theoretical results on base station movement problem for sensor network yi shi ( 石毅 ) and y....
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Theoretical Results on Base Station Movement Problem for Sensor Network
Yi Shi ( 石毅 ) and Y. Thomas Hou ( 侯一釗 )Virginia Tech, Dept. of ECE
IEEE Infocom 2008
Outline Introduction Problem
Constrained Mobile Base Station (C-MB) Problem Unconstrained Mobile Base Station (U-MB) Problem
Approach C-MB Problem
Optimal Solution U-MB Problem
(1-)-approximate solution Numerical Results Conclusion
Introduction Sensor Networks
Sensors: gather data, transmit and relay data packets
Low computation power Battery power Small storage space
Base station: data collector Network Lifetime
The first time instance when any of the sensors runs out of energy.
The first time instance when half of the sensors runs out of energy.
The first time instance when the network connectivity is broken up.
BS
VCLAB ezLMS references: [1] [2]
Problem Network Model
The BS is movable Each sensor node i generates data at rate ri Data is transmitted to base station via multi-hop Initial energy at sensor node i is ei
Energy Consumption Modeling
Transmission power modeling
where
Receiving power modeling
i jdij: distance
fij: data rate
ifki: data rate
Unconstrained Mobile Base Station Problem (U-MB)
Goal: Find an optimal moving path for the base station such t
hat the network lifetime is maximized. Optimize base station location (x, y)(t) at any time t such that the net
work lifetime is maximized.
Problem Formulation
flow balance:
i
gki
incoming data rate
ri
outgoing data rategij
Energy constraint
The position of the BS
Time-dependent
Network lifetime
Constrained Mobile Base Station Problem (U-MB) The base station is only allowed to be prese
nt at a finite set of pre-defined points. For example: (x, y)(t) p1, p2, p3, p4, p5 }
p1
p2
p3
p4
p5
Goal:
Find an optimal time-dependent location sequences such that the network lifetime is maximized.
Time-dependent location sequences:
t1
p1
t2
p4
t3
p3
t4
p2
t
p1
……
The Roadmap of the Theoretical Analysis
C-MB Problem1. Transform the problem from time domain to space d
omain Theorem 1
2. Linear programming Optimal Solution
U-MB Problem Change infinite search space to finite search points
U-MB C-MB (1-)-approximate solution by solving C-MB on the fin
ite search points Theorem 2 and 3
t1
p1
t2
p4
t3
p3
t4
p2
……
p1
t1
p2
t2
p3
t3
p4
t4
……
C-MB ProblemFrom time-domain to space-domain
Time Domain: [0, 50]
p1
[50, 90]
p2
[90, 100]
p2
[100, 130]
p1
C-MB ProblemFrom time-domain to space-domain Space domain:
p1
[0, 50] + [100, 130]
p2
[50, 100]
From Time Domain to Space Domain (cont’d) Data routing only depends on base
station location; not time
TheoremThe optimal location-dependent solution can achieve the same maximum network lifetime as that by the optimal time-dependent solution
Linear programming Formulation
Location-dependent
W(p): the cumulative time periods for the BS to be present at location p.
fki(p): normalized data rate
Linear programming FormulationLet
A (1 − ε) Optimal Algorithm to the U-MB ProblemSearch Space
Claim: Optimal base station movement must be within the Smallest Enclosing Disk (SED).
Reference [19]
SED The smallest disk that
covers all sensor nodes Can be found in polynomial-
time
Still infinite search space!
A (1 − ε) Optimal Algorithm Roadmap
1. Discretize transmission cost and distance with (1-Ɛ) optimality guarantee Get a set of distance D[h]
2. Divide SED into subareas By the sequence of circles with radius D[h] at
each sensor3. Represent each subarea by a fictitious cost
point (FCP)4. Compute the optimal total sojourn time and
routing topology for each FCP (or subarea) A linear program
Step 1: Discretize Transmission Cost and Distance
Step 1: Discretize Transmission Cost and Distance
Discretize transmission cost in a geometric sequence, with a factor of (1+Ɛ)
C[1]
C[2]C[3]
C[1] c4B C[2]
c4B: the transmission cost between sensor I and the base station
Step 2: Division on SED SED is divided by the sequence of circles with
radius D[h] and center sensor node i
C[1] c1B C[2]
C[2] c2B C[3]
C[1] c3B C[2]
C[2] c4B C[3]
Step 3: Represent Each Subarea by A Fictitious Cost Point (FCP)
Define a FCP pm for each subarea Am
N-tuple cost vector Pm = (C[2], C[3], C[2], C[3]) Define
C[1] c1B C[2]
C[2] c2B C[3]
C[1] c3B C[2]
C[2] c4B C[3]
Pm
Step 3: Represent Each Subarea by A Fictitious Cost Point (FCP)
Properties: A fictitious point pm is a virtual point, not a physic
al point in the space.
The transmission cost from each sensor node i to pm is the worst case cost for all points in Am
For any point p in this subarea, we have CiB(p)≤CiB (pm)
For any point pAm, we have
Step 4: Finding a (1-Ɛ) Optimal Solution
Find the best total sojourn time W(pm) and routing topology fij(pm) and fiB(pm) for each FCP pm Solve a linear program
(Linear programming for the C-MB problem)
Base station should stay at each subarea Am for total W(pm) of time
Whenever base station is in subarea Am, routing topology should be fij(pm) and fiB(pm)
The (1-) Optimality
(1)
By Theorem 2
(2)
By Theorem 3
Example
=0.2, OA=(0.61, 0.57), RA=0.51 =1, =0.5, =1 17 subareas A1, A2,…,A17
1
2
3
Final solution: T =190.37
Stay A3 for 157 time
Stay A6 for 33.37 time
Numerical Results Settings
Randomly generated networks: 50 and 100 nodes in 1x1 area (all units are normalized)
Data rate at each sensor randomly generated in [0.1, 1]
Initial energy at each sensor randomly generated in [50, 500]
Parameters in energy consumption model: α=β=ρ=1, n=2
Result The obtained network lifetime is at least 95% of the
optimum, i.e., Ɛ is set to 0.05
Result – 50-node Network
Tε=122.30
A Sample Base Station Movement Path
Base station movement path is not unique
Moving time (from one subarea to another) is much smaller than network lifetime
Each sensor can buffer its data when base station is moving and transmit when base station arrives the next subarea
Network lifetime will not change
Result – 100-node Network
Summary Investigated base station movement
problem for sensor networks Developed a (1-Ɛ) approximation
algorithm with polynomial complexity Transform the problem from time domain to
space domain Change infinite search space to finite search
space with (1-Ɛ) optimal guarantee Proved (1-Ɛ) optimality
Proof of Theorem 1
Time domain
Space domain
C-MBC-MB
Proof of Theorem 1
Time domain
Space domain
C-MBC-MB
Indicator function
Proof of Theorem 1
Proof of Lemma 1
(1)
(2)
(3)
Proof of Lemma 1(1)
Proof of Lemma 1 (2)
Proof of Lemma 1(3)
Proof of Theorem 2
Proof of Theorem 2
C-MBU-MB
Proof of Theorem 3
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