1. 1 Maximizing Lifetime of Sensor Surveillance Systems IEEE/ACM TRANSACTIONS ON NETWORKING Authors: Hai Liu, Xiaohua Jia, Peng-Jun Wan, Chih- Wei Yi, S. Kami Makki, and Niki Pissinou

2. 2 Outline Introductions System Model and Problem Statement Our Solutions –Find Maximal Lifetime –Decompose Workload Matrix –Determine Surveillance Tree Experiments and Simulations Conclusion

3. 3 Introductions Given a set of targets, a set of sensors (at most watch one target at a time) are used to watch the targets and collect sensed data to the BS. Lifetime –Duration until one target can no longer be watched by any sensor or data can’t be forward to the BS. Problems –Schedule a subnet of sensors –Find the routes for the active sensor to send data back to BS

4. 4 Introductions BS sensor target

5. 5 System Model and Problem Statement B base station S set of sensors, n = T set of targets, m = S(j) set of sensors that can watch target j T(i) set of targets that are within the surveillance range of sensor i N(i) set of neighbors of sensor i initial energy of sensor i distance between sensor i and j energy for transmitting and receiving one unit data energy for watching a target per unit time R data rate generated from sensors while watching targets

6. 6 System Model and Problem Statement S(1) = {S1, S2, S3} T(1) = {T1, T2, T3} s1 s2 s3 T1 s1 T1 T2 T3

7. 7 System Model and Problem Statement Two requirements for sensors watching targets –Each sensor can watch at most one target at a time –Each target should be watched by one sensor at any time The problem is to find a schedule that meets the above two requirements for sensors watching targets, such that the lifetime of network is maximized.

8. 8 Our Solutions Find Maximal Lifetime Linear Programming (LP) – total time sensor i watching target j – amount of data transmitted from sensor i to sensor j

9. 9 Our Solutions Find Maximal Lifetime We call matrix the workload matrix –The sum of column is equal to L (each column) –The sum of row is less than or equal to L (each row)

10. 10 Our Solutions Decompose Workload Matrix are schedule matrices –Elements are either “0” or –Each column has exactly one non-zero element –Each row has at most one non-zero element The number of sensors is grater than or equal the number of targets ( n >= m) – n = m – n > m

11. 11 Our Solutions Special Case n = m and denotes the sum of row I and the sum of column j in workload matrix ∵ and ∴ =>

12. 12 Our Solutions Special Case n = m Bipartite graph –Left hand side : sensors –Right hand side : targets –Edges : Since n = m, every sensor has a target to watch in each session Find perfect matchings… S1 S2 Sn T1 T2 Tm ……….

13. 13 Our Solutions Special Case n = m

14. 14 Our Solutions Special Case n = m s1 s2 s3 t1 t2 t3 1 2 2 1 1 1 1 1 2 1 1 1 1 1 1 G P1 G ……

15. 15 Our Solutions Special Case n = m Lema1: For of nonnegative real numbers, if = for 1 i,j n, =>exist perfect matching Proof: – –There doesn’t exist n positive entries in that no two entries in the same column or row. –By Konig theorem, we can cover all the positive entries in the matrix with e rows and f columns, such that e+f<n –The sum of all lines of is equal to 1, n e+f <= (L is sum of all elements in a row)

16. 16 Our Solutions Special Case n = m Theorem 1: The DecomposeMatrix-nn algorithm can always find a perfect matching –From lemma 1 Theorem 2: The time complexity of DecomposeMatrix-nn algorithm is O( ), where is the number of non-zero elements in –At most number of rounds to remove all edges in G –Find a perfect matching is O( )

17. 17 Our Solutions Special Case n > m Let be the dummy matrix

18. 18 Our Solutions Special Case n > m

19. 19 Our Solutions Special Case n > m =3, =5 =3, =2 =5, =3 =3, =2 = =3 =3 Let and record the sum of remaining undetermined elements of row i and column j

20. 20 Our Solutions Special Case n > m Theorem 3: the FillMatrix Algorithm can compute the filled matrix –Given row sums and column sums of a matrix –Induction method –1. n=1, m=1, since, we have –2. when n p-1, m q-1, can compute –3. when n=p, m=q, we first compare with A. =, =>, according 2. B. >, =>, monotonously decreases after each round and, there must exist in round l, =>, according 2. C similar to B.

21. 21 Our Solutions Special Case n > m Let denotes the matrix contains the first m columns in

22. 22 Our Solutions Special Case n > m Theorem 4: The time complexity of FillMatrix Algorithm is O( ) Theorem 5: The time complexity of DecomposeMatrix algorithm is O( )

23. 23 Our Solutions Determine Surveillance Tree Root : BS Leaf nodes : active sensors Suppose sensor i has l downstream nodes (i.e. are non-zero), let sensor i forward its outgoing data first to until is saturated, then switch to until the value of is met, and finally forward the last flow to.

24. 24 Experiments and Simulations Numeric Example 10x10 region Surveillance range: 0.4 * 10 Maximal transmission range: 0.8*10 Initial energy is randomly generated form [0, 100] with mean at 50 =0.12, =0.1 =0.1 R=1 α=2

25. 25 Experiments and Simulations Numeric Example

26. 26 Experiments and Simulations Numeric Example

27. 27 Experiments and Simulations Simulations 1) Linear Growth of Decomposition Steps –

28. 28 Experiments and Simulations Simulations 2) Comparison with Greedy Method: –Use the maximum matching algorithm in the sensor-target bipartite graph to find the pairs of sensor and target –For each active sensor we find the minimal energy cost path from it to the BS.

29. 29 Experiments and Simulations Simulations N=100, M=10

30. 30 Experiments and Simulations Simulations The number of steps for decomposing the workload matrix is linear to the size of the system Our algorithm has better performance when –Large surveillance range –Large transmission range –Sensors are density deployed The increase of surveillance range is more effective than the increase of the maximal transmission range M=50

31. 31 Conclusion We have presented the maximal lifetime scheduling problem in sensor surveillance systems. This is the first time in the literature that the problem of maximizing lifetime of sensor surveillance systems was formulated and the optimal solution was obtained.