1 HEINZ NIXDORF INSTITUTE University of Paderborn Algorithms and Complexity Christian Schindelhauer...

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HEINZ NIXDORF INSTITUTEUniversity of Paderborn

Algorithms and ComplexityChristian Schindelhauer

Algorithms for Radio NetworksWinter Term 2005/2006

25 Jan 200613th Lecture

Stefan Rührup

[email protected]

Algorithms for Radio Networks 2



Modeling Worst Case Mobility

• Problem:

– Mobile users

– positions are known at the moment (t=0), but not in the future (t=∆).

– How to adjust the transmission range?

• Reasonable restriction for the worst case

– velocity bound pedestrian model

– acceleration bound vehicular model

transmission range





V: Pedestrian Model ↔ Maximum velocity ≤ vmax

A: Vehicular Model ↔ Maximum accelaration ≤ amax





V: Pedestrian Model ↔ Maximum velocity ≤ vmax

A: Vehicular Model ↔ Maximum acceleration ≤ amax




The Mobile Ad Hoc Network

• Basic idea: Maintain the network for a time interval Δ

• As a start: synchronous round model

• In every round of duration Δ

– Determine positions (speed vectors) of possible

communication partners

– Establish (stable) communication links

– Update routing information

– Do the job, i.e. packet delivery, video streams, telephone,…




The Mobility Model

• Velocity bounded mobility model Pedestrians move with a (known) bounded velocityFor all i{1,…,n}:

• Acceleration bounded mobility model Cars move with a (known) bounded accelerationFor all i{1,…,n}:

• Technical assumptions:– polynomial distances and speeds– complete knowledge of position




Taming Mobility in the Link Layer

• General Concept

– Increase transmission distance

to guarantee a certain life span Δ for each link

– Apply the mobility model (i.e. amax or vmax is known)

transmission range

start position (t0)

end position (t0+Δ)




vx

x

Example

• Trains moving in opposite directions

• Transmission range only sufficient in the static case

• If we take the velocity into account:

• We need a modified distance measure




• Given the positions u,w and the velocity bound vmax

• Maximum distance after the time interval ∆:

Velocity bounded (pedestrian) model

uncertainty

u w




• Transmission range for the velocity bounded (pedestrian) model:

end

Walking range

start startend

Sa-tisfies

triangleinequality

Pedestrian Model

transmission rangefor node u and t=0




• Acceleration bound amax

• Positions u,v and speed vectors u’,v’ known

• Maximum distance after time interval ∆:

• With given distances andwe can approximate by a constant factor.

Vehicular Model

uncertainty due to acceleration

u w

velocity




end

start start

end

Sa-tisfies

triangleinequality

Vehicular Model

• Transmission range for the acceleration bounded (vehicular) model:




• Communication links (edges) can interfere

– edges interfere, if one node is located within the transmission radius of a node of the other edge

• Velocity bounded model:

interference, if

• Acceleration bounded model:

interference, if and

interference

pq e

e’

Interferences




• An edge g interferes with edge e in the

1. pedestrian (v) model

2. vehicular (a) model

No interference Interference

gg

e e e e

g

e

g

p

q

qp

Interferences more formally




Interference number of a network

• What is the maximum number of interfering edges in a network?

• Interference number is influenced by the transmission rangeand the positions of the nodes

• How many nodes can meet in one place and form a crowd?

Definition of crowdedness

• Crowdedness gives a lower bound for interferences

• In both mobility models we observe for all connected graphs G(V,E): Int(G) crowd(S) - 1




• Crowdedness of node set – natural lower bound on network parameters (like diversity)

• Pedestrian (v) model:– Maximum number of nodes that can collide with a given node

in time span [0,Δ]

• Vehicular (a) model:– Maximum number of nodes that may move to node u meeting

it with zero relative speed in time span [0,Δ]

• crowd(S) := maxuS crowd(u)

Crowdedness more formally




• Transmission ranges defined for two mobility models

• Basis for

– the definition of interferences

– construction of a network topology

– analyisis: interference number, congestion

Mobility Models




The Hierarchical Grid Graph(pedestrian model)

• Start with grid of box size Δ vmax

• For O(log n) rounds do– Determine a cluster head per

box– Build up star-connections

from all nodes to their cluster heads

– Erase all non cluster heads– Connect neighbored cluster

heads– Increase box size by factor 2

od




• Interference number of the network G: Int(G) := maxeE(G) |Int(e)|

• Load and path system

– Set of all message paths defines path system P

– l(e) := # number of packets sent over edge e according to P

• Congestion of an edge (with respect to a path system P )

– Congestion of a network G:

• Mobile spanner

– A network is a mobile spanner, if for all u,w exists a path

connecting u and v with bounded length

Load, Congestion and Spanners




• A graph G is called a mobile spanner, if for all nodes there is a path in G with

• Congestion of a path system:

• A mobile spanner G approximates an optimal path system

Mobile Spanner




Results

Lemma

In both mobility models α{v,a} every mobile spanner is also a mobile power spanner, i.e. for some ß≥1 for all u,w S there exists a path (u=p0,p1,…,pk=w) in G such that:




Results

Theorem

Given a mobile spanner G for any of our mobility models then

– for every path system P in a complete network C

– there exists a path system P‘ in G such that

Theorem

The Hierarchical Grid Graph constitutes a mobile spanner with at most O(crowd(V) + log n) interferences (for both mobility models).

The Hierarchical Grid Graph can be built up in O(crowd(V) + log n) parallel steps using radio communication




The Hierarchical Grid Graph(vehicular model)

• Algorithm:

– Consider coordinates (x(si),y(si),x(s‘i),y(s‘i))

– Start with four-dimensional grid with rectangular boxes of size (6Δ²amax, 6Δ²amax,2Δvmax,2Δvmax)

– Use the same algorithm as before

x

vx t=0

x

vx t=Δ

x

vx t=2Δ




Stable Basic Networks under Worst Case Mobility

CorollaryThere exist distributed algorithms that construct a mobile network G for velocity bounded and acceleration bounded model with the following properties:

1. G allows routing approximating the optimal congestion by O(log² n)

2. Energy-optimal routing can be approximated by a factor of O(1)

3. G approximates the minimal interference number by O(log n)4. The degree is O(crowd(S)+ log n)5. The diameter is O(log n)

• Still no routing can satisfy small congestion and energy at the same time!




Applications of mobility models

• Various protocols for mobile networks

• more or less influenced by mobility (esp. routing)

• How to evaluate and compare protocols?

– simulation of mobile networks

– mobility models as benchmarks

• traces or synthetic models?




Requirements

• imitation of realistic motion

• mobility model should not be too complex

⇒ simplifications (depends on the network model)

e.g., cellular network:

– random walk on network cells

– speed and direction vectors can be neglected

• many application-specific mobility models




Parameters

• Bounds on velocity, acceleration or the change of direction

• Is the motion independent of other mobile hosts?

– entity mobility models

– group mobility models

• Degree of randomness

– sharp turns possible or smoothed motion?

• Granularity

– macroscopic view (→ cellular networks)

– microscopic view (→ ad hoc networks)




• Brownian Motion (microscopic view)

– speed an direction are chosen randomly in each time step (uniformly from [vmin,vmax] and [0,])

• Random Walk

– macroscopic view

– memoryless

– e.g., for cellular networks

– movement from cell to cell

– choose the next cell randomly

– residual probability

[Camp et al. 2002]

Brownian Motion, Random Walk




[Camp et al. 2002]

[Johnson, Maltz 1996]Random Waypoint Mobility Model

• move directly to a randomly chosen destination

• choose speed uniformly from [vmin,vmax]

• stay at the destination for a predefined pause time




• adjustable degree of randomness

• velocity:

• direction:

Gauss-Markov Mobility Model[Liang, Haas 1999]

mean random variablegaussian distribution

tuning factor

[Camp et al. 2002]

α=0.75




Concept map of mobility models[Bettstetter 2001]

32



Thanks for your attention!End of 13th lectureNext lecture: We 1 Feb 2006, 4pm, F1.110Next exercise class: Th 26 Jan 2006, 1.15 pm, F2.211 or Tu 31 Jan 2006, 1.15 pm, F1.110Next mini exam Mo 13 Feb 2006, 2pm, FU.511

1 HEINZ NIXDORF INSTITUTE University of Paderborn Algorithms and Complexity Christian Schindelhauer...

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