1. 1 Sensor Placement and Lifetime of Wireless Sensor Networks: Theory and Performance Analysis Ekta Jain and Qilian Liang, Department of Electrical Engineering, University of Texas at Arlington IEEE GLOBECOM 2005

2. 2 Outline  Introduction  Preliminaries  Node Lifetime Evaluation  Network Lifetime Analysis Using Reliability Theory  Simulation  Conclusion

3. 3 Introduction (1/3)  Sensor networks have limited network lifetime.  energy-constrained  Most applications have pre-specified lifetime requirement.  Example: [4] has a requirement of at least 9 months  Estimation of lifetime becomes a necessity. [4] A. Mainwaring, J. Polastre, R. Szewczyk, D. Culler, J. Anderson, ” Wireless Sensor Networks for Habitat Monitoring ”

4. 4 Introduction (2/3)  Sensor Placement vs. Lifetime Estimation  Two basic placement schemes: Square Grid, Hex-Grid.  Bottom-up approach lifetime evaluation.  Theoretical Result vs. Actual Result  by extensive simulations

5. 5 Introduction (3/3)  Bottom-up approach to lifetime evaluation of a network. Lifetime Behavior Analysis (single sensor node) Lifetime Behavior Analysis (sensor networks using two basic placement schemes)

6. 6 Preliminaries Basic Model  r s : the sensing range  r c : the communication range  neighbors  distance of separation r ≤ r c rsrs r assume r s = r c

7. 7 Preliminaries Basic Model  The maximum distance between two neighboring nodes:  r max = r c = r s  A network is said to be deployed with minimum density when:  the distance between its neighboring nodes is r = r max

8. 8 Preliminaries Placement Schemes Placement Schemes 2-neighbor group 3-neighbor group4-neighbor group Hex-Grid Square Grid described in [1] [1] K. Kar, S. Banerjee, ” Node Placement for Connected Coverage in Sensor Networks ”

9. 9 Preliminaries Placement Scheme in Reference [1] 2-neighbor group and provides full coverage!! [1] K. Kar, S. Banerjee, ” Node Placement for Connected Coverage in Sensor Networks ”

10. 10 Preliminaries Placement Schemes  Square Grid  Hex-Grid

11. 11 Preliminaries Coverage and Connectivity  Various levels of coverage may be acceptable.  depends on the application requirement  In our analysis …  require the network to provide complete coverage  only 100% connectivity is acceptable  the network fails with loss of connectivity

12. 12 Preliminaries Lifetime  consider basic placement schemes Square- GridHex- Grid

13. 13 Preliminaries Lifetime  Tolerate the failure of a node all of whose neighbors are functioning.  Define minimum network lifetime as the time to failure of any two neighboring nodes.  i.e. the first loss of coverage

14. 14 Node Lifetime Evaluation (1/5)  A sensor node is said to have:  m possible modes of operation  at any given time, the node is in one of these m nodes  w i : fraction of time that a node spends in i-th mode 12m …… w1w1 w2w2 wmwm

15. 15 Node Lifetime Evaluation (2/5)  W i are modeled as random variables.  take values from 0 to 1  probability density function (pdf)  E total : total energy  P i : power spended in the i-th mode per unit time  T node : lifetime of the node  E th : threshold energy value

16. 16 Node Lifetime Evaluation (3/5)  The lifetime of a single node can be represented as a random variable.  takes different values by its probability density function (pdf), ft (t)

17. 17 Node Lifetime Evaluation (4/5)  Assume that the node has two modes of operation.  Active: Pr (node is active) = p, w 1  Idle: Pr (node is idle) = 1-p, w 2 = 1- w 1  Observe the node over T time units.  binomial distribution

18. 18 Node Lifetime Evaluation (5/5)  As T becomes large:  binomial distribution ~ N(μ, σ)  μ(mean) = Tp, σ(variance) = Tp(1-p)  The fraction of time ( w 1 and w 2 ) follows the normal distribution.  The reciprocal of the lifetime of a node is normally distributed.

19. 19 Network Lifetime Analysis Reliability Theory  The network lifetime is also a random variable.  Using Reliability Theory to find the distribution of the network lifetime.

20. 20 Reliability Theory  Concerned with the duration of the useful life of components and systems.  We model the lifetime as a continuous non-negative variable T.  pdf, cdf, Survivor Function, System Reliability, RBD.

21. 21 Reliability Theory pdf and cdf  Probability Density Function  f(t): the probability of the random variable taking a certain value  Cumulative Distribution Function  F(t): the proportion of the entire population that fails by time t.

22. 22 Reliability Theory Survivor Function  Survivor Function: S(t)  the probability that a unit is functioning at any time t  survivor function vs. pdf S(0) = 1, S(t) is non-decreasing

23. 23 Reliability Theory System Reliability  To consider the relationship between components in the system.  using RBD distribution of the components distribution of the system single node entire network

24. 24 Reliability Theory Reliability Block Diagram (RBD)  Any complex system can be realized in the form of combination blocks, connected in series and parallel.  S 1 (t) and S 2 (t) are the survivor functions of two components. S1(t) S2(t) S1(t) S2(t)

25. 25 Network Lifetime Analysis  minimum network lifetime: the time to failure of two adjacent nodes  Assume that:  All sensor nodes have the same survivor function.  Each sensor node fails independent of one another.

26. 26 Network Lifetime Analysis Square Grid  Square Grid Placement Analysis Region 1 Region 2 Region 1 a b c d Region 2 x y xy or

27. 27 Network Lifetime Analysis Square Grid a b c d a b c Region 1 Block 1 : RBD for Region 1 ∵ sensors are identical

28. 28 Network Lifetime Analysis Square Grid or x x y y x y Region 2 Block 2 : RBD for Region 2 ∵ sensors are identical, have the same survivor function

29. 29 Network Lifetime Analysis Network Survivor Function for Square Grid  block 1 ’ s  block 2 ’ s  connect in series

30. 30 Network Lifetime Analysis Hex-Grid  Hex-Grid Placement Analysis Block : RBD for Hex-Grid a b c d a b c d ∵ sensors are identical, have the same survivor function

31. 31 Network Lifetime Analysis Network Survivor Function for Hex-Grid  blocks connect in series. Why ?

32. 32 Simulation Flow Chart Survivor Function (single node) Survivor Function (network) p.d.f. (network) p.d.f. (single node) theoretical vs. actual Network Lifetime Analysis Node Lifetime Analysis Given Network Protocol Distribution of W i Node Lifetime Calculation p.d.f. (single node) theoretical vs. actual

33. 33 Simulation Node Lifetime Distribution theoretical p.d.f. actual p.d.f.

34. 34 Simulation Network Lifetime Distribution  Square Grid Placement Scheme theoretical p.d.f. actual p.d.f. closely match!

35. 35 Simulation Network Lifetime Distribution  Hex-Grid Placement Scheme theoretical p.d.f. actual p.d.f. closely match!

36. 36 Conclusion  The analytical results based on the application of Reliability Theory.  We came up not with any particular value, but a p.d.f. for minimum network lifetime.  The theoretical results and the methodology used will enable analysis of:  other sensor placement scheme  tradeoff between lifetime and cost  performance of energy efficiency algorithm