1. Trade-offs Between Mobility and Density for Coverage in Wireless Sensor Networks Wei Wang, Vikram Srinivasan and Kee-Chaing Chua National University of Singapore 2007 Mobicom

2. Outline Introduction Coverage with mobile sensors Coverage of hybrid networks Mobility algorithm Numerical results Conclusion

3. Introduction Coverage problem Important research problem in WSNs k-covered Network Deployment Mobility

4. Introduction- deployment Metric: over-provisioning factor Indicates the efficiency of a network deployment strategy Consider a random deployment strategy What is the sensor density to guarantee k-coverage?

5. Introduction- mobility Mobile sensors can relocate themselves to heal coverage holes Over-provisioning factor for a network with all mobile sensors can be Θ(1) Consumes more energy Mobile sensors Limited mobility: move once, over a short distance Maximum distance?

6. Coverage with mobile sensors Sensing field: L=l*l Num. of static sensors: N = λL Uniformly and independently scattered in the network. Number of static sensors in a region with area of A: n A Sensing range: r = 1 /√π 1=πr 2 1 Density

7. Over-Provisioning Factor Optimal over-provisioning factor:Θ(1) d s = √2r Density of mobile sensor K-coverage r = 1 /√π

8. Over-Provisioning Factor Randomly deployed static sensor networks Density λ Total expected area which is uncovered is e −λ L. Random coverage processes Large enough λ, e −λ can be made arbitrarily small Probability approaches one for a network with constant sensor density λ when the network size L→∞. Exist a connected coverage hole larger than unit area

9. Over-Provisioning Factor To achieve k-coverage in a large network, the static sensor density needs to grow with the network size λ = logL +(k + 2) log log L + c(L) c(L) → +∞ as L → +∞

10. All Mobile Networks η m = Θ(1). key question what is the maximum distance that each sensor has to move? Limit the maximum moving distance for each mobile

11. All Mobile Networks maximum distance Theorem1: Network can provide k-coverage with an over-provisioning factor of η m = π/ 2 and the maximum distance moved by any mobile sensor is O( 1 √klog 3/4 (kL)) w.h.p.

12. All Mobile Networks Sensing field into square grids with side length of d a =√2r/√k Number of nodes in the sensing range πr 2 /(√2r/√k) 2 =πk/2 η m =(πk/2) / k = π/2

13. All Mobile Networks By the lower bounds on lattice points covered by a circle, there are at least W(k) lattice points of side length of d a covered by a circle of radius r d a =√2r/√k Increasing function

14. All Mobile Networks W(k) > k when k ≥ 25 ->k coverage W(k)=25.13274 Network is at least k-covered when 1 ≤ k < 25.

15. All Mobile Networks l × l square, L = l 2 points in the region there exists a perfect match between the L random points and the L grid points with maximum distance between any matched pairs of O(log3/4 L). Grid points (k/2r 2 )*L O(log 3/4 (kL)) Grid size is d a =√2r /√k O( 1/√k log 3/4 (kL)) 1=πr 2 1/r 2 ＝ π η m =Densty/k Densty= η m *k= πk/2 =k/2r 2

16. Coverage of hybrid networks Over-provisioning factor is O(1) Fraction of mobile sensors required is less than 1 /√2πk Maximum distance that any mobile sensor will have to move is O(log 3/4 L)

17. Density of Mobile Sensors Static sensor density at λ =2πk. Divide the network into square cells equal side length of d h = r/√2. Average number of static sensors in each cell will be 2πkd 2 h = k.

18. Density of Mobile Sensors The network will be k-covered if all cells contain at least k sensors. cell i has v i = k−n i vacancies, If a cell i contains n i < k static sensors Poisson approximation

19. Density of Mobile Sensors The random variable v i = [k − n i ] +, will be distributed as: The expected number of vacancies in a cell will be:

20. Density of Mobile Sensors Using Stirling’s approximation Density of mobile sensor Density of Static sensor Fraction of mobile sensors required is less than r = 1 /√π d h = r/√2.

21. Maximum distance for mobiles A grid with side length of 1/ √Λ Maximum distance Decreasing function Matching distance

22. Mobility Algorithm Problem Formulation Movement cost Initial number of mobile sensor Number of mobile sensor from cell i to cell j

23. Distribution Solution A distributed algorithm Maximum flow problem Assume Sensor knows Its location Which cell it is located in. v i and m i Each cell elects a mobile or static sensor as the delegate Communicate and exchange information with its neighbors in graph G

24. Distribution Solution- push-relabel algorithm a bc io oioi Cell a Cell c Distance D v-m=3 v-m=-2 v-m=-1

25. Distribution Solution- push-relabel algorithm a bc io oioi Cell a Cell c h(i)=0 e(i)=0 h(i) =0 e(i) =0 h(i) =0 e(i) =0 h(o)=0 e(o)=3 h(o) =0 e(o) =-2 h(o)=0 e(o) =-1 Zero cost cici v-m=3 v-m=-2v-m=-1

26. Distribution Solution- push-relabel algorithm a bc io oioi Cell a Cell c h(i)=0 e(i)=0 h(i) =0 e(i) =0 h(i) =0 e(i) =0 h(o)=0 e(o)=3 h(o) =0 e(o) =-2 h(o)=0 e(o) =-1 v-m=3 v-m=-2v-m=-1 h(o)=1 e(o)=3

27. Distribution Solution- push-relabel algorithm a bc io oioi Cell a Cell c h(i)=0 e(i)=0 h(i) =0 e(i) =0 h(i) =0 e(i) =1 h(o) =0 e(o) =-2 h(o)=0 e(o) =-1 v-m=3 v-m=-2v-m=-1 h(o)=1 e(o)=2 h(o)=1 e(o)=1 h(i) =0 e(i) =1

28. Distribution Solution- push-relabel algorithm a bc io oioi Cell a Cell c h(i)=0 e(i)=0 h(i) =0 e(i) =0 h(o) =0 e(o) =-1 h(o)=0 e(o) =1 v-m=3 v-m=-2v-m=-1 h(o)=1 e(o)=1 h(i) =0 e(i) =0

29. Distribution Solution- push-relabel algorithm a bc io oioi Cell a Cell c h(i)=0 e(i)=0 h(i) =0 e(i) =0 h(o) =0 e(o) =-1 h(o)=0 e(o) =1 v-m=3 v-m=-2v-m=-1 h(o)=1 e(o)=1 h(i) =0 e(i) =1

30. Numerical results Mobile Sensor Networks only consider the maximum matching distance for 1-coverage in our simulations M = ΛL mobiles Λ=π/2 d s= √2 r 10 5 randomly generated topologies Probability that no feasible matching exists for a given maximum moving distance D.

31. dsds

32. Numerical results Hybrid Networks Cells with side length of d h = r/√2 N = λL static sensors, λ = 2πk M = ΛL mobiles M is selected so that there are exactly enough mobiles to fill all vacancies Moving distance D

33. k=10 d h =0.5 d s

34. Cells=900

35. Performance of Push-Relabel Algorithm Execution process is divided into rounds 10 3 randomly generated topologies Total number of messages Rounds

36. Conclusion Investigate the distance that a mobile sensor will have to move Mobile sensor networks Hybrid sensor networks Results prove that Mobility has significant advantages in providing coverage