Arindam K. DasCIA Lab

University of Washington

Seattle, WA

LIFETIME MAXIMIZATION IN ENERGY CONSTRAINED WIRELESS NETWORKS

with Robert J. Marks II &

M.A. El-Sharkawi (UW CIA)

Payman Arabshahi & Andrew Gray (JPL/NASA)

LIFETIME MAXIMIZATION IN ENERGY CONSTRAINED WIRELESS NETWORKS

Example Static Multicast

Assumptions (1)

• We assume that there is a fixed source node which wants to communicate with some/all (multicast/broadcast) the other nodes in the wireless network.

• The source uses a fixed connection tree for a certain multicast group.

• All nodes have omni-directional antennas.• Each node in the network is provided with a

finite amount of battery energy which is used for signal transmission.

The Problem

• For a continuous packet transmission process at a constant bit rate and a stationary channel with no bandwidth constraints, we ask the question: “For a given multicast group, how long can the source continue to use the connection tree before any node in the system runs out of battery power”?

• We define the system lifetime as the time from t = 0 to the instant at which the first node in the system runs out of battery power.

Assumptions (2)

• Power is expended for signal transmission only. No power expenditure for signal reception or processing.

• The transmitter power is modeled as the ‘’ power of its distance from the receiver (2 4).

rPT

The Problem

• Define the initial node energy vector, E(0), as an N-element vector which specifies the initial battery energy at each node.

• The battery lifetime of node i for a particular choice of the connection tree is defined as:

i

ii

nodeoflevelpower

nodeofenergyinitialnodeoflifetimebattery

• The static case system lifetime is the minimum of all elements in the battery lifetime vector.

)say()0(

min

nodeoflevelpower

nodeofenergyinitialminLifetimeSystem )(

si

ii

istatic

Y

E

i

i

The Problem

The Problem

Objective of the static energy constrained multicasting (SECM) problem

For a given multicast group, find the connection tree which maximizes the system lifetime.

The Problem

maximize (system lifetime)

= maximize {min (battery lifetime vector)}

s

s

i

ii

i

i

E

Y

i

i

i

i

minimize1

minimize

)0(maxminimize

nodeofenergyinitial

nodeoflevelpowermaxminimize

nodeoflevelpower

nodeofenergyinitialminaximizem

Proposed Approach

• We propose a GA based approach for solving the minimax optimization problem.

• Key question: Encoding of chromosomes

Some Definitions

• Power matrix, P: The (i,j)th element of the power matrix is defined as

where rij is the Euclidean distance between nodes i and j.

• Rank matrix, R: The rank matrix is obtained by ranking each row of the power matrix from smallest to largest.

Pij = rij

Examples

Some Definitions

• Cut vector, R: The cut vector, referenced to R, is an N-element integer vector, where N is the number of nodes in the network. It indicates the location of an element on each row of the rank matrix.

• Threshold vector, t : An N-element vector of the elements of R specified by the cut vector R. Elements of the threshold vector represent power settings of the individual nodes.

Examples

R = [3 4 5 1 2], t = [8.01 14.06 16.73 0 9.55]

Some Definitions

• Viability of a cut vector: A cut is viable if it allows all destination nodes to be reached. Otherwise, it is non-viable. A viable cut vector has an associated connection tree.

Outline of the Proposed Approach

• GA based

• Chromosome encoding : cut vectors

• Crossover : standard (e.g., random 1-point crossover) , subject to a certain crossover probability.

• Parent selection : standard (e.g., roulette wheel)

Outline of the Proposed Approach

• Fitness function : s = maxi [ti / Ei(0)]

• Mutation : 1, on randomly chosen elements of the chromosomes, subject to a certain mutation probability.

• Elitism : yes

Viability of the Children

• Randomly generated cut vectors need not be viable the children created after crossover and mutation need not correspond to viable connection trees.

• Use the computationally simple Viability Lemma to determine the viability of a child.

- If viable, accept it.

- If not, reject it, or, apply a repair operator.

Viability of the ChildrenA Repair Strategy

• Suppose a node (say n) is not reached by a cut vector, R.

• Compute the connection tree corresponding to R.

• Identify the node closest to n (say m).• Augment the power level of node m so that node

n is reached.• Recompute the cut vector corresponding to the

modified connection tree.

Example Static Multicast Search

The Dynamic Energy Constrained Broadcast (DECB) Problem

• In the Static Energy Constrained Broadcast (SECB) problem, we assumed that the source uses only one connection tree.

• In the DECB problem, we assume that the source uses a set of {: 1 } connection trees, each for a certain duration of time, (called the duty cycle of tree ), such that:

1θθ 1π

The Dynamic Energy Constrained Broadcast (DECB) Problem

• Clearly, the system lifetime in this case is a function of the trees used in the dictionary set and their corresponding duty cycles.

• The best trees need not be the best trees.

For a given multicast group, find the connection trees and the set of duty cycles { : 1 } which maximize the system lifetime.

DECB Optimization Function

• Proposed optimization approach : Team optimization of co-operating systems (TOCS)

1θ

θθ τ/π/1minaximizem

τaximizem

lifetimesystemaximizem

ii

The TOCS Approach

Summary

• Presented optimization models for static and dynamic energy constrained broadcast / multicast problems.

• Outlined a GA based approach for solving the static problem.

• Proposed a TOCS approach for solving the dynamic problem.