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