Algorithms for Radio Networks Exercise 11

1. Algorithms for Radio NetworksExercise 11 Stefan Rührup sr@upb.de

2. Exercise 22 • Consider a multistory building of height 50 m. At each floor of height 2.5 m a sensor node is attached to the wall. Now, every 1 second a sensor is dropped from the top of the building. • Calculate the transmission radius of the falling sensors which is needed to maintain a connection to the static nodes. Use the acceleration bounded (vehicular) mobility model with acceleration g ≈ 10m/s2 = amax and assume a time interval of ∆ = 1 sec. • Draw the location-velocity-diagram of the scenario.

3. Exercise 22 distance d = 50 m 20 sensors acceleration: g ≈ 10m/s2 = amaxtime interval ∆ = 1 s 50m

4. Vehicular Model • Acceleration bound amax • Positions u,v and speed vectors u’,v’ known • Maximum distance after time interval ∆ ( transmission range): uncertainty due to acceleration u w velocity

5. Exercise 22 • Location-Velocity-Diagram: y vy

6. Exercise 23 • Consider a quadratic area which is divided into n squares of size 1m x 1m. Now, n pedestrians are placed randomly and uniformly in this area. • What is the expected number of pedestrians per square? • What is the relation between the crowdedness and the maximum number of pedestrians per square? • What is the probability that exactly k pedestrians are in one square? • What is the probability at leastk pedestrians are in one square? • For which k is this probability smaller than 1/n?

7. Velocity bounded (pedestrian) model • Given the positions u,w and the velocity bound vmax • Maximum distance after time interval ∆ ( transmission range): • Crowdedness: Maximum number of nodes that can collide with a given node in time span [0,Δ]: uncertainty u w

8. Exercise 23 • The relation between the crowdedness and the maximum number of pedestrians per square • Consider the radius 2vmax ∆ for vmax = 1/2 m/s and ∆ = 1 s. • Crowdedness is linear in the maximum number of pedestrians per square.

9. Exercise 23 • Random placement: • What is the probability that at least k pedestrians are in one square? • For which k is this probability smaller than 1/n? • Balls into Bins: • Assume n balls are thrown sequentially into n bins (randomly and uniformly distributed) • What is the maximum nuber of balls per bin?

10. Balls into Bins Theorem: The probability that at least t log n/log log n balls fall into a single bin is at most O(1/nc) for constants t and c.With high probability (P = 1 - 1/n(1)) at most O(log n/log log n) balls fall into one bin. Proof: • Determine the Probability (generally) that at least k out of n Balls fall into a certain bin. • Consider the case that at least k out of n balls fall into any of the n bins • Choose k such that this holds with probability 1/nc.

11. Balls into Bins Probability that exactly k balls fall into a certain bin: Probability that at least k of n balls fall into a certain bin: We use follows from Sterling´sformula:

12. Balls into Bins

13. Balls into Bins Probability that at least k of n balls fall into a certain bin: For which k is the probability We only consider the dominant terms: For which k holds ?

14. Balls into Bins ... For which k holds ? Inverse of k ln k? So, we choose k as follows: Probability that at least k of n balls fall into a certain bin: Probability that at least k of n balls fall into any of the n bins: for a constant c = t - 1 + o(1) i.e.