Satish T Eddhu Oct 17, 2002

1. Birthday Protocols for Energy Efficient Deployment and Powerful Neighbor Discovery in Static Ad Hoc Wireless Networks Satish T Eddhu Oct 17, 2002

2. Agenda • Static ad hoc wireless networks • Need for power conservation • Deployment phase • Adjacent neighbor/Network discovery phase • Birthday protocols for energy efficiency • Mathematical model of protocol • Analysis of different protocol modes

3. Why is power conservation important? • Nodes usually deployed with small batteries. For example, sensor network whose nodes are dropped by a plane into a forest • On power failure, the node essentially disappears with dire consequences for the data path it helped to support. • In dynamic networks, like the headsets of a squadron of soldiers, the consequences are less critical as the nodes can be recharged.

4. Node deployment phase • Large number of wireless battery-powered sensors are released from an airplane into a forest • Several drops are required before all sensors are deployed • About a week before all sensors are in place • Network not yet discovered!

5. Node deployment phase (cont) • Nodes listening constantly for about a week exhaust batteries very soon as listening is power-intensive • Saving energy during this long period is a goal

6. Network discovery phase • Nodes discover their adjacent neighbors • Discovery phase lasts usually a couple of minutes • Energy efficiency not a goal • Nodes to participate vigorously to maximize hearing and to be heard

7. Alternate solutions to power conservation • Currently, no way to transmit or receive a special wakeup broadcast. Hence, only 2 approaches possible. 1. Transmit at low power 2. Do not transmit • We explore the 2nd approach

8. Birthday protocols – A probabilistic approach • Saves a great deal of energy during the deployment of nodes • Simultaneously allows a high probability of discovering neighbor nodes • Can be parameterized to meet varying goals • Does not deal with data transmission phase

9. Inspiration for birthday protocol • Birthday paradox Probability that at least 2 people in a room have the same birthday exceeds ½ even when as few as 23 people are present!

10. The hearing scenario • Over a period of n slots, two wireless nodes independently and randomly select k slots. • The 1st node transmits a message on its k slots • The 2nd node listens on its k slots • The probability that the 2nd node hears the 1st is Q(n,k) = 1 – C(n-k,k)/C(n,k) which is nearly 1 when the ratio k/n is relatively small. • Q(1000,70) ≈ 0.995. Though each node is idle 93% of the time, still there is a high probability that one hears the other

11. Model of the wireless network • Time is slotted. n timeslots of total interval length I. Length of timeslot = I/n • n is large • The total number of nodes is known • Nodes do not coordinate their actions in any way • Nodes are randomly placed in some area • Nodes are distinguishable by an ID such as a MAC address • Each node has some internal memory to record local topology (neighboring nodes)

12. States of a node • A node can be in one of three states • Transmit(T): node broadcasts a discovery message advertising itself. Consists of, at a minimum, the address of broadcaster • Listen(L): node listens for discovery messages. On hearing, records source address in its local topology table • Energy-saving(S): node spends zero energy • Assumption: Only a small but positive time needed to switch between states

13. State switching dynamics of a node • Each node randomly chooses to enter one of the three states at the beginning of each timeslot depending on its mode and the choice of globally fixed parameters Πt and Πl • The choice of state for one timeslot is independent of choices made for other timeslots

14. Modes of operation • Birthday-listen-and-transmit(BLT): Node can be in either of the 3 states L,T or S. In each timeslot, states are chosen with probabilities p(T) = pt = Πt p(L) = pl = Πl p(S) = ps = 1- Πt- Πl

15. Modes of operation • Birthday-listen (BL): Node is either in L or S state in each timeslot. Does not transmit. p(T) = pt = 0 p(L) = pl = Πl p(S) = ps = 1- Πl

16. Terminologies and definitions • Node X hears node Y on a timeslot if X is in state L, Y is in state T, and no other node in the hearing range of X is in state T • X may hear Y on many timeslots. But X “discovers” Y on the first such timeslot • Links are not bi-directional. “X hears Y” is different from “Y hears X”.

17. Terminologies and definitions • There are N nodes all in the radio range of each other, forming a clique of size N • Total number of discoveries possible in interval I = P(N,2) = N(N-1) • Actual number of discoveries = U • Fraction of links discovered, F = U/[N(N-1)]; to be maximized in discovery phase

18. N=4,n=9 pt = pl = 1/6. ps = 2/3 Links discovered, U = 4 Fraction discovered, F = 4/12 Energy gain, G = 1/(pt +pl) = 3 Discovery example

19. Nodes to be deployed in BL mode to conserve power Upon initiation of discovery, change to BLT mode to maximize the fraction of neighbors (and thereby U) discovered Customer specifies Maximum duration of discovery, I Expected fraction of links discovered, E(F) Expected energy gain, E(G ) Given any two of the above three requirements, optimize the third An implementation proposal

20. Intuition from the two-node case, N = 2 • Two nodes X and Y, both in BLT mode for some interval of length I • P(X discovers Y in one timeslot) = plpt • P(X discovers Y over an interval of n timeslots) = [1 – (1 – plpt)n] • Results unchanged even if X is in BL mode and Y is in BLT mode • By symmetry, expected number of links discovered, E(U) = 2[1 – (1 – plpt)n]

21. What if N > 2? • Each node has a fixed probability of transmission at deployment • More the nodes, higher the probability that two transmissions collide, wasting a timeslot • Does the protocol still work for large N?

22. Analysis : N > 2 • Denote the number of nodes in each state in a timeslot by T, L and S • T, L + S = N • E(number of hearings in a timeslot), E(h)= E(L | T = 1)*P(T = 1) = (N-1) pl/(pl+ps) * C(N,1) pt(1- pt) N-1 = N(N-1) plpt(1- pt) N-2

23. Analysis : N > 2 (cont)

24. Analysis : N > 2 (cont) • For large N, the binomial distribution can be approximated to a Poisson distribution with mean, λ = nE(h)/[N(N-1)] = npl pt(1- pt) N-2 • Fraction of discovered links, F = 1 – e-λ = 1 – e –n pl pt(1- pt) N-2 • For fixed n, even for large N, by proper choice of pl =pt , F = 1/e can be achieved • Can maximize fraction of links discovered by choosing the “right” transmission probability

25. Probabilistic Round Robin (PRR) mode • To maximize discovery, i.e., to maximize E(h) • Set pl = C – pt ; 0 < pt < C ≤ 1. C is inversely related to energy savings. • So, E(h) = N(N-1) pt (C – pt ) (1 – pt ) N - 2 • E(h) is maximized at pt = 1 / N and C = 1 • This models the PRR mode with the parameters pt = 1/N pl = 1 - 1/N ps = 0

26. E(h) = N(N-1) pt (C – pt ) (1 – pt ) N – 2 = (N-1) (1 – 1/N) N – 1 F = 1 – e-λ = 1 – e –n/N(1 – 1/N) N – 1 Flexibility When estimated N differs from expected N, the protocol degrades gracefully Graphical results show that it is better to overestimate N when we are not sure PRR mode (cont)

27. PRR mode (cont)

28. Mixed-mode scenario • Some nodes in BL mode, rest in PRR • In the worst case of hearing (when only one neighbor is in PRR mode) P(hear in 1 slot) = 1 – [P(not hear in 1 slot)]n = 1 – (1 – plpt)n = 1 – (1 – pl/N)n • If more neighbors in PRR mode, higher the probability of hearing in 1 slot

29. Suggested deployment and discovery technique • Deploy nodes in BL mode to conserve power • During discovery, trigger exactly one node into PRR mode for a small fixed duration • When a BL node hears for the 1st time(discovery), it switches to PRR mode for a small fixed duration and then returns to BL mode • “Chain reaction” triggered and many links discovered • Discovery ends when all nodes return to BL mode

30. Example • Expected fraction of links discovered, F = 95% • Expected energy gain, G = 100 • So, pl = 100 during BL mode • Estimated N = 10 • P(hear in 1 slot) = 1 – (1 – pl /N)n ≥ 0.95n ≥ log(0.05) / log(1 - pl /N) n ≥ 2994 • This discovered 3068/3080 possible links, and 494/497 nodes (Of the 500 nodes deployed, 3 were out of radio range)

31. BL-PRR protocol: Simulation

32. Extensions to this approach • Energy conservation during the data transmission phase is not considered. This phase is much longer than deployment and discovery. • Can the global parameters Πt ,Πl and Πs be altered? Can the values be localized for each node?

33. References • Birthday protocols for low energy deployment and flexible neighbor discovery in ad hoc wireless networks (Michael J. McGlynn, Steven A. Borbash: Proceedings of the 2001 ACM International symposium on mobile ad hoc networking and computing)