1

HEINZ NIXDORF INSTITUTEUniversity of Paderborn

Algorithms and ComplexityChristian Schindelhauer

Algorithms for Radio NetworksWinter Term 2005/2006

01 Feb 200614th Lecture

Christian Schindelhauerschindel@upb.de

Algorithms for Radio Networks 2

HEINZ NIXDORF INSTITUTEUniversity of Paderborn

Algorithms and ComplexityChristian Schindelhauer

Mobility in Wireless Networks

Models of Mobility– Cellular– Random Trip– Group– Combined– Non-Recurrent – Particle based

Discussion– Mobility is Helpful– Mobility Models and Reality

Algorithms for Radio Networks 3

HEINZ NIXDORF INSTITUTEUniversity of Paderborn

Algorithms and ComplexityChristian Schindelhauer

Models of MobilityRandom Trip Mobility

Random WalkRandom WaypointRandom DirectionBoundless Simulation AreaGauss-MarkovProbabilistic Version of the Random Walk MobilityCity Section Mobility Model

Zur Anzeige wird der QuickTime™ Dekompressor „TIFF (LZW)“

benötigt.

[Bai and Helmy in Wireless Ad Hoc Networks 2003]

Algorithms for Radio Networks 4

HEINZ NIXDORF INSTITUTEUniversity of Paderborn

Algorithms and ComplexityChristian Schindelhauer

Brownian Motion (microscopic view)– speed and direction are chosen randomly in each time step

(uniformly from and [0, 2] )

Random Walk – macroscopic view– memoryless– e.g., for cellular networks– movement from cell to cell– choose the next cell randomly– residual probability

Models of MobilityBrownian Motion, Random Walk

[Camp et al. 2002]

Algorithms for Radio Networks 5

HEINZ NIXDORF INSTITUTEUniversity of Paderborn

Algorithms and ComplexityChristian Schindelhauer

move directly to a randomly chosen destination choose speed uniformly from stay at the destination for a predefined pause time

Models of MobilityRandom Waypoint Mobility Model

[Camp et al. 2002]

[Johnson, Maltz 1996]

Algorithms for Radio Networks 6

HEINZ NIXDORF INSTITUTEUniversity of Paderborn

Algorithms and ComplexityChristian Schindelhauer

move directly to a randomly chosen destination

choose speed uniformly from

stay at the destination for a predefined pause time

Problem:– If vmin=0 then the average

speed decays over the simulation time

Random Waypoint Considered Harmful[Yoon, Liu, Noble 2003]

Algorithms for Radio Networks 7

HEINZ NIXDORF INSTITUTEUniversity of Paderborn

Algorithms and ComplexityChristian Schindelhauer

Random Waypoint Considered Harmful

The Random Waypoint (Vmin,Vmax, Twait)-Model– All participants start with random position (x,y) in [0,1]x[0,1]– For all participants i {1,...,n} repeat forever:

• Uniformly choose next position (x’,y’) in [0,1]x[0,1] • Uniformly choose speed vi from (Vmin, Vmax]

• Go from (x,y) to (x’,y’) with speed vi

• Wait at (x’,y’) for time Twait.• (x,y) (x’,y’)

What one might expect– The average speed is (Vmin + Vmax)/2– Each point is visited with same probability– The system stabilizes very quickly

All these expectations are wrong!!!

Algorithms for Radio Networks 8

HEINZ NIXDORF INSTITUTEUniversity of Paderborn

Algorithms and ComplexityChristian Schindelhauer

Random Waypoint Considered Harmful

What one might expect– The average speed is

(Vmin + Vmax)/2

– Each point is visited with same probability

– The system stabilizes very quickly

All these expectations are wrong!!!

Reality– The average speed is much

smaller• Average speed tends to 0 for

Vmin = 0

– The location probability distribution is highly skewed

– The system stabilizes very slow

• For Vmin = 0 it never stabilizes

Why?

Algorithms for Radio Networks 9

HEINZ NIXDORF INSTITUTEUniversity of Paderborn

Algorithms and ComplexityChristian Schindelhauer

Random Waypoint Considered HarmfulThe average speed is much smaller

Assumption to simplify the analysis:

1. Assumption: Replace the rectangular

area by an unbounded plane Choose the next position

uniformly within a disk of radius Rmax with the current position as center

2. Assumption: Set the pause time to 0:

Twait = 0 This increases the average

speed supports our argument

Algorithms for Radio Networks 10

HEINZ NIXDORF INSTITUTEUniversity of Paderborn

Algorithms and ComplexityChristian Schindelhauer

Random Waypoint Considered HarmfulThe average speed is much smaller

The probability density function of speed of each node is then for

given by

since fV(v) is constant and

Algorithms for Radio Networks 11

HEINZ NIXDORF INSTITUTEUniversity of Paderborn

Algorithms and ComplexityChristian Schindelhauer

Random Waypoint Considered HarmfulThe average speed is much smaller

The Probability Density Function (pdf) of travel distance R:

The Probability Density Function (pdf) of travel time:

12

HEINZ NIXDORF INSTITUTEUniversity of Paderborn

Algorithms and ComplexityChristian Schindelhauer

Thanks for your attention!End of 12th lectureNext lecture: We 08 Feb 2006, 4pm, F1.110Next exercise class: Th 09 Feb 2006, 1.15 pm, F2.211 or Tu 14 Jan 2006, 1.15 pm, F1.110Next mini exam Mo 13 Feb 2006, 2pm, FU.511