1. Maximal Lifetime Scheduling in Sensor Surveillance Networks Hai Liu 1, Pengjun Wan 2, Chih-Wei Yi 2, Siaohua Jia 1, Sam Makki 3 and Niki Pissionou 4 Dept of Computer Science 1 City University of Hong Kong, 2 Illinois Institute of Technology Dept of Electrical Engineering & Computer Science 3 University of Toledo Telecommunications & Information Technology Institute 4 Florida 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 the stringent power budget of wireless sensor nodes.  It is important to prolong the lifetime of sensor networks.  In this paper, we discuss a scheduling problem in 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 E i = 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 x ij ： 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 x ij 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 z i, 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 X nxm by L and denote the new matrix by Y nxn, that is, y ij = x ij / L, for i,j =1,2,…,n.  For matrix Y nxn, we have :  Matrix Y nxn is a Doubly Stochastic Matrix.

12. A special case n=m   when n=m,workload matrix X nxm 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  X nxm is no longer a square matrix, the idea is to “fill” the matrix X nxm with dummy column to make it a doubly stochastic matrix of order n.  Let Z nx(n-m) be the dummy matrix. By appending the column of the dummy matrix to to the right hand side of X nxm.

14. General case n>m  To make matrix W nxn having the feature of (3) and (6), the dummy matrix Z nx(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>m FillMatrix Algorithm  If

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

18. General case n>m FillMatrix Algorithm   W nxn can be decomposed as ： simply denote c i L as c i, 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 c i 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. S T S1S1 S5S5 S2S2 S4S4 S3S3 T1T1 T3T3 T2T2 T4T4 T5T5 M ： { （ S 1,T 1 ） } M-path ： { （ S 2,T 1 ）（ T 1,S 1 ）（ S 1,T 2 ） } PerfectMatching algorithm M ： { （ S2,T1 ）（ S1,T2 ） }

23. DecomposeMatric Algorithm

24. 

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 W 6x6.

29. Experiment cont ’ Then, run the DecomposeMatrix Algorithm to decompose W 6x6 into a sequence of schedule matrices, such that W 6x6 =c 1 P 1 +c 2 P 2 +…+c 5 P 5 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.