Maximizing Lifetime of Sensor
Surveillance Systems
IEEE/ACM TRANSACTIONS ON NETWORKING
Authors: Hai Liu, Xiaohua Jia, Peng-Jun Wan, ChihWei Yi, S. Kami Makki, and Niki Pissinou
1
Outline
• Introductions
• System Model and Problem Statement
• Our Solutions
– Find Maximal Lifetime
– Decompose Workload Matrix
– Determine Surveillance Tree
• Experiments and Simulations
• Conclusion
2
Introductions
• Given a set of targets, a set of sensors (at most
watch one target at a time) are used to watch
the targets and collect sensed data to the BS.
• Lifetime
– Duration until one target can no longer be watched by
any sensor or data can’t be forward to the BS.
• Problems
– Schedule a subnet of sensors
– Find the routes for the active sensor to send data
back to BS
3
Introductions
BS
sensor
target
4
System Model and Problem
Statement
•
•
•
•
•
B
S
T
S(j)
T(i)
• N(i)
•
•
•
•
• R
targets
base station
set of sensors, n =
set of targets, m =
set of sensors that can watch target j
set of targets that are within the surveillance range
of sensor i
set of neighbors of sensor i
initial energy of sensor i
distance between sensor i and j
energy for transmitting and receiving one unit data
energy for watching a target per unit time
data rate generated from sensors while watching
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System Model and Problem
Statement
• S(1) = {S1, S2, S3}
s2
s1
T1
s3
• T(1) = {T1, T2, T3}
T1
T2
s1
T3
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System Model and Problem
Statement
• Two requirements for sensors watching
targets
– Each sensor can watch at most one target at a
time
– Each target should be watched by one sensor
at any time
• The problem is to find a schedule that meets
the above two requirements for sensors
watching targets, such that the lifetime of
network is maximized.
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Our Solutions
Find Maximal Lifetime
• Linear Programming (LP)
–
–
total time sensor i watching target j
amount of data transmitted from sensor i to sensor j
8
Our Solutions
Find Maximal Lifetime
• We call matrix
the workload matrix
– The sum of column is equal to L (each column)
– The sum of row is less than or equal to L (each
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row)
Our Solutions
Decompose Workload Matrix
•
are schedule matrices
– Elements are either “0” or
– Each column has exactly one non-zero element
– Each row has at most one non-zero element
• The number of sensors is grater than or equal the
number of targets ( n >= m)
– n=m
– n>m
10
Our Solutions
Special Case n = m
•
and
matrix
∵
denotes the sum of row I and the sum of column j in workload
and
∴
=>
=>
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Our Solutions
Special Case n = m
• Bipartite graph
– Left hand side : sensors
– Right hand side : targets
– Edges :
T1
S2
T2
……….
……….
• Since n = m, every sensor has a
target to watch in each session
• Find perfect matchings…
S1
Sn
Tm
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Our Solutions
Special Case n = m
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Our Solutions
Special Case n = m
G
2
1
1 1
s2
1
1
s3
2
21 0
1 1 1 



0 1 2

s1
t1
t2
t3
1 0 0 

P1 0 0 1 



0 1 0 
s1
1
t1
s2
1
1
t2
1 1 0 


1
1
0




0 0 2 
s1
G
s3
s2
s3
t3
1
1
1
1
t1
t2
2
t3
……
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Our Solutions
Special Case n = m
• Lema1: For
= for 1 i,j
• Proof:
of nonnegative real numbers, if
n, =>exist perfect matching
–
(L is sum of all elements in a row)
– There doesn’t exist n positive entries in
that no two
entries in the same column or row.
– By Konig theorem, we can cover all the positive entries in
the matrix with e rows and f columns, such that e+f<n
– The sum of all lines of
is equal to 1, n e+f<n,
=><=
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Our Solutions
Special Case n = m
• Theorem 1: The DecomposeMatrix-nn
algorithm can always find a perfect matching
– From lemma 1
• Theorem 2: The time complexity of
DecomposeMatrix-nn algorithm is O(
),
where is the number of non-zero elements
in
– At most
number of rounds to remove all
edges in G
– Find a perfect matching is O( )
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Our Solutions
Special Case n > m
• Let
be the dummy matrix
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Our Solutions
Special Case n > m
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Our Solutions
Special Case n > m
=3,
=5
=
=5,
=3,
Let
and
and column j
=2
=3,
=3
=3
=2
=3
record the sum of remaining undetermined elements of row i
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Our Solutions
Special Case n > m
• Theorem 3: the FillMatrix Algorithm can compute
the filled matrix
– Given row sums
and column sums
– Induction method
– 1. n=1, m=1, since
, we have
of a matrix
– 2. when n p-1, m q-1, can compute
– 3. when n=p, m=q, we first compare
with
• A.
=
, =>
, according 2.
• B.
>
, =>
,
monotonously decreases after
each round and
, there must exist
in round l , =>
, according 2.
• C similar to B.
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Our Solutions
Special Case n > m
• Let
denotes the matrix contains the first m
columns in
21
Our Solutions
Special Case n > m
• Theorem 4: The time complexity of
FillMatrix Algorithm is O( )
• Theorem 5: The time complexity of
DecomposeMatrix algorithm is O(
)
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Our Solutions
Determine Surveillance Tree
• Root : BS
• Leaf nodes : active sensors
• Suppose sensor i has l downstream nodes
(i.e.
are non-zero), let sensor i
forward its outgoing data first to until
is
saturated, then switch to until the value of
is met, and finally forward the last flow
to .
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Experiments and Simulations
Numeric Example
• 10x10 region
• Surveillance range: 0.4 *
10
• Maximal transmission
range: 0.8*10
• Initial energy is randomly
generated form [0, 100]
with mean at 50
•
=0.12,
=0.1
•
=0.1 R=1
• α=2
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Experiments and Simulations
Numeric Example
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Experiments and Simulations
Numeric Example
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Experiments and Simulations
Simulations
• 1) Linear Growth of Decomposition Steps
–
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Experiments and Simulations
Simulations
• 2) Comparison with Greedy Method:
– Use the maximum matching algorithm in the
sensor-target bipartite graph to find the pairs
of sensor and target
– For each active sensor we find the minimal
energy cost path from it to the BS.
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Experiments and Simulations
Simulations
N=100, M=10
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Experiments and Simulations
Simulations
M=50
•
•
The number of steps for decomposing the workload matrix is linear to the size of the
system
Our algorithm has better performance when
–
–
–
•
Large surveillance range
Large transmission range
Sensors are density deployed
The increase of surveillance range is more effective than the increase of the maximal
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transmission range
Conclusion
• We have presented the maximal lifetime
scheduling problem in sensor surveillance
systems.
• This is the first time in the literature that
the problem of maximizing lifetime of
sensor surveillance systems was
formulated and the optimal solution was
obtained.
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