1. Theoretical Results on Base Station Movement Problem for Sensor Network Yi Shi ( 石毅 ) and Y. Thomas Hou ( 侯一釗 ) Virginia Tech, Dept. of ECE IEEE Infocom 2008

2. Outline Introduction Problem Constrained Mobile Base Station (C-MB) Problem Unconstrained Mobile Base Station (U-MB) Problem Approach C-MB Problem Optimal Solution U-MB Problem (1-  )-approximate solution Numerical Results Conclusion

3. Introduction Sensor Networks Sensors: gather data, transmit and relay data packets Low computation power Battery power Small storage space Base station: data collector Network Lifetime The first time instance when any of the sensors runs out of energy. The first time instance when half of the sensors runs out of energy. The first time instance when the network connectivity is broken up. BS VCLAB ezLMS references: [1] [2]12

4. Problem Network Model The BS is movable Each sensor node i generates data at rate r i Data is transmitted to base station via multi-hop Initial energy at sensor node i is e i Energy Consumption Modeling Transmission power modeling where Receiving power modeling ij d ij : distance f ij : data rate i f ki : data rate

5. Unconstrained Mobile Base Station Problem (U-MB) Goal: Find an optimal moving path for the base station such that the network lifetime is maximized. Optimize base station location (x, y)(t) at any time t such that the network lifetime is maximized. Problem Formulation flow balance: i g ki incoming data rate riri outgoing data rate g ij Energy constraint The position of the BS  Time-dependent Network lifetime

6. Constrained Mobile Base Station Problem (U-MB) The base station is only allowed to be present at a finite set of pre-defined points. For example: ( x, y ) (t)  p 1, p 2, p 3, p 4, p 5 } p1p1 p2p2 p3p3 p4p4 p5p5 Goal: Find an optimal time-dependent location sequences such that the network lifetime is maximized. Time-dependent location sequences: t1p1t1p1 t2p4t2p4 t3p3t3p3 t4p2t4p2 tp1tp1 ……

7. The Roadmap of the Theoretical Analysis C-MB Problem 1. Transform the problem from time domain to space domain Theorem 1 2. Linear programming Optimal Solution U-MB Problem Change infinite search space to finite search points U-MB  C-MB (1-  )-approximate solution by solving C-MB on the finite search points Theorem 2 and 3 t1p1t1p1 t2p4t2p4 t3p3t3p3 t4p2t4p2 …… p1t1p1t1 p2t2p2t2 p3t3p3t3 p4t4p4t4

8. C-MB Problem From time-domain to space-domain Time Domain: [0, 50] p 1 [50, 90] p 2 [90, 100] p 2 [100, 130] p 1

9. C-MB Problem From time-domain to space-domain Space domain: p 1 [0, 50] + [100, 130] p 2 [50, 100]

10. From Time Domain to Space Domain (cont ’ d) Data routing only depends on base station location; not time Theorem The optimal location-dependent solution can achieve the same maximum network lifetime as that by the optimal time-dependent solution

11. Linear programming Formulation Location-dependent W(p): the cumulative time periods for the BS to be present at location p. f ki (p): normalized data rate

12. Linear programming Formulation Let

13. A ( 1 − ε ) Optimal Algorithm to the U-MB Problem Search Space Claim: Optimal base station movement must be within the Smallest Enclosing Disk (SED). Reference [19] SED The smallest disk that covers all sensor nodes Can be found in polynomial-time Still infinite search space!

14. A ( 1 − ε ) Optimal Algorithm Roadmap 1. Discretize transmission cost and distance with ( 1- Ɛ ) optimality guarantee Get a set of distance D[h] 2. Divide SED into subareas By the sequence of circles with radius D[h] at each sensor 3. Represent each subarea by a fictitious cost point (FCP) 4. Compute the optimal total sojourn time and routing topology for each FCP (or subarea) A linear program

15. Step 1: Discretize Transmission Cost and Distance

16. Discretize transmission cost in a geometric sequence, with a factor of (1+ Ɛ ) C[1] C[2] C[3] C[1]  c 4B  C[2] c 4B : the transmission cost between sensor I and the base station

17. Step 2: Division on SED SED is divided by the sequence of circles with radius D[h] and center sensor node i C[1]  c 1B  C[2] C[2]  c 2B  C[3] C[1]  c 3B  C[2] C[2]  c 4B  C[3]

18. Step 3: Represent Each Subarea by A Fictitious Cost Point (FCP) Define a FCP p m for each subarea A m N-tuple cost vector P m = (C[2], C[3], C[2], C[3]) Define C[1]  c 1B  C[2] C[2]  c 2B  C[3] C[1]  c 3B  C[2] C[2]  c 4B  C[3] PmPm

19. Step 3: Represent Each Subarea by A Fictitious Cost Point (FCP) Properties: A fictitious point p m is a virtual point, not a physical point in the space. The transmission cost from each sensor node i to p m is the worst case cost for all points in A m For any point p in this subarea, we have C iB (p)≤C iB (p m ) For any point p  A m, we have

20. Step 4: Finding a ( 1- Ɛ ) Optimal Solution Find the best total sojourn time W(p m ) and routing topology f ij (p m ) and f iB (p m ) for each FCP p m Solve a linear program (Linear programming for the C-MB problem) Base station should stay at each subarea A m for total W(p m ) of time Whenever base station is in subarea A m, routing topology should be f ij (p m ) and f iB (p m )

21. The (1-  ) Optimality (1) By Theorem 2 (2) By Theorem 3

22. Example  =0.2, O A =(0.61, 0.57), R A =0.51  =1,  =0.5,  =1 17 subareas A 1, A 2, …,A 17 1 2 3 Final solution: T =190.37 Stay A 3 for 157 time Stay A 6 for 33.37 time

23. Numerical Results Settings Randomly generated networks: 50 and 100 nodes in 1x1 area (all units are normalized) Data rate at each sensor randomly generated in [0.1, 1] Initial energy at each sensor randomly generated in [50, 500] Parameters in energy consumption model: α = β = ρ =1, n=2 Result The obtained network lifetime is at least 95% of the optimum, i.e., Ɛ is set to 0.05

24. Result – 50-node Network T ε =122.30

25. A Sample Base Station Movement Path Base station movement path is not unique Moving time (from one subarea to another) is much smaller than network lifetime Each sensor can buffer its data when base station is moving and transmit when base station arrives the next subarea Network lifetime will not change

26. Result – 100-node Network

27. Summary Investigated base station movement problem for sensor networks Developed a (1- Ɛ ) approximation algorithm with polynomial complexity Transform the problem from time domain to space domain Change infinite search space to finite search space with (1- Ɛ ) optimal guarantee Proved (1- Ɛ ) optimality

28. Proof of Theorem 1 Time domain Space domain C-MB

29. Proof of Theorem 1 Time domain Space domain C-MB Indicator function

30. Proof of Theorem 1

31. Proof of Lemma 1 (1) (2) (3)

32. Proof of Lemma 1 (1)

33. Proof of Lemma 1 (2)

34. Proof of Lemma 1 (3)

35. Proof of Theorem 2

36. C-MB U-MB

40. Proof of Theorem 3