1 / 14

Algorithms for Radio Networks Exercise 11

Algorithms for Radio Networks Exercise 11. Stefan Rührup sr@upb.de. 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.

nerea-hale
Download Presentation

Algorithms for Radio Networks Exercise 11

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript


  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.

More Related