1 a simple asymptotically optimal energy allocation and routing scheme in rechargeable sensor...
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A Simple Asymptotically Optimal Energy Allocation and Routing Scheme in Rechargeable Sensor Networks
Shengbo Chen, Prasun Sinha, Ness Shroff, Changhee JooElectrical and Computer Engineering & Computer Science and Engineering
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Introduction[Rechargeable sensor networks]
Applications Environment monitoring: earthquake, structural, soil, glacial
Unattended operability for long periods
Opportunity Harvesting and storing renewable energy (like solar or wind)
Challenges Full battery means no opportunity to harvest renewable energy Unpredictable and time-varying renewable energy
Goal: develop control mechanism to maximize the total utility for a sensor network with energy replenishment
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Model[Rechargeable sensor node]
M
( 1) min max ( ) ( ),0 ( ),B t B t e t r t M
r(t) B(t) e(t)B(t+1)
M: Battery size
B(t): Battery level in time slot t
e(t): allocated energy in time slot t
r(t): harvested energy in time slot t
Rechargeable sensor node
Rate-power functionNondecreasing and strictly concaveAmount of data transmitted with spending units of energy
( )e
e e
( )e
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*
1
1( ) max ( )
s.t. Routing constraints
Energy constraints
Ts
ss t
J T U x tT
Problem Statement
Sensor network with renewable energy Assume the date rate is low
Ignore interference from other nodes
Problem: utility maximization
amount of data from source to destination in slot t
is a strictly concave utility function
( )sx t
1. Convex optimization problem: Joint energy allocation and routing
2. Requires full knowledge of the replenishment profile
3. Time coupling property: have to optimize all time slots simultaneously
Flow 1
Flow 2
, ,
:( , ) :( , ) : ,
:( , )
*
1
1( ) max ( )
s.t. ( ) ( ) ( ) 0 for all ,
( ) ( ( )) for all ,
e x
j i j L j i j L s f i d ds s
j i j L
Ts
ss t
d d sij ji
t t t
dij i
d
J T U x tT
t t x t d i
t e t t i
666666666666666666666666666666666666666666
sU
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Related Literature Finite horizon
[S. Chen, P. Sinha and N. B. Shroff], INFOCOM, 2011. [A. Fu, E. Modiano and J. Tsitsiklis], TON, 2003.
Dynamic programming Infinite horizon
[L. Lin, N. B. Shroff, and R. Srikant], TON, 2007. Asymptotically optimal competitive ratio
[Z. Mao, C. E. Koksal, N. B. Shroff ], TAC, 2011 Finite battery size
[M. Gatzianas, L. Georgiadis, and L. Tassiulas], TWC, 2010. Maximize a function of the long-term rate per link
[L. Huang, M. Neely], Mobihoc, 2011 Asymptotically optimal utility
Our focus: Infinite horizon
Lyapunov optimizati
on technique
Our contribution: develop a low-complexity solution
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Our approach Construct a fictitious infeasible energy allocation and routing
scheme
Prove that its performance forms an upper bound on
Develop a low-complexity online scheme
Prove that the performance achieved by the online scheme can get arbitrarily close to the upper bound as tends to infinity
*( )J T
T
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Assumption Replenishment process has a finite mean value Infinite battery capacity
Upper bound for the optimum Jensen’s Inequality: is an upper bound
1
1max ( ( ))
T
et
e tT
1
1lim ( )
T
Tt
r r tT
( )r
Single node case [Throughput maximization]
Spending energy at the average rate is the best
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Single node case (cont’d) [Throughput maximization]
Consider the energy allocation scheme In each time slot, the estimated average replenishment rate
The allocated energy in each slot
where is an arbitrary parameter
1
1ˆ( ) ( )
t
r t rt
ˆ ˆ(1 ) ( ), ( ) ( ) (1 ) ( ),( )
( ) ( ),
r t if B t r t r te t
B t r t Otherwise
Theorem 1: The scheme above achieves the throughput performance arbitrarily close to by choosing
to be sufficiently small as tends to infinity
( )rT
ˆ( ) tr t r
Intuition: spend energy at a rate close to the mean
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Upper bound on the optimum Consider a fictitious infeasible scheme
For each node i, energy allocation in each slot
Routing decision in each slot
*( )J T
( ) (1 )i ie t r
:( , ) :( , ) : ,
:( , )
( ) argmax ( )
s.t. ( ) ( ) ( ) 0 for all ,
( ) (1 ) for all
j i j L j i j L s f i d ds s
j i j L
s sub s
s
d d sij ji
dij i
d
x t U x t
t t x t d i
t r i
1. Energy allocation and routing decoupled
2. Time decoupled
3. Time homogeneous
Theorem 2: is upper bounded by *( )J T ( ) ( )ub ss ubcs
J T U x
( )s sub ubcx t x
Network Case[fictitious scheme]
Spend a little more energy than the
average harvested
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Consider the online scheme Energy allocation (same as the single node case)
The estimated average replenishment rate
The allocated energy in each slot
Routing decision in each slot
:( , ) :( , ) : ,
:( , )
max ( )
s.t. ( ) ( ) ( ) 0 for all ,
( ) ( ) for all
j i j L j i j L s f i d ds s
j i j L
ss
s
d d sij ji
dij i
d
U x t
t t x t d i
t e t i
1
1ˆ ( ) ( )
t
i ir t rt
ˆ ˆ(1 ) ( ), ( ) ( ) (1 ) ( ),( )
( ) ( ),i i i i
ii i
r t if B t r t r te t
B t r t Otherwise
Theorem 3: The scheme achieves the performance
arbitrarily close to
by choosing to be sufficiently
small as tends to infinity
( )ubJ T
Network Case (cont’d)[Online scheme]
T
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Distributed algorithm Duality based
At each time slot, source s generates data at rate by solving
Routing
Lagrange multipliers are updated as
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Simulation Setup Network topology:
100 nodes and three flows in 1×1 field Link available if distance is less than 0.2
Using real traces of solar energy and wind energy [3] June 5th, 2011-July 5th, 2011
[3]. “National Renewable Energy Laboratory,” http://www.nrel.gov.
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Simulation results
ESA: Infinite-horizon based scheme in [1][1] L. Huang, M. Neely, “Utility Optimal Scheduling in Energy Harvesting Networks,” in Proceedings of Mobihoc, May 2011.
minute minute
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Conclusion Study the joint problem of energy allocation and
routing to maximize total utility in a sensor network with energy replenishment.
Develop a low-complexity online solution that is asymptotically optimal with general energy replenishment profiles.
Evaluate the performance using real traces
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Simulation results for one node
ESA: Infinite-horizon based scheme in [1][1] L. Huang, M. Neely, “Utility Optimal Scheduling in Energy Harvesting Networks,” in Proceedings of Mobihoc, May 2011.