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
1
Outline






Introduction
Preliminaries
Node Lifetime Evaluation
Network Lifetime Analysis Using
Reliability Theory
Simulation
Conclusion
2
Introduction (1/3)

Sensor networks have limited
network lifetime.


Most applications have pre-specified
lifetime requirement.


energy-constrained
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”
3
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
4
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)
5
Preliminaries
Basic Model



rs : the sensing range assume rs = rc
rc : the communication range
neighbors

distance of separation r ≤ rc
r
rs
6
Preliminaries
Basic 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 is r = rmax
7
Preliminaries
Placement Schemes
Placement Schemes
2-neighbor group
3-neighbor group
4-neighbor group
described in [1]
Hex-Grid
Square Grid
[1] K. Kar, S. Banerjee, ”Node Placement for Connected Coverage in
Sensor Networks”
8
Preliminaries
Placement Scheme in Reference [1]
2-neighbor group
and provides full
coverage!!
[1] K. Kar, S. Banerjee, ”Node Placement for Connected Coverage in
Sensor Networks”
9
Preliminaries
Placement Schemes

Square Grid

Hex-Grid
10
Preliminaries
Coverage 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
11
Preliminaries
Lifetime

consider basic placement schemes
Square- Grid
Hex- Grid
12
Preliminaries
Lifetime


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
13
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-th mode
w
i
i
1
i  1,2...m
1
2
……
m
w1
w2
……
wm
14
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
E total -  w i Pi Tnode  E th
i
E total
Tnode 
 w i Pi
i
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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
density function (pdf), ft (t)
Tnode 
Etotal
i wi Pi
16
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
P(w 1  x)  C Tx p x (1 - p) T-x
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Node Lifetime Evaluation (5/5)

As T becomes large:
2




binomial distribution ~ N(μ, σ)
2
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.
18
Network Lifetime Analysis
Reliability Theory


The network lifetime is also a random
variable.
Using Reliability Theory to find the
distribution of the network lifetime.
19
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
Reliability, RBD.
20
Reliability Theory
pdf and cdf

Probability Density Function


f(t): the probability of the random variable
taking a certain value
Cumulative Distribution Function

F(t): the proportion of the entire
population that fails by time t.
t
F(t)   f(s)ds
0
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Reliability Theory
Survivor Function

Survivor Function: S(t)

the probability that a unit is functioning
at any time t
S(0) = 1,
S(t)  P [T  t]
t0
lim t  S(t)  0,
S(t) is non-decreasing

survivor function vs. pdf
t
S(t)  1 - F(t)  1 -  f(s)ds
0
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Reliability Theory
System Reliability
distribution of the components
distribution of the system

single node
entire network
To consider the relationship between
components in the system.

using RBD
23
Reliability Theory
Reliability 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.
S1(t)
S1(t)
S2(t)
S2(t)
Sseries (t)  S1 (t)S 2 (t)
S parallel (t)  1 - [(1 - S1 (t))(1 - S 2 (t))]
24
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.
25
Network Lifetime Analysis
Square Grid

Square Grid Placement Analysis
Region 1
Region 1
a
b
c
d
Region 2
x
y
or
x
y
Region 2
26
Network Lifetime Analysis
Square Grid
Region 1
a
b
c
d
Block 1 : RBD for Region 1
a
b
c
s block1  1 - (1 - s a )(1 - s bs c )
∵ sensors are identical
s block1  1 - (1 - s)(1 - s 2 )  s  s 2 - s 3
27
Network Lifetime Analysis
Square Grid
Region 2
x
y
Block 2 : RBD for Region 2
x
y
x
or
y
s block2  1 - (1 - s x )(1 - s y )
∵ sensors are identical, have the same survivor function
s block2  1 - (1 - s)(1 - s)  2s - s 2
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Network Lifetime Analysis
Network Survivor Function for Square Grid
N min - 1

( N min - 1 ) 2 block 1’s
block 2’s
connect in series
 2 * ( N min - 1 )

N min - 1
s network  (s block1 )
( N min - 1)2
(s block 2 )
2( N min - 1)
29
Network Lifetime Analysis
Hex-Grid

Hex-Grid Placement Analysis
Block : RBD for Hex-Grid
b
a
a
c
d
b
c
d
s block  1 - (1 - s a )(1 - s bs cs d )
∵ sensors are identical, have the same survivor function
s block  1 - (1 - s)(1 - s 3 )
30
Network Lifetime Analysis
Network Survivor Function for Hex-Grid
N

2
blocks connect in series.
s network  (s block )
N
2
N
Why 2 ?
31
Simulation
Flow Chart
Node Lifetime Analysis
Network Lifetime Analysis
Given Network Protocol
p.d.f. (single node)
Distribution of Wi
Survivor Function (single node)
Node Lifetime Calculation
Survivor Function (network)
p.d.f. (single node)
p.d.f. (network)
theoretical vs. actual
theoretical vs. actual
32
Simulation
Node Lifetime Distribution
theoretical p.d.f.
actual p.d.f.
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Simulation
Network Lifetime Distribution

Square Grid Placement Scheme
theoretical p.d.f.
actual p.d.f.
closely match!
34
Simulation
Network Lifetime Distribution

Hex-Grid Placement Scheme
theoretical p.d.f.
actual p.d.f.
closely match!
35
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
36