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1. Monitoring Churn in Wireless Networks TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: AAAAAAA

2. Motivation / Intro Network of sensor nodes: • measuring certain properties of their environment • wireless, communicating on several channels • battery powered Nodes might fail / nodes may be added All nodes should be aware of all present nodes • with small delay • with little energy consumption • using few channels for communication

3. Model 1 2 5 3 12 • n nodes with IDs • single-hop t • synchronized time slots t t r s r s s s t s time • transmit / receive / sleep 0 0 1 1 1 0 1 0 0 1 • energy 1 / 1 / 0 channel 1 s s t s s node 1 • local computations free t t t channel 2 r s s s r • k channels r r r r channel 1 node 2 channel 2 • no collision detection s s s s r s r t t channel 1 node 3 • bounded message size channel 2 O(1) IDs

4. Model cont’d 1 1 2 2 5 5 3 3 12 Nodes may join or crash at any time churn = joins and crashes 7 burst = large number of joins and crashes in short time Adversary: May let nodes crash or join in order to make an algorithm fail

5. Goal Every node must know the IDs of all nodes currently in the network

6. Simple Lower Bounds What time / energy is at least necessary in this model? • every node can only receive one message per time slot containing at most a constant number of IDs  on average only constant rate of churn tolerable per time slot  Ω(b) time slots necessary to learn about b joins / crashes  every node needs Ω(b) energy units to learn about b joins / crashes

7. Results Our Monitoring Algorithm: • tolerates churn bursts in any order of magnitude • is deterministic except for detection of joining nodes • handles asymptotically maximum average rate of churn tolerable in this model • after each burst of size b it takes • O(b + log n) time slots and • O(b + log n) energy per node • until all nodes have updated their ID table (optimal up to additive logarithmic term) • needs Θ(n/log n) channels

8. Results cont‘d Our Monitoring Algorithm: • can get by with less than Θ(n/log n) channels: • k channels available  time .

9. Monitoring Algorithm • burst size is assumed to be b'=log n • nodes partitioned into n/(2b'+2)-1 sets Θ(n/b') sets 2b'+2 nodes in each set • each set detects crashed and joined nodes on its own channel • disseminate information to all nodes • all nodes update ID table 1 ... 7 ... 3 2 12 34 13 ... ... • double b' if algo did not work

10. Crash Detection • nodes send „I am here“ messages I am still here! I am still here! I am still here! • min(2b'+2,n) time slots necessary

11. Join Detection • joiners send join requests to with S1 with probability 1/b' I want to join! Set S1 I want to join! • b' in Ω(b)  in constant number of rounds at least 1 joiner

12. Information Dissemination • every set becomes vertex of balanced binary tree depth log n • every set forwards information on node v with smallest ID first • information on v disseminated after O(log n) time slots • all information disseminated after O(log n + b') time slots

13. Monitoring Algorithm Time Step O(1) • b' = log n O(1) • partitioning • crash detection O(min(b', n)) log(b/log n) times runtime O(b + log n) O(b' + log n) • join detection O(b' + log n) • dissemination O(1) • update ID table 1 ... 2 34 7 13 12 ... 3 ... ... O(1) • double b'

14. What if critical nodes crash? • in each set node which is responsible for communication with other sets • = representative • all other nodes replacements • replacements take over if representative does not send • delay of at most b' • still runtime of O(b' + log n) per round

15. Conclusions & Future Work • Model t t r s r s s s t s • Lower Bounds • Monitoring Algorithm • Future Work: Multi-hop

16. Thank You!Questions & Comments? TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: AAAAAAA