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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 1 Outline Introduction Preliminaries Node Lifetime Evaluation Network Lifetime Analysis Using Reliability Theory Simulation Conclusion 2 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” 3 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 4 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) 5 Preliminaries Basic Model rs : the sensing range assume rs = rc rc : the communication range neighbors distance of separation r ≤ rc rs r 6 Preliminaries Basic Model The maximum distance between two neighboring nodes: rmax = rc = rs A network is said to be deployed with minimum density when: the distance between its neighboring nodes is r = rmax 7 Preliminaries Placement Schemes Placement Schemes 2-neighbor group 3-neighbor group 4-neighbor group described in [1] Hex-Grid Square Grid [1] K. Kar, S. Banerjee, ”Node Placement for Connected Coverage in Sensor Networks” 8 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” 9 Preliminaries Placement Schemes Square Grid Hex-Grid 10 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 11 Preliminaries Lifetime consider basic placement schemes Square- Grid Hex- Grid 12 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 13 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 wi : fraction of time that a node spends in i-th mode w i i 1 i 1,2...m 1 2 …… m w1 w2 …… wm 14 Node Lifetime Evaluation (2/5) Wi are modeled as random variables. take values from 0 to 1 probability density function (pdf) Etotal: total energy Pi: power spended in the i-th mode per unit time Tnode: lifetime of the node Eth: threshold energy value E total E total - w i Pi Tnode E th Tnode i w i Pi i 15 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) Etotal Tnode i wi Pi 16 Node Lifetime Evaluation (4/5) Assume that the node has two modes of operation. Active: Pr (node is active) = p, w1 Idle: Pr (node is idle) = 1-p, w2 = 1- w1 Observe the node over T time units. binomial distribution P(w 1 x) C T p x (1 - p) T- x x 17 Node Lifetime Evaluation (5/5) As T becomes large: 2 binomial distribution ~ N(μ, σ) μ(mean) = Tp, σ(variance) = Tp(1-p) 2 The fraction of time (w1 and w2) follows the normal distribution. The reciprocal of the lifetime of a node is normally distributed. 18 Network Lifetime Analysis Reliability Theory The network lifetime is also a random variable. Using Reliability Theory to find the distribution of the network lifetime. 19 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. 20 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. t F(t) f(s)ds 0 21 Reliability Theory Survivor Function Survivor Function: S(t) the probability that a unit is functioning at any time t S(0) = 1, S(t) P [T t] t 0 lim t S(t) 0, S(t) is non-decreasing survivor function vs. pdf t S(t) 1 - F(t) 1 - f(s)ds 0 22 Reliability Theory System Reliability distribution of the components single node distribution of the system entire network To consider the relationship between components in the system. using RBD 23 Reliability Theory Reliability Block Diagram (RBD) Any complex system can be realized in the form of combination blocks, connected in series and parallel. S1(t) and S2(t) are the survivor functions of two components. S1(t) S1(t) S2(t) S2(t) Sseries (t) S1 (t)S 2 (t) S parallel(t) 1 - [(1 - S1 (t))(1 - S 2 (t))] 24 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. 25 Network Lifetime Analysis Square Grid Square Grid Placement Analysis Region 1 Region 1 a b c d Region 2 x x y or y Region 2 26 Network Lifetime Analysis Square Grid Region 1 Block 1 : RBD for Region 1 a b a c d b c s block1 1 - (1 - s a )(1 - s bs c ) ∵ sensors are identical s block1 1 - (1 - s)(1 - s 2 ) s s 2 - s 3 27 Network Lifetime Analysis Square Grid Region 2 Block 2 : RBD for Region 2 x x y x y or y s block2 1 - (1 - s x )(1 - s y ) ∵ sensors are identical, have the same survivor function s block2 1 - (1 - s)(1 - s) 2s - s 2 28 Network Lifetime Analysis Network Survivor Function for Square Grid Nmin - 1 ( N min - 1 ) 2 block 1’s block 2’s 2 * ( Nmin - 1 ) connect in series Nmin - 1 ( Nmin - 1) 2 s network (sblock1 ) 2( N min - 1) (sblock 2 ) 29 Network Lifetime Analysis Hex-Grid Hex-Grid Placement Analysis Block : RBD for Hex-Grid b a a c d b c d s block 1 - (1 - s a )(1 - s bs cs d ) ∵ sensors are identical, have the same survivor function s block 1 - (1 - s)(1 - s 3 ) 30 Network Lifetime Analysis Network Survivor Function for Hex-Grid N blocks connect in series. 2 N N Why 2 ? s network (s block ) 2 31 Simulation Flow Chart Node Lifetime Analysis Network Lifetime Analysis Given Network Protocol p.d.f. (single node) Distribution of Wi Survivor Function (single node) Node Lifetime Calculation Survivor Function (network) p.d.f. (single node) p.d.f. (network) theoretical vs. actual theoretical vs. actual 32 Simulation Node Lifetime Distribution theoretical p.d.f. actual p.d.f. 33 Simulation Network Lifetime Distribution Square Grid Placement Scheme theoretical p.d.f. actual p.d.f. closely match! 34 Simulation Network Lifetime Distribution Hex-Grid Placement Scheme theoretical p.d.f. actual p.d.f. closely match! 35 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 36