick Thiran

… but not always.

Differences with the boolean model: The existence of an edge depends on every other node’s position The node degree is bounded (it was Poisson for the Boolean model) by 1

+ 1/A necessary condition for percolation is thus 1/

= 0 > 0


Patrick Thiran

Beyond connectivity: MAC, Broadcast, routing ?

Simple TDMA ensures as much connecitivty as performant CDMA shemes (Dousse, Baccelli, Thiran 2003)

Routing is more challenging close to percolation threshold (graph is looks like a labyrinth) (Kuhn, Wattenhofer, Zollinger 2003)

Probabilistic broadcast (Sasson, Cavin and Schiper 2003)


Patrick Thiran

Conclusion

Percolation theory is very useful for sensor and ad hoc networks: connectivity and beyond.

Boolean model with circular grains Importance of dimensionality of the area covered by the

network In 2 dim: connectivity coverage Phase transition is key to explain connectivity in 2 dim at low

density

Beyond Boolean model with circular grains : Elongated grains (directional antennas) helpful for connectivity Physical model (STIRG): Percolation occurs despite

interferences Proof for L(x) with a compact support. Probably not a necessary

condition.

Beyond connectivity : MAC, routing, broadcast, …

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