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?