1. 1 HEINZ NIXDORF INSTITUTE University of Paderborn Algorithms and Complexity Christian Schindelhauer Algorithms for Radio Networks Winter Term 2005/2006 01 Feb 2006 14th Lecture Christian Schindelhauer schindel@upb.de

2. Algorithms for Radio Networks 2 HEINZ NIXDORF INSTITUTE University of Paderborn Algorithms and Complexity Christian 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

3. Algorithms for Radio Networks 3 HEINZ NIXDORF INSTITUTE University of Paderborn Algorithms and Complexity Christian Schindelhauer Models of Mobility Random Trip Mobility  Random Walk  Random Waypoint  Random Direction  Boundless Simulation Area  Gauss-Markov  Probabilistic Version of the Random Walk Mobility  City Section Mobility Model [Bai and Helmy in Wireless Ad Hoc Networks 2003]

4. Algorithms for Radio Networks 4 HEINZ NIXDORF INSTITUTE University of Paderborn Algorithms and Complexity Christian 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 Mobility Brownian Motion, Random Walk [Camp et al. 2002]

5. Algorithms for Radio Networks 5 HEINZ NIXDORF INSTITUTE University of Paderborn Algorithms and Complexity Christian Schindelhauer  move directly to a randomly chosen destination  choose speed uniformly from  stay at the destination for a predefined pause time Models of Mobility Random Waypoint Mobility Model [Camp et al. 2002] [Johnson, Maltz 1996]

6. Algorithms for Radio Networks 6 HEINZ NIXDORF INSTITUTE University of Paderborn Algorithms and Complexity Christian Schindelhauer  move directly to a randomly chosen destination  choose speed uniformly from  stay at the destination for a predefined pause time  Problem: –If v min =0 then the average speed decays over the simulation time Random Waypoint Considered Harmful [Yoon, Liu, Noble 2003]

7. Algorithms for Radio Networks 7 HEINZ NIXDORF INSTITUTE University of Paderborn Algorithms and Complexity Christian Schindelhauer Random Waypoint Considered Harmful  The Random Waypoint (V min,V max, T wait )-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 v i from (V min, V max ] Go from (x,y) to (x’,y’) with speed v i Wait at (x’,y’) for time T wait. (x,y)  (x’,y’)  What one might expect –The average speed is (V min + V max )/2 –Each point is visited with same probability –The system stabilizes very quickly  All these expectations are wrong!!!

8. Algorithms for Radio Networks 8 HEINZ NIXDORF INSTITUTE University of Paderborn Algorithms and Complexity Christian Schindelhauer Random Waypoint Considered Harmful  What one might expect –The average speed is (V min + V max )/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 V min = 0 –The location probability distribution is highly skewed –The system stabilizes very slow For V min = 0 it never stabilizes  Why?

9. Algorithms for Radio Networks 9 HEINZ NIXDORF INSTITUTE University of Paderborn Algorithms and Complexity Christian Schindelhauer Random Waypoint Considered Harmful The 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 R max with the current position as center 2.Assumption:  Set the pause time to 0: T wait = 0  This increases the average speed  supports our argument

10. Algorithms for Radio Networks 10 HEINZ NIXDORF INSTITUTE University of Paderborn Algorithms and Complexity Christian Schindelhauer Random Waypoint Considered Harmful The average speed is much smaller  The probability density function of speed of each node is then for  given by  since f V (v) is constant and

11. Algorithms for Radio Networks 11 HEINZ NIXDORF INSTITUTE University of Paderborn Algorithms and Complexity Christian Schindelhauer Random Waypoint Considered Harmful The average speed is much smaller  The Probability Density Function (pdf) of travel distance R:  The Probability Density Function (pdf) of travel time:

12. 12 HEINZ NIXDORF INSTITUTE University of Paderborn Algorithms and Complexity Christian Schindelhauer Thanks for your attention! End of 12th lecture Next lecture:We 08 Feb 2006, 4pm, F1.110 Next exercise class: Th 09 Feb 2006, 1.15 pm, F2.211 or Tu 14 Jan 2006, 1.15 pm, F1.110 Next mini examMo 13 Feb 2006, 2pm, FU.511