1 Sensor Placement and Lifetime of Wireless Sensor Networks: Theory and Performance Analysis Ekta Jain and Qilian Liang, Department of Electrical Engineering,

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Sensor Placement and Lifetime of Wireless Sensor Networks:Theory and Performance Analysis

Ekta Jain and Qilian Liang, Department of Electrical Engineering,

University of Texas at Arlington

IEEE GLOBECOM 2005

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Outline

Introduction Preliminaries Node Lifetime Evaluation Network Lifetime Analysis Using

Reliability Theory Simulation Conclusion

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Introduction (1/3)

Sensor networks have limited network lifetime. energy-constrained

Most applications have pre-specified lifetime requirement. Example: [4] has a requirement of at

least 9 months Estimation of lifetime becomes a

necessity.[4] A. Mainwaring, J. Polastre, R. Szewczyk, D. Culler, J. Anderson, ”Wireless Sensor Networks for Habitat Monitoring”

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Introduction (2/3)

Sensor Placement vs. Lifetime Estimation Two basic placement schemes: Square

Grid, Hex-Grid. Bottom-up approach lifetime evaluation.

Theoretical Result vs. Actual Result by extensive simulations

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Introduction (3/3)

Bottom-up approach to lifetime evaluation of a network.

Lifetime Behavior Analysis

(single sensor node)

Lifetime Behavior Analysis

(sensor networks using two basic placement schemes)

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PreliminariesBasic Model

rs : the sensing range rc : the communication range neighbors

distance of separation r ≤ rc

rsr

assume rs = rc

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PreliminariesBasic Model

The maximum distance between two neighboring nodes: rmax = rc = rs

A network is said to be deployed with minimum density when: the distance between its neighboring nodes i

s r = rmax

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PreliminariesPlacement Schemes

Placement Schemes

2-neighbor group 3-neighbor group 4-neighbor group

Hex-Grid Square Grid described in [1]

[1] K. Kar, S. Banerjee, ”Node Placement for Connected Coverage in Sensor Networks”

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PreliminariesPlacement Scheme in Reference [1]

2-neighbor group and provides full coverage!!

[1] K. Kar, S. Banerjee, ”Node Placement for Connected Coverage in Sensor Networks”

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PreliminariesPlacement Schemes

Square Grid Hex-Grid

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PreliminariesCoverage and Connectivity

Various levels of coverage may be acceptable. depends on the application requirement

In our analysis… require the network to provide complete

coverage only 100% connectivity is acceptable the network fails with loss of connectivity

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PreliminariesLifetime

consider basic placement schemes

Square- Grid

Hex- Grid

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PreliminariesLifetime

Tolerate the failure of a node all of whose neighbors are functioning.

Define minimum network lifetime as the time to failure of any two neighboring nodes. i.e. the first loss of coverage

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Node Lifetime Evaluation (1/5)

A sensor node is said to have: m possible modes of operation at any given time, the node is in one of these

m nodes wi : fraction of time that a node spends in i-t

h mode

1,2...m i 1 wi

i 1 2 m……

w1

w2

wm……

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Node Lifetime Evaluation (2/5)

Wi are modeled as random variables. take values from 0 to 1 probability density function (pdf)

Etotal: total energy Pi: power spended in the i-th mode per unit time Tnode: lifetime of the node Eth: threshold energy value

iii

totalnode Pw

E Tthnodei

iitotal E TPw - E

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Node Lifetime Evaluation (3/5)

The lifetime of a single node can be represented as a random variable. takes different values by its probability densi

ty function (pdf), ft (t)

i ii

totalnode Pw

ET

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Node Lifetime Evaluation (4/5)

Assume that the node has two modes of operation. Active: Pr (node is active) = p, w1

Idle: Pr (node is idle) = 1-p, w2 = 1- w1

Observe the node over T time units. binomial distribution

x-TxTx1 p)-(1p C x) P(w

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Node Lifetime Evaluation (5/5)

As T becomes large: binomial distribution ~ N(μ, σ) μ(mean) = Tp, σ(variance) = Tp(1-p)

The fraction of time (w1 and w2) follows the normal distribution.

The reciprocal of the lifetime of a node is normally distributed.

2

2

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Network Lifetime AnalysisReliability Theory

The network lifetime is also a random variable.

Using Reliability Theory to find the distribution of the network lifetime.

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Reliability Theory

Concerned with the duration of the useful life of components and systems.

We model the lifetime as a continuous non-negative variable T. pdf, cdf, Survivor Function, System Reliabili

ty, RBD.

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Reliability Theorypdf and cdf

Probability Density Function f(t): the probability of the random variable taki

ng a certain value Cumulative Distribution Function

F(t): the proportion of the entire population that fails by time t.

t

0f(s)ds F(t)

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Reliability TheorySurvivor Function

Survivor Function: S(t) the probability that a unit is functioning at a

ny time t

survivor function vs. pdf

t

0f(s)ds - 1 F(t) - 1 S(t)

0 tt][T P S(t) S(0) = 1,

S(t) is non-decreasing

0, S(t) lim t

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Reliability TheorySystem Reliability

To consider the relationship between components in the system. using RBD

distribution of the components

distribution of the system

single node

entire network

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Reliability TheoryReliability Block Diagram (RBD)

Any complex system can be realized in the form of combination blocks, connected in series and parallel.

S1(t) and S2(t) are the survivor functions of two components.

(t)(t)SS (t)S 21series (t))]S-(t))(1S-[(1 - 1 (t)S 21parallel

S1(t) S2(t)S1(t)

S2(t)

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Network Lifetime Analysis

minimum network lifetime: the time to failure of two adjacent nodes

Assume that: All sensor nodes have the same survivor

function. Each sensor node fails independent of

one another.

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Network Lifetime AnalysisSquare Grid

Square Grid Placement AnalysisRegion 1

Region 2

Region 1

a b

c d

Region 2

x

y

x yor

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Network Lifetime AnalysisSquare Grid

a b

c d

a

b c

Region 1 Block 1 : RBD for Region 1

)ss-)(1s-(1 - 1 s cbablock1

322 block1 s - s s )s-s)(1-(1 - 1 s

∵ sensors are identical

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Network Lifetime AnalysisSquare Grid

or

xx

y

y x

y

Region 2 Block 2 : RBD for Region 2

)s-)(1s-(1 - 1 s yx block2

∵ sensors are identical, have the same survivor function2

block2 s - 2s s)-s)(1-(1 - 1 s

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Network Lifetime AnalysisNetwork Survivor Function for Square Grid

block 1’s block 2’s connect in series

1 - Nmin

1 - Nmin 2min ) 1 - N(

) 1 - N(*2 min

1) - N2(2block

1) - N(1block network

min2

min )(s)(s s

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Network Lifetime AnalysisHex-Grid

Hex-Grid Placement AnalysisBlock : RBD for Hex-Grid

a

b c d

a

b

c d

)s - s)(1-(1 - 1 s 3block

)sss - )(1s-(1 - 1 s dcbablock

∵ sensors are identical, have the same survivor function

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Network Lifetime AnalysisNetwork Survivor Function for Hex-Grid

blocks connect in series.2

N

2

N

blocknetwork )(s s Why ?

2

N

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SimulationFlow Chart

Survivor Function (single node)

Survivor Function (network)

p.d.f. (network)

p.d.f. (single node)

theoretical vs. actual

Network Lifetime AnalysisNode Lifetime Analysis

Given Network Protocol

Distribution of Wi

Node Lifetime Calculation

p.d.f. (single node)

theoretical vs. actual

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SimulationNode Lifetime Distribution

theoretical p.d.f. actual p.d.f.

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SimulationNetwork Lifetime Distribution

Square Grid Placement Scheme

theoretical p.d.f. actual p.d.f.

closely match!

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SimulationNetwork Lifetime Distribution

Hex-Grid Placement Scheme

theoretical p.d.f. actual p.d.f.

closely match!

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Conclusion

The analytical results based on the application of Reliability Theory.

We came up not with any particular value, but a p.d.f. for minimum network lifetime.

The theoretical results and the methodology used will enable analysis of: other sensor placement scheme tradeoff between lifetime and cost performance of energy efficiency algorithm