1. CS Dept, City Univ.1 Maximal Lifetime Scheduling for Wireless Sensor Surveillance Networks Prof. Xiaohua Jia Dept. of Computer Science City University of Hong Kong

2. CS Dept, City Univ.2 Wireless Sensor Networks A sensor node has three basic components: A sensor network consists of many low-cost and low- powered sensor devices. A wireless sensor node has three basic components: A processorA processor A set of radio communication devicesA set of radio communication devices Sensing devicesSensing devices

3. CS Dept, City Univ.3 Maximum Lifetime Target Surveillance Systems Given a set of sensors to watch a set of targets: Each sensor has a given energy reserve. It can watch at most one target at a time. A target can be inside several sensors’ watching range. It should be watched by at least one sensor at any time. find a schedule for sensors to watch the targets in turn, such that the lifetime is maximized. Problem: find a schedule for sensors to watch the targets in turn, such that the lifetime is maximized. Lifetime is the duration up to when a target can no longer be watched by any sensor.

4. CS Dept, City Univ.4 Solving the Maximum Lifetime Problem Our solution consists of three steps: 1) compute the upper bound of the maximal lifetime and a workload matrix of sensors. 2) decompose the workload matrix into a sequence of schedule matrices. 3) obtain a target watching timetable for each sensor.

5. CS Dept, City Univ.5 Finding Maximum Lifetime S / T = set of sensors / targets, n=|S|, m=|T|. E i = initial energy reserve of sensor i. S(j) = set of sensors able to watch target j. T(i) = set of targets within watching range of sensor i. x ij : the total time sensor i watching target j. Objective: Max L (1) (2) }, min{ )( i iT j ij ELx    Lx jSi    )(

6. CS Dept, City Univ.6 The Workload Matrix X n×m is a workload matrix, specifying the total time a sensor watching a target: the sum of all elements in each column is equal to L (from eq. (1) in the LP formulation). the sum of all elements in each row is less than or equal to L (from ineq. (2) in the LP formulation). X n×m = targets sensors

7. CS Dept, City Univ.7 Decompose Workload Matrix into a Sequence of Scheduling Matrices A scheduling matrix specifies the schedule of sensors to watch targets during a session: only one non-zero number in each column (i.e., a target is watched by only one sensor during the session). at most one non-zero number in each row (i.e., a sensor can watch at most one target at a time and there is no switching in a session). all non-zero elements having the same value, which is the duration of the session.

8. CS Dept, City Univ.8 A Special Case of n=m When n = m, we have: R i = C j = L, for 1 ≤i, j ≤n. (R i : sum of row i, C j : sum of column j). Because: The workload matrix X n×n can be represented as: X n×n = L ×Y n×n Y n×n is a Doubly Stochastic Matrix. The sum of each row and each column is equal to 1. and L L

9. CS Dept, City Univ.9 A Special Case of n=m (cont ’ d) Theorem 1. Matrix Y n×n can be decomposed as: Y n×n = c 1 P 1 + c 2 P 2 + … + c t P t, where t≤(n - 1) 2 +1, each P i, 1≤i≤t, is a permutation matrix; and c 1, c 2, …, c t, are positive real numbers and c 1 +c 2 + … +c t =1.

10. CS Dept, City Univ.10 Convert to Perfect Matching 1) Represent X n×m as a bipartite graph, with x ij as edge weight. 2) Compute a perfect matching in the graph. Let c i be the smallest weight of the n edges in the matching. 3) Deduct c i from the weight of the n matching-edges and remove the edges whose weight is zero. 4) Repeat step 2) & 3) until there is no edge in the bipartite graph.

11. CS Dept, City Univ.11 Complete Decomposibility Does there exist a perfect matching in every round of the decomposition process? Theorem 5. For any square matrix W n×n of nonnegative numbers, if R i = C j for 1 ≤i, j ≤n, there exists a perfect matching in the corresponding bipartite graph. The workload matrix can be exactly decomposed into a sequence of schedule matrices!

12. CS Dept, City Univ.12 General case of n>m Fill matrix X n×m with dummy columns to transform to the case of n = m: nn mnnnnnmnn mnm mnm nn zzzxxx zzxxx zzxxx W ×    ×           ............z z 2121 2222122221 1121111211

13. CS Dept, City Univ.13 Fill Matrix Record the remaining numbers of row-sums and column-sums. Determine dummy matrix Z n×(n-m) from z 11. Assign z ij to the largest possible number without violating the above two constraints R ’ i and C ’ j. nn mnnnnnmnn mnm mnm nn zzzxxx zzxxx zzxxx W ×    ×           ............z z 2121 2222122221 1121111211 L L

14. CS Dept, City Univ.14 An example for filling matrix 47674767 8 8 84 8 8 07670767 03670367 0 8 8

15. CS Dept, City Univ.15 DecomposeMatrix Algorithm Input: workload matrix X n×m. Output: a sequence of schedule matrices. Begin if n>m then Fill matrix X n×m to obtain a square matrix W n×n ; Construct a bipartite graph G from W n×n ; while there exist edges in G do Find a perfect matching M (i.e., P i ) on G; Let c i be smallest weight in M; Deduct c i from all edges in M and remove edges with weight 0; endwhile Output W n×n = c 1 P 1 + c 2 P 2 + … + c t P t ; End

16. CS Dept, City Univ.16 A Walkthrough Example Sensors123 EiEi 15.692634.262724.8717 Sensors456 EiEi 21.784746.686534.5310 6 sensors (clear color) and 3 targets (grey color) Tab. 1. Energy reserve of sensors

17. CS Dept, City Univ.17 Compute the LP formulation L = 40.5643 Workload matrix:

18. CS Dept, City Univ.18 Fill X n×m to a square matrix

19. CS Dept, City Univ.19 Decompose the workload matrix W 6×6 = c 1 P 1 + c 2 P 2 + … + c 5 P 5. By removing the dummy columns, we have:

20. CS Dept, City Univ.20 Obtain scheduling timetable for sensors SensorsWatching Duty (time duration and watching targets) 1 0~4.0936 Target 1 4.0936~28.9653 Turn off 28.8953~40.5643 Target 1 2 0~10.2454 Target 2 10.2454~28.9653 Target 3 28.8953~40.5643 Turn off 3 0~4.0936 Turn off 4.0936~28.9653 Target 1 28.8953~40.5643 Turn off 4 0~10.2454 Turn off 10.2454~16.5589 Target 2 16.5589~28.8953 Turn off 28.8953~40.5643 Target 2 5 0~10.2454 Target 3 10.2454~16.5589 Turn off 16.5589~28.8953 Target 2 28.8953~40.5643 Target 3 6 0~40.5643 Turn off Tab. 2. The schedule timetable for 6 sensors

21. CS Dept, City Univ.21 Simulation Results Fig. 2(a). t versus N when M=10Fig. 2(b). t versus M when N=100

22. CS Dept, City Univ.22 Simulation Results Fig. 3(a). Lifetime versus surveillance rangeFig. 3(b). Lifetime versus N when M=10

23. CS Dept, City Univ.23 Summary Discussed the maximal lifetime scheduling problem in sensor surveillance networks. Proposed an optimal solution to the max lifetime scheduling problem. The number of decomposition steps for finding the optimal schedule is linear to the network size.

24. CS Dept, City Univ.24 Thank You !