Optimal Radius for Connectivity in Duty-Cycled Wireless Sensor - - PowerPoint PPT Presentation

Optimal Radius for Connectivity in Duty-Cycled Wireless Sensor Networks

• A. Bagchi1

Cristina M. Pinotti 2

• S. Galhotra 1
• T. Mangla 1
• 1Dept. of Comp. Sci. and Engg., IIT Delhi, India
• 2Dept. of Comp. Sci. and Math.,University of Perugia, Italy

16th ACM International Conference on Modeling, Analysis and Simulation

• f Wireless and Mobile Systems,

November 4, 2013

Cristina M. Pinotti (University of Perugia) Optimal Connectivity Radius for DC-WSN November 4th 1 / 19

Outline

In a functioning network, sensors may follow a duty-cycle to save energy, but they must preserve

time coverage, i.e., data generated at any time must be sensed and relayed by the network connectivity, i.e., every node should be connected to any other node and to the sink

For this goal, we propose:

a weak condition derived from Random Geometric Graph (RGG) a strong condition derived from Vertex-Based Random Geometric Graph (VB-RGG)

We present the contiguous and the random selection duty-cycle schemes We test by simulation the connectivity of such schemes using both radii

Cristina M. Pinotti (University of Perugia) Optimal Connectivity Radius for DC-WSN November 4th 2 / 19

Network Model

Random Geometric Graph RGG(n,r):

There are n sensors distributed uniformly at random in the unit circle centred at the origin, where the sink is placed. Each sensor is modeled by a vertex. Two sensors u and v communicate if they are at distance not greater than

• r. Each possible transmission is modeled by an edge.

Sensors are synchronized and follow a pattern of sleeping and waking period:

L: length of the duty-cycle d: the number of waking slots

δ = d

L : the duty-cycle ratio

DC-WSNA (n,r,δ,L) :

the same vertex set as in RGG(n,r) each sensor u selects an awake configuration Au ⊆ {0,1,...,L − 1} ∈ A with |Au| = d E = {(u,v) : d(u,v) ≤ r,Au ∩ Av = /

0}. For two vertices u and v that are

within transmission range of each other to be connected, they must share a slot where they are both awake.

Cristina M. Pinotti (University of Perugia) Optimal Connectivity Radius for DC-WSN November 4th 3 / 19

Property DC-WSNA (n,r,δ,L): Time Coverage

Any duty-cycle scheme A that we consider guarantees:

For each k ∈ {0,1,...,L − 1}, the probability that a node u is awake in slot k is δk > 0, where δk may be a function of d and L, independent of the number of sensors in the network

Uniform time coverage:

δk = d/L for 0 ≤ k ≤ L−1, i.e., the probability does not depend on the slot

Cristina M. Pinotti (University of Perugia) Optimal Connectivity Radius for DC-WSN November 4th 4 / 19

Property DC-WSNA (n,r,δ,L): Connectivity

If δ > 1/2 then

DC-WSNA (n,r,δ,L) is connected whenever RGG(n,r) is connected.

If δ ≤ 1/2 to guarantee connectivity we need the Reachability property:

There is a finite sequence of awake configurations that leads in the duty-cycle scheme A from any configuration to another. Example

L = {0,1,...,5}, A = {A1 = {0,1,2},A2 = {3,4,5}} does not satisfy the reachability condition L = {0,1,...,9}, A = {A1 = {1,2,3},A2 = {2,3,4},A3 = {3,4,5},A4 =

{4,5,6},...,A10 = {0,1,2}} satisfies the reachability condition

Cristina M. Pinotti (University of Perugia) Optimal Connectivity Radius for DC-WSN November 4th 5 / 19

The weak connectivity result

Based on the connectivity condition for RGG(n,r), we prove that for δ ≤ 1/2: Theorem The probability that DC-WSNA (n,r(n),δ,L) is connected tends to 1 as n → ∞ if r2(n) = (lnn+ c(n))/n

πδ =

rRGG

δ ,

with c(n) → ∞ as n → ∞.

Cristina M. Pinotti (University of Perugia) Optimal Connectivity Radius for DC-WSN November 4th 6 / 19

Two specific duty-cycle schemes A

The contiguous model: DC-C-WSN Each sensor u independently chooses an integer iu from the set

{0,1,...,L− 1}

u is awake for the all the time slots iu to iu + d − 1 u is asleep from all the remaing L− d time slots The independent random selection model: DC-R-WSN Each sensor u chooses a set of awake slots Au of size d at random from

{0,1,...,L− 1}

Cristina M. Pinotti (University of Perugia) Optimal Connectivity Radius for DC-WSN November 4th 7 / 19

Simulations

n sensors are placed uniformly at random in a unit disk with 105 ≤ n ≤ 106

δ varying between 0.05 and 0.5:

to find the connected component, we make use of the Union Find algorithm each experiment is repeated at least five times

Cristina M. Pinotti (University of Perugia) Optimal Connectivity Radius for DC-WSN November 4th 8 / 19

Percentage of sensors in the largest connected component

In our experiments, we set c(n) to lnlnn if not otherwise stated.

2 4 6 8 10 x 10

5

99.95 99.955 99.96 99.965 99.97 99.975 99.98 99.985 99.99 99.995 100 Number of nodes (n) Connectivity Percentage DC−C−WSN, L = 100 δ = 0.25 δ = 0.05 δ = 0.10

For DC-R-WSN, the percentage of connectivity is almost 100% ! Is the weak radius the minimum radius able to guarantee connectivity?

Cristina M. Pinotti (University of Perugia) Optimal Connectivity Radius for DC-WSN November 4th 9 / 19

The vertex-based random geometric graph

To improve on the radius, we introduce a generalization of the random connection model by Meester and Roy [1996]: The vertex set V is the same as RGG With each u ∈ V, we associate an independent random variable Zu which represents the awake configuration The edge (u,v) exists with the probability γ that Zu ∩ Zv = /

0 if and only if

d(u,v) ≤ r Theorem P(VB-RGG(n,r,γ) is connected) → 1 as n → ∞ if and only if r(n)2 = (lnn+ c(n))/n

πγ =

rRGG

γ ,

where limn→∞ c(n) = ∞ as n → ∞.

Cristina M. Pinotti (University of Perugia) Optimal Connectivity Radius for DC-WSN November 4th 10 / 19

The probability γ of sharing a slot

DC-C-WSN

γ = 2δ − 1/L

In DC-C-WSNs, a sensor v will share at least one slot with node u if v chooses as its starting point any of the slots iu −(d − 1) mod L,...,iu,...iu +

(d − 1) mod L.

r2(n) =

rRGG 2δ−1/L

DC-R-WSN

γ >

• 1−(1−δ)d

In DC-R-WSNs, when a node u has chosen d slots, another node v has d possibilities to choose one slot in common with u and the probability of doing that is at least δ each time. r2(n) =

rRGG

(1−(1−δ)d )

Cristina M. Pinotti (University of Perugia) Optimal Connectivity Radius for DC-WSN November 4th 11 / 19

Percentage of sensors in the largest connected component when L = 100, n and δ vary

2 4 6 8 10 x 10

5

97.5 98 98.5 99 99.5 100 Number of nodes (n) Connectivity Percentage DC−C−WSN, L=100 δ = 0.25 δ = 0.10 δ = 0.05 2 4 6 8 10 x 10

5

75 80 85 90 95 100 Number of nodes (n) Connectivity Percentage DC−R−WSN, L =100 δ = 0.25 δ = 0.10 δ = 0.05

Cristina M. Pinotti (University of Perugia) Optimal Connectivity Radius for DC-WSN November 4th 12 / 19

The ratio of the weak to the strong radius in DC-C-WSN and DC-R-WSN for different δ and L

L = 200 L = 100

δ

DC-C-WSN DC-R-WSN DC-C-WSN DC-R-WSN 0.05 1.378 2.832 1.341 2.127 0.10 1.396 2.963 1.378 2.552 0.15 1.402 2.572 1.390 2.466 0.20 1.405 2.236 1.396 2.249 Recalling that the energy is proportional to nδr2(n), the strong radius halves the energy spent in DC-C-WSNs and divides at least by four the energy spent in DC-R-WSNs

Cristina M. Pinotti (University of Perugia) Optimal Connectivity Radius for DC-WSN November 4th 13 / 19

Is the strong radius optimum?

Percentage of connectivity with different values of c(n)

0.5 1 1.5 2 2.5 3 x 10

6

0.05 0.1 0.15 0.2 0.25 Number of nodes (n) Connectivity Percentage DC−C−WSN, d=10, L=100 c(n) = −2 sqrt(log(n)) c(n) = −2.5

• (log(n))

c(n) = −(log(log(n)))2 2 4 6 8 10 x 10

5

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 Number of nodes (n) Connectivity Percentage DC−R−WSN, d = 10, L =100 c(n) = −5 c(n) = −4.5 c(n) = −4 c(n) = −3.5 c(n) = −3

Cristina M. Pinotti (University of Perugia) Optimal Connectivity Radius for DC-WSN November 4th 14 / 19

Conclusions

We have studied the optimal radius for achieving connectivity in DC-WSNs Two specific duty-cycled schemes, the DC-C-WSN and DC-R-WSN, have been proposed to stress the differences between the weak and the strong radii We have presented a general and foundational contribution because these results apply also to:

the key-predistribution for secure communication L: the size of the overall set of keys d: the size of the subset of keys assigned to each node the study of connectivity in WSNs with directional antennas where the direction is fixed at random independently at each node

Cristina M. Pinotti (University of Perugia) Optimal Connectivity Radius for DC-WSN November 4th 15 / 19

Questions

Cristina M. Pinotti (University of Perugia) Optimal Connectivity Radius for DC-WSN November 4th 16 / 19

Proof of Theorem

Vi,0 ≤ i < L consists of the sensors awake at time slot i

Ei = {(u,v) : u,v ∈ Vi,d(u,v) ≤ r}. E(|Vi|) ≥ δn

By Chernoff bounds, for any ε > 0, lim

n→∞

• P(|Vi| < (1−ε)δ|V|) ≤ e−(ε2δn)/2

= 0 Hence, by Gupta and Kumar result for RGG [1998], Gi = (Vi,Ei) is connected with high probability

The probability that Gi = (Vi,Ei) and Gi+1 = (Vi+1,Ei+1) are disconnected can be expressed as: P

• Γu ∩ V Bi+1 = /

0||Γu| ≥ log(n)+c(n)

δmin

• + o(1)

≤ (1−βi+1)

log(n)+c(n)

δmin

+ o(1) ≤ exp−

• βi+1·(log(n)+c(n))

δmin

• + o(1),

which tends to 0 when n → ∞

Cristina M. Pinotti (University of Perugia) Optimal Connectivity Radius for DC-WSN November 4th 17 / 19

Percentage of sensors in the largest connected component when L = 100, n = 106 and δ vary

0.1 0.2 0.3 0.4 0.5 0.5 92 93 94 95 96 97 98 99 100 δ Connectivity Percentage DC−C−WSN, n = 1000000, L = 100 0.1 0.2 0.3 0.4 0.5 91 92 93 94 95 96 97 98 99 100 DC−R−WSN, n = 1000000, L = 100 δ Connectivity Percentage

Cristina M. Pinotti (University of Perugia) Optimal Connectivity Radius for DC-WSN November 4th 18 / 19

Is the strong radius optimum?

Number of isolated points and size of the second largest component

1 2 3 4 5 x 10

6

1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 Number of nodes (n) Number of isolated components DC-C-WSN, d=10, L=100 c(n) = 1 c(n) = −1 1 2 3 4 5 x 10

6

0.5 1 1.5 2 2.5 3 x 10

4

Number of nodes (n) Size of second largest component DC−C−WSN, d=10, L=100 c(n) = −1 c(n) = −log(log(n))2 c(n) = −2 ∗

• (log(n))

Cristina M. Pinotti (University of Perugia) Optimal Connectivity Radius for DC-WSN November 4th 19 / 19