Maximal Lifetime Scheduling in Sensor Surveillance Networks

1. Maximal Lifetime Scheduling in Sensor Surveillance Networks Hai Liu1, Pengjun Wan2, Chih-Wei Yi2, Siaohua Jia1,Sam Makki3 and Niki Pissionou4 Dept of Computer Science 1City University of Hong Kong, 2Illinois Institute of Technology Dept of Electrical Engineering & Computer Science 3University of Toledo Telecommunications & Information Technology Institute 4Florida International University Infocom 2005

2. Outline • Introduction • System model and problem statement • Solutions • Experiment and simulation • Conclusions

3. Introduction • One important characteristic of sensor networks is thestringent power budgetof wireless sensor nodes. • It is importantto prolong the lifetimeof sensor networks. • In this paper, we discussa scheduling problemin sensor surveillance networks. • Sensor surveillance networks • Given a set of sensors and targets in a Euclidean plane, all targets should be watched by sensors at any time • A sensor can watch only one target at a time

4. System model and problem statement • Notations： • S = the set of sensors • T = the set of targets • n = |S| the number of sensors • m = |T| the number of sensors • S(j) = the set of sensors that are able to watch target j, j=1,…m • T(i) = the set of targets that are within the surveillance range of sensor i, i=1,…n • Ei = initial energy reserve of sensor i, i=1,…n

5. System model and problem statement • There 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 anytime. • The problem is to find a schedule that meets the above two requirements for sensors watching targets, such that lifetime is maximized. • Lifetime is defined as the length of time until there exists a target j such that all sensors in S(j) run out their energy.

6. Solutions • Tackle the problem in three steps. • Step1, compute the upper bound on the maximal lifetime of the system and a workload matrix of sensors. • Step2, decompose the workload matrix into a sequence of schedule matrices. • Step3, obtain a target watching timetable for each sensor.

7. Find maximal lifetime • Use linear programming technique to find the maximum lifetime of the system. • L：the lifetime of the surveillance system • xij：the total time sensor i watching target j • The maximum lifetime for sensors watching targets can be formulated：

8. Find maximal lifetime • Find a schedule for each sensor • The value of xij can be represented as a workload matrix： • There two important features about this matrix： • the sum of all elements in each column is equal to L（from eq.(1)） • the sum of all elements in each row is less than or equal to L (from ieq(2))

9. Decompose workload matrix • The lifetime can be divided into of a sequence of sessions. Thus the schedule of sensors during a session can be represented as a matrix. • There is only one positive number in each column, and at most one positive number in each row. • All non-zero elements in this matrix have the same value. • Decompose the workload matrix into a sequence of session schedule matrices： where zi , i=1,2,…,t ,is the length of time of session i.

10. A special case n=m • according eq.(1) and ineq.(2) • we have： • combining (4) and (5), we have:

11. A special case n=m • (3) and (6) imply that the workload matrix the sum of each column is the same as the sum of each row, all equal to L. • divide the workload matrix Xnxm by L and denote the new matrix by Ynxn , that is, yij = xij / L, for i,j =1,2,…,n . • For matrix Ynxn , we have : • Matrix Ynxn is a Doubly Stochastic Matrix.

12. A special case n=m • when n=m ,workload matrix Xnxm can be decomposed into a sequence of matrices： (theorem 3 in [14] T. Inukai, “An Efficient SS/TDMA Time Slot Assignment Algorithm” Algorithm”, IEEE Trans.)

13. General case n>m • Xnxm is no longer a square matrix, the idea is to “fill” the matrix Xnxm with dummy column to make it a doubly stochastic matrix of order n. • Let Znx(n-m) be the dummy matrix. By appending the column of the dummy matrix to to the right hand side of Xnxm .

14. General case n>m • To make matrix Wnxn having the feature of (3) and (6), the dummy matrix Znx(n-m) should satisfy the following conditions： • Let record the sum of the remaining undetermined element of row i and column j ,for i=1,…,n and j= 1,…,n-m • Initially,

15. General case n>m • FillMatrix Algorithm

16. General case n>mFillMatrix Algorithm • If • If

17. General case n>mFillMatrix Algorithm • If ,we can determine elements in both row I and in column j by .

18. General case n>mFillMatrix Algorithm • Wnxn can be decomposed as： simply denote ciL as ci , Proof

19. General case n>m • Let denote the matrix which contains the first m columns in ,i=1,…,t , we have • The matrices ,i=1,…,t, are the schedule matrices • In session i, sensors are scheduled to watch their respective targets according to the position of “1” elements in for the period of ci time. • Workload matrix is decomposable to a sequence of schedule matrices such that the optimal lifetime can be achieved.

20. Algorithm for decomposing workload matrix • The basic idea of the algorithm is to represent the filled workload matrix as a bipartite graph where one side are sensors and other are targets. • Decomposing the filled workload matrix is transformed into the problem of finding perfect matchings in a bipartite graph. Proof

21. PerfectMatching algorithm • M denote a set of edges of a perfect matching. • M-path is a path in the bipartite graph. It starts with a S node that not in M and end with a T node that is also not in M. • By replacing M-edges in the M-path by the non M-edge, the number of edges in M is incremented by 1. • Finding the M-path and increasing the size of M, until a perfect matching is found.

22. PerfectMatching algorithm S1 M：{（ S1,T 1）} T1 S2 M-path： {（S2,T1）（T1,S1）（S1,T2）} T2 S3 T3 S4 T4 M： {（S2,T1）（S1,T2）} S5 T5 S T

23. DecomposeMatric Algorithm

24. DecomposeMatric Algorithm

25. Obtain schedule timetable • Simply take the i-th row of all the schedule matrices, and combine the time of the consecutive sessions that it watches the same target. • Then, we have an independent timetable for each sensor.

26. Experiment and Simulation • Experiment • Place 6 sensors and 3 targets in a 50x50 two-dimensional free-space region. • The survelliance range is set to 20.(the solution can work for any system with non-uniform surveillance range) • The initial energy reserves of sensors are random number generated in the range of [0,50] with the mean at 25.

27. Experiment cont’ • Step1, using the linear programming to compute the maximum lifetime and the workload matrix. L：40.5643 hr.

28. Experiment cont’ • Step2, run the FillMatrix Algorithm ,to append a dummy matrix to the workload to make it a square matrix W6x6 .

29. Experiment cont’ • Then, run the DecomposeMatrix Algorithm to decompose W6x6 into a sequence of schedule matrices , such that W6x6 =c1P1+c2P2+…+c5P5 • By removing the dummy columns of the schedule matrices, we have:

30. Experiment cont’ • Finally, obtain target watch timetables for sensors based on the above schedule matrix.

31. Simulations • Growth of decomposition steps is linear • ⇒the actual number of steps for decomposing the matrix is linear to the size of system in real runs.

32. Simulations cont’ • Comparison with a greedy method（surveillance range） • Set N=100 and M=10

33. Simulations cont’ • Comparison with a greedy method（sensor density）

34. Conclusions • Solution consists of three steps: • compute the maximal lifetime of the system and a workload matrix by using linear programming method. • decompose the workload matrix into a sequence of schedule matrices by using perfect matching method. • obtain target watching timetable for sensors. • The solution is the optimum in the sense that it can find the schedules that achieve the maximum lifetime. • the steps of decomposition is linear to the size of system. • This method can take more advantages in the situtation that senses are densely deployed or sensors have larger coverage ranges.

35. Theorem 3

38. Theorem 5 konig

39. Konig theorem • The number of edges in a maximum matching of a bipartite graph G=(X,Y,E) is equal to |X|-σ(G), where σ(G) is the deficiency of G.