Barrier Coverage With Wireless Sensors

Santosh Kumar, Ten H. Lai, Anish Arora

The Ohio State University

Presented at Mobicom 2005

Barrier Coverage

USA

Belt Region

Two special belt regions

Rectangular:

Donut-shaped:

How to define a belt region?

Parallel curves Region between two parallel curves

Crossing Paths

A crossing path is a path that crosses the complete width of the belt region.

Crossing paths Not crossing paths

k-Covered

A crossing path is said to be k-covered if it intersects the sensing disks of at least k sensors.

3-covered 1-covered 0-covered

k-Barrier Covered

A belt region is k-barrier covered if all crossing paths are k-covered.

1-barrier covered

Not barrier covered

Barrier vs. Blanket Coverage Barrier coverage

Every crossing path is k-covered Blanket coverage

Every point is covered (or k-covered) Blanket coverage Barrier coverage

1-barrier covered but not 1-blanket covered

Question 1

Given a belt region deployed with sensors Is it k-barrier covered?

Is it 4-barrier covered?

Reduced to k-connectivity problem Given a sensor network over a belt region Construct a coverage graph G(V, E)

V: sensor nodes, plus two dummy nodes L, RE: edge (u,v) if their sensing disks overlap

Region is k-barrier covered iff L and R are k-connected in G.

L R

Be Careful!

Assumption:

If D1 ∩ D2 ≠ Φ, then (D1 U D2) ∩ B is connected.

Global algorithm for testing k-barrier coverage

Given a sensor network Construct a coverage graph Using existing algorithms

To test k-connectivity between two nodes

Question: what about donut-shaped regions? Question: can it be done locally?

Is it k-barrier covered? Still an open problem for donut-shaped

regions.

Is it k-barrier covered? Cannot be determined locally k-barrier covered iff k red sensors exist

In contrast, it can be locally determined if a region is not k-blanket covered.

Question 2

Assuming sensors can be placed at desired locationsWhat is the minimum number of sensors to

achieve k-barrier coverage?k x L / (2R) sensors, deployed in k rows

Question 3 If sensors are deployed randomly

How many sensors are needed to achieve k-barrier coverage with high probability (whp)?

Desired are A sufficient condition to achieve barrier coverage whp A sufficient condition for non-barrier coverage whp Gap between the two conditions should be as small

as possible

Conjecture: critical condition for k-barrier coverage whp

If , then k-barrier covered whp

If , non-k-barrier covered whp

s1/s

Expected # of sensors in the r-neighborhood of path

r r

k-barrier covered whp

k-barrier covered whp lim Pr( belt region is k-barrier covered ) = 1

not (k-barrier covered whp) lim Pr( belt region is k-barrier covered ) < 1

non-k-barrier covered whp lim Pr( belt region is not k-barrier covered ) = 1 lim Pr( belt region is k-barrier covered ) = 0

L(p) = all crossing paths congruent to p

p

p

Weak Barrier Coverage

A belt region is k-barrier covered whp if

lim Pr(all crossing paths are k-covered) = 1or lim Pr( crossing paths p, L(p) is k-covered ) = 1

A belt region is weakly k-barrier covered whp if

crossing paths p, lim Pr( L(p) is k-covered ) = 1

Conjecture: critical condition for k-barrier coverage

If , then k-barrier covered whp

If , not k-barrier covered whp

What if the limit equals 1?

weakly

weakly

weak

Determining #Sensors to Deploy

Given: Length (l), Width (w), Sensing Range (R), and

Coverage Degree (k), To determine # sensors (n) to deploy, compute

s2 = l/wr = (R/w)*(1/s)Compute the minimum value of n such that

2nr/s ≥ log(n) + (k-1) log log(n) + √log log(n)

s1/s

Simulations

Using this formula to determine n, The n randomly deployed sensors

provide weak k-barrier coverage with probability ≥0.99.

They also provide k-barrier coverage with probability close to 0.99.

Summary Barrier coverage

Basic results

Open problemsBlanket coverage: extensively studiedBarrier coverage: further research needed