trade-offs between mobility and density for coverage in wireless sensor networks wei wang, vikram...
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Trade-offs Between Mobility and Density for Coverage in
Wireless Sensor Networks
Wei Wang, Vikram Srinivasan and Kee-Chaing Chua
National University of Singapore
2007 Mobicom
Outline
Introduction Coverage with mobile sensors Coverage of hybrid networks Mobility algorithm Numerical results Conclusion
Introduction
Coverage problem Important research problem in WSNs k-covered Network Deployment Mobility
Introduction- deployment
Metric: over-provisioning factor Indicates the efficiency of a network deployment s
trategy Consider a random deployment strategy
What is the sensor density to guarantee k-coverage?
Introduction- mobility
Mobile sensors can relocate themselves to heal coverage holes Over-provisioning factor for a network with all
mobile sensors can be Θ(1) Consumes more energy
Mobile sensors Limited mobility: move once, over a short distance
Maximum distance?
Coverage with mobile sensors
Sensing field: L=l*l Num. of static sensors: N = λL
Uniformly and independently scattered in the network. Number of static sensors in a region with area of A:
nA
Sensing range: r = 1 /√π 1=πr2 1
Density
Over-Provisioning Factor
Optimal over-provisioning factor:Θ(1) ds= √2r
Density of mobile sensor
K-coverage
r = 1 /√π
Over-Provisioning Factor
Randomly deployed static sensor networks Density λ
Total expected area which is uncovered is e−λL. Random coverage processes Large enough λ, e−λ can be made arbitrarily small
Probability approaches one for a network with constant sensor density λ when the network size L→∞. Exist a connected coverage hole larger than unit area
Over-Provisioning Factor
To achieve k-coverage in a large network, the static sensor density needs to grow with the network size λ = logL +(k + 2) log log L + c(L)
c(L) → +∞ as L → +∞
All Mobile Networks
ηm = Θ(1). key question
what is the maximum distance that each sensor has to move?
Limit the maximum moving distance for each mobile
All Mobile Networks
Theorem1: Network can provide k-coverage with an over-provisioning factor of ηm= π/ 2 and the maximum distancemaximum distance moved by any mobile sensor is O( 1 √klog3/4(kL)) w.h.p.
All Mobile Networks
Sensing field into square grids with side length of da =√2r/√k Number of nodes in the sensing range
πr2/(√2r/√k)2=πk/2
ηm=(πk/2) / k = π/2
All Mobile Networks
By the lower bounds on lattice points covered by a circle, there are at least W(k) lattice points of side length of da covered by a circle of radius r
da =√2r/√k Increasing function
All Mobile Networks
W(k) > k when k ≥ 25 ->k coverage W(k)=25.13274
Network is at least k-covered when 1 ≤ k < 25.
All Mobile Networks
l × l square, L = l2 points in the region there exists a perfect match between the L rando
m points and the L grid points with maximum distance between any matched pairs of O(log3/4 L).
Grid points (k/2r2)*L O(log3/4 (kL))
Grid size is da =√2r /√k O( 1/√k log3/4(kL))
1=πr2
1/r2= πηm =Densty/kDensty= ηm*k= πk/2=k/2r2
Coverage of hybrid networks
Over-provisioning factor is O(1) Fraction of mobile sensors required is less th
an 1 /√2πk Maximum distance that any mobile sensor wil
l have to move is O(log3/4L)
Density of Mobile Sensors
Static sensor density at λ =2πk. Divide the network into square cells
equal side length of dh = r/√2.
Average number of static sensors in each cell will be 2πkd2
h = k.
Density of Mobile Sensors
The network will be k-covered if all cells contain at least k sensors. cell i has vi = k−ni vacancies, If a cell i contains ni
< k static sensors
Poisson approximation
Density of Mobile Sensors
The random variable vi = [k − ni]+ , will be distributed as:
The expected number of vacancies in a cell will be:
Density of Mobile Sensors
Using Stirling’s approximation
Density of mobile sensor
Density of Static sensor
Fraction of mobile sensors required is less than
r = 1 /√πdh = r/√2.
Maximum distance for mobiles
A grid with side length of 1/ √Λ Maximum distance
Decreasing function Matching distance
Mobility Algorithm
Problem Formulation Movement cost
Initial number of mobile sensor
Number of mobile sensor from cell i to cell j
Distribution Solution
A distributed algorithm Maximum flow problem
Assume Sensor knows
Its location Which cell it is located in. vi and mi
Each cell elects a mobile or static sensor as the delegate Communicate and exchange information with its neighbors in
graph G
Distribution Solution-push-relabel algorithm
a
b c
i o
o i o i
Cell a
Cell a Cell c
Distance D v-m=3
v-m=-2 v-m=-1
Distribution Solution-push-relabel algorithm a
b c
i o
o i o i
Cell a
Cell a Cell c
h(i)=0e(i)=0
h(i) =0e(i) =0
h(i) =0e(i) =0
h(o)=0e(o)=3
h(o) =0e(o) =-2
h(o)=0e(o) =-1
Zero cost
ci
v-m=3
v-m=-2 v-m=-1
Distribution Solution-push-relabel algorithm a
b c
i o
o i o i
Cell a
Cell a Cell c
h(i)=0e(i)=0
h(i) =0e(i) =0
h(i) =0e(i) =0
h(o)=0e(o)=3
h(o) =0e(o) =-2
h(o)=0e(o) =-1
v-m=3
v-m=-2 v-m=-1h(o)=1e(o)=3
Distribution Solution-push-relabel algorithm a
b c
i o
o i o i
Cell a
Cell a Cell c
h(i)=0e(i)=0
h(i) =0e(i) =0
h(i) =0e(i) =1
h(o) =0e(o) =-2
h(o)=0e(o) =-1
v-m=3
v-m=-2 v-m=-1h(o)=1e(o)=2h(o)=1e(o)=1
h(i) =0e(i) =1
Distribution Solution-push-relabel algorithm a
b c
i o
o i o i
Cell a
Cell a Cell c
h(i)=0e(i)=0
h(i) =0e(i) =0
h(o) =0e(o) =-1
h(o)=0e(o) =1
v-m=3
v-m=-2 v-m=-1h(o)=1e(o)=1
h(i) =0e(i) =0
Distribution Solution-push-relabel algorithm a
b c
i o
o i o i
Cell a
Cell a Cell c
h(i)=0e(i)=0
h(i) =0e(i) =0
h(o) =0e(o) =-1
h(o)=0e(o) =1
v-m=3
v-m=-2 v-m=-1h(o)=1e(o)=1
h(i) =0e(i) =1
Numerical results
Mobile Sensor Networks only consider the maximum matching distance for
1-coverage in our simulations M = ΛL mobiles
Λ=π/2 ds= √2 r 105 randomly generated topologies
Probability that no feasible matching exists for a given maximum moving distance D.
ds
Numerical results
Hybrid Networks Cells with side length of dh = r/√2 N = λL static sensors , λ = 2πk M = ΛL mobiles
M is selected so that there are exactly enough mobiles to fill all vacancies
Moving distance D
k=10
dh=0.5 ds
Cells=900
Performance of Push-Relabel Algorithm
Execution process is divided into rounds 103 randomly generated topologies
Total number of messagesRounds
Conclusion
Investigate the distance that a mobile sensor will have to move Mobile sensor networks Hybrid sensor networks
Results prove that Mobility has significant advantages in providing
coverage