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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