Topology Control –power control

Outlineintroduction History ReviewK-neighbor graph

Power control

Adjust transmission power of nodes such that the resulting network is connected and energy consumption is optimized

Motivation

Limited energy in wireless network

Energy can be saved if the topology itself is energy efficient

Power saving

Physical layer MAC layer Network layer

Power control Awake-sleep Routing

History review

Energy ModelMetrics Main Methods

Energy Model

Omni-directional antennas + Uniform power detection thresholds(t) Signal power falls inversely proportional to dk

1<K<5

P=t* dk P=t

Observation 1

Transmission through small hops is more power efficient than through big hops.

d1+d2+d3

d1 d2 d3

Interference Model

Transmission area: a disk centered at the node with radii equal to it’s transmission range

Transmit /Receive mode

Sleep /Idle mode y is not interfered if X is in transmit mode and all other y’s

neighbors is in sleep/indle mode.

yx

Observation 2

Because there could be more simultaneous transmission with small hops than big hops, using small hops can improve throughput.

History review

Energy ModelMetrics Main Methods

Metrics

Energy efficiencyThroughputAverage DegreeDelay

2

Source MS

Destination MS1

Big hops

2

3

4

Source MS

Destination MS

1

Small hops

Small hop VS Big hop

Minimum transmission range obtain optimal performance?

History review

Energy Model Metrics Main Methods

Main Methods

Homogeneous transmission range-a common value for all nodes

Node-based transmission range-each node has a different

transmission range

Homogeneous transmission range

Assumption: every node knows the positions of other nodes (GPS)Basic Idea: take the longest edge in the minimum spanning tree(MST)weakness: centralized

Node-based transmission range

Feature: fully distributed, localizedWell-known Proximity graphs:

1. Relative neighborhood graph(RNG)2. Gabriel graph(GG)3. Yao graph(YG)

Common: all these graphs are well- known sparse spanners. In addition, they all contain the Euclidean Minimum Spanning Tree (EMST) as a subgraph. However, all of these graphs have no constant degree.

Relative neighborhood graph(RNG)

RNG has an edge between u and v, if there is no node w such that

Gabriel graph(GG)

GG graph has an edge between two nodes u and v such that there is no node w

Given a set of nodes in 2-dimensional space, suppose we partition the space around each node into k(k>=6) sectors of a fixed angle and connect the node to the nearest neighbor in each sector.

Yao Graph

The disk can be broken

arbitrarily

Pros & Cons

ProsI. simple and easy to implementII. average node degree is bounded by a

constant Cons

The maximum degree can be as large as n-1

Vi-1

V1

Vi u

V2

Vi

Question!

Can we keep the number of neighbors of a node around an optimal (minimum) value k?

Less->increase transmission rangeMore->decrease transmission range

What’s the minimum number k than can ensure connectivity?

K-Neighbors Graph

Asymmetric Connectivity

Strongly connected

Nodes transmit messages within a range depending on their battery power, e.g., ab cb,d gf,e,d,a

a

1

2

3

1

11

1

b

d

g

f

e

c

b

a

c

d

g

f

e

Range radii

Message from “a” to “b” has multi-hop acknowledgement route

a2

3

11

b

d

g

f

e

c

1

1

1

Symmetric Connectivity

Two nodes are symmetrically connected iff they are within transmission range of each other

Node “a” cannot get acknowledgement directly from “b”

a2

3

11

b

d

g

f

e

c

1

1

1

Asymmetric Connectivity

Increase range of “b” by 1 and decrease “g” by 2

a 2

1

11

b

d

g

f

e

c

1

1

2

Symmetric Connectivity

Symmetric K-Neighbors Graph Definition 1. The symmetric super-graph of G is

defined as the undirected graph G+ obtained from G by adding the undirected edge (i, j) whenever edge [i, j] or [j, i] is in G. Formally, G+ = (N,E+), where E+ = {(i, j)|([i, j] ∈ E) or ( [j, i] ∈ E)}.

Definition 2. The symmetric sub-graph of G- is definedas the undirected graph G- obtained from G by removing All the non-symmetric edges. Formally, G- = (N,E-), whereE-={(i, j)|([i, j] ∈ E) and ( [j, i] ∈ E)}.

Theorem k???

K-Neighbors Protocol

Assumption:1. Nodes are stationary2. The maximum transmission power is the same for all

the nodes3. Given n, P is chosen in such a way that the

communication graph that results is connected with w.h.p

4. A distance estimation mechanism, possibly error prone, is available to every node

5. The nodes initiate the k-Neigh protocol at different time. However, the difference between nodes wake up

time is upper bounded by a known constant

More……

1. Node i wakes up at time ti, with ti ∈ [0, ]. At random time t1,i chosen in the interval [ti + ,ti + +d], node i announces its ID at maximum power.

2. For every message received from other nodes, i stores the identity and the estimated distance of the sender

3. At time ti +2 +d, i orders the list of its neighbors (i.e.,of the nodes from which it has received the announcement message) based on the estimated distance; let Li be the list of the k nearest neighbors of node i (if i has less than k neighbors, Li is the list of all its neighbors). ex

Simple Example

a

c

d

f

e

b

La: f d b e

Lb: c d a f

Lc: b

Ld: b a

Le: a

Lf: a b

Lsa f d e

LSb c d a

LScb

LSd b a

LSe a

LSf a

More….4. At random time t2 i chosen in the interval [ti +2 +d +τ, ti +2 +2d+τ] (τ is an upper bound on the duration of step 3), node i announces its ID and the list Li at maximum power.

5. At time ti + 3 +2d +τ node i, based on the lists Lj received from its neighbors, calculates the set of symmetric neighbors in Li. Let LSi be the list of symmetric neighbors of node i, and let j be the farthest node in LSi .

6. Node i sets its transmitting power Pi to the power needed to transmit at distance δe(ij), where δe(ij) is the estimated distance

between nodes i and j. ex

Some results

Future Work

Adapt k-neighbor to mobility?