Silence is Golden with High Probability: Maintaining a Connected Backbone in Wireless Sensor Networks Paolo Santi*

1. Silence is Golden with High Probability: Maintaining a Connected Backbone in Wireless Sensor NetworksPaolo Santi* Janos Simon† * Istituto di Informatica e Telematica del CNR, Pisa, Italy † Dept. of Computer Science, Univ. of Chicago, USA

2. Introduction • Energy conservation is fundamental in WSNs • Wireless transceiver major source of energy consumption in the node • Considerable energy can be saved by turning the radio off Example: (Medusa II node [Raghunathan et al. 02]) sleep:idle:rx:tx ratios = 0.25:1:1.006:1.244 Silence is Golden: 1/15

3. Cooperative strategies • Node transceivers’ sleeping times must be coordinated Why? To preserve connectivity • Cooperative strategy: distributed protocol that coordinates nodes’ sleeping times GOAL: Saving as much energy as possible while preserving connectivity Silence is Golden: 2/15

4. Application scenario Connectivity: WHY? Active Sleeping Silence is Golden: 3/15

5. Cell-based strategies • Introduced by Xu et al. in [Xu et al. – Mobicom 01], analyzed by Blough and Santi in [BloughSanti – Mobicom 02] • Idea (GAF protocol [Xu et al. 01]): • divide the deployment region into equal cells; • leave an active node for each cell If the cell size is correctly chosen, connectivity is ensured Silence is Golden: 4/15

6. Our contribution • Two simple cell-based coordination algorithms: deterministic and randomized • The algorithms use: • location information (as in GAF) • loose synchronization (additional requirement) Good news: these features are likely to be available in WSNs • The algorithms are asymptotically energy optimal: • in the worst case (deterministic algorithm) • on the average (randomized algorithm) • Knowledge vs. performance tradeoff Silence is Golden: 5/15

7. The energy model A node can be: • Energy for sensing and receiving GPS signal comparable to energy in the sleep state • Our model: sleeping 0.25 idle 1 receiving 1.006 transmitting 1.244 C units of energy per time unit when idle/rx/tx c << C units of energy per time unit when sleeping (and sensing). For simplicity, c=0 Silence is Golden: 6/15

8. The system model • n sensors are deployed in a square region of side l • All sensors have the same transmitting range r << l • Deployment region divided into N = 8 l2/r2 square cells of r/22 With this setting, any two nodes in adjacentcells (horizontal, vertical, diagonal) can communicate directly Silence is Golden: 7/15

9. The FULL protocol Assumptions: Every node knows: • its cell ID; • the ID of every other node in its cell The leader election process starts at time Tr . Every step lasts Ts Protocol for node i: At time Tr + (i-1)Ts: • turn radio on and receive message M = (Emax,m) from node i -1 • estimate available energy Ei • Emax = max {Emax, Ei} • if Emax = Ei then m i At time Tr + iTs: • send message (Emax ,m) • turn radio off Silence is Golden: 8/15

10. The FULL protocol (2) At time Tr + niTs: (niis the number of nodes in the cell of node i ) • turn radio on and receive message M=(Emax,m); m is the leader for the next sleep period • if i <> m turn the radio off • The protocol is re-executed after a certain sleep period Why the sleep period? To balance energy consumption Silence is Golden: 9/15

11. Choosing the sleep period Time diagram of FULL execution with different sleep periods: Emax/2C and Emax/C Emax/C : cell lifetime = 391 time units; average per node lifetime: 248.5 Emax/ 2C : cell lifetime = 324 time units; average per node lifetime: 321.75 Silence is Golden: 10/15

12. Analysis Assumptions: • initialization cost is disregarded • no “external factor” I.e., we estimate “The Cost of Silence” Theorem 1: Assume cell i contains ni nodes. Using the FULL protocol with sleep period set to Emax/2C, the cell lifetime is (ni Tb), where Tb is the baseline cell lifetime (with no cooperative strategy). The FULL protocol is (worst-case) energy optimal Silence is Golden: 11/15

13. The RANDOM protocol Assumptions: Every node: • knows its cell ID; • can detect conflicts on the wireless channel The leader election process starts at time Tr . Every iteration lasts Ts Protocol for node i: At time Tr : Repeat until TERMINATION • flip a coin with probability of success p • if SUCCESS send message (Ei , i ) • if nobody sent a message or COLLISION, go to next iteration • TERMINATION=True; if not SUCCESS turn the radio off Silence is Golden: 12/15

14. Setting the value of p Ideally, we should set p=1/ni, where ni is the number of nodes in the cell WHY? Because with this setting the expected number of iterations #S is minimized: E [#S] = e ( 2.718) (Optimal in expectation) What if ni is not known? If n is known, and nodes are distributed uniformly at random, we still have E [#S] = e Theorem 2: Assume n nodes are distributed uniformly in [0, l ]2, and set p to the expected number of nodes in a cell. If r is appropriately chosen, then limn, l E [#S] = e . Silence is Golden: 13/15

15. Network-wide analysis How many iterations are needed to elect the leader in every cell? (Average-case analysis; N is the number of cells) Assume p = 1/ni • N/e cells elect the leader in the first iteration; • N(1-1/e)/e cells elect the leader in the second iteration; …. After k steps, Nk = N (1-(1-1/e)k) cells have elected the leader With k = 10 we have N10 = 0.9898 N Silence is Golden: 14/15

16. Conclusion and future work • We have presented two simple algorithms for cell-based node coordination in WSNs • Our algorithms can be a starting point for real implementations of cell-based energy conservation • Future work: • change p depending on the node’s energy level • change p depending on the duration of the previous election Silence is Golden: 15/15