A Minimum Cost Heterogeneous Sensor Network with a Lifetime Constraint

1. A Minimum Cost Heterogeneous Sensor Network with a Lifetime Constraint Vivek P. Mhatre, Catherine Rosenberg, Daniel Kofman, Ravi Mazumdar and Ness Shroff IEEE Transactions on Mobile Computing, 2005 Presented by Manu Shukla Virginia Tech CS 6204 - Fall 2006

2. Outline • Introduction • Previous work • Problem • Solution to random deployment scenario • Solution to grid deployment scenario • Numerical Results • Conclusions

3. Introduction • Sensor networks are dense low cost network of wireless nodes • Sense certain phenomenon in the area of interest and report observations to a central base station • In the paper authors study a scenario where an aircraft or a LEO satellite passes over an area periodically and collects updates from deployed nodes • Nodes are organized as clusters and cluster heads aggregate the data

4. Consider a heterogeneous network with two types of nodes, type 0 deployed with intensity λ0 and battery energy E0 and type 1 with intensity λ1 and energy E1 • Type 0 nodes do basic sensing as well as relaying of packets • Type 1 nodes are the cluster heads that do data aggregation and transmit to the aircraft • Type 1 nodes have more complex hardware • Every visit of the aircraft triggers a sensing and data gathering cycle

6. Objective is to determine the optimum node deployment parameters that will ensure a certain minimum number of data gathering cycles before sensor nodes become unusable • Each type of node has a cost function associated with it that take into account its hardware and battery life • Minimize overall network cost • Reduce waste of residual energy • Study two deployment scenarios, grid and random deployment and obtain results for λ0, E0, λ1 and E1

7. Previous Work • In authors approach, they observe that energy drainage is not uniform over entire network • Cluster heads and nodes close to them have highest energy burden • Focus on heterogeneous networks unlike previous work • Authors take into account conditions for connectivity from previous work “Unreliable Sensor Grids: Coverage, Connectivity and Diameter”, IEEE INFOCOM ‘03 • Minimize overall network cost, not just battery energy • Assume reliable nodes but extend for unreliable nodes

8. Problem • Deploy more nodes over regions of frequent updates • Redundant nodes stay inactive and save battery • Join cluster when other nodes start to expire • Nodes can be deployed two ways • Nodes are thrown from an aircraft and can be modeled using a two-dimensional homogeneous Poisson point process for each type of nodes • Nodes a deterministically placed along grid points

9. In random deployment, clustering leads to the formation of Voronoi cells with type 1 nodes being the nuclei of these cells • In grid deployment, topology is λ1 equally spaced type 1 nodes and λ0 equally spaced type 0 nodes along grid points • Cost per node is C0 and C1 for each type of node • Simple model for a cost function is Ci=αi+βEi where α and β are constants that depend on the manufacturing process

10. The overall cost of the network as function of is • For sensor network, necessary that conditions for node connectivity and area coverage be met • For the case of deployment over unit area with two-dimensional homogeneous Poisson point process • Sensing radius of each node is r • r is also critical distance between two nodes for successful transmission • r depends on allowable signal to noise ratio for successful packet reception, modulation scheme, propagation loss exponent etc. • Probability of connectedness of nodes and coverage of area is where λ0 and λ1 are intensity of type 0 and type 1 nodes

11. Probability Equation • Lemma • Use bin-packing argument where a square of unit area with circles of radii θr(λ) which are shifted by γr(λ) where γ+2θ = 1 • Probability that there is at least one active node in each circle

12. Coverage and Connectivity

13. is the probability that there are no active nodes in circle of area x • If all k nodes fail independently of each other • For Ps(λ) • For n nodes deployed along grid points

14. Problem Contd… • In a network dimensioning problem, designers provide parameter ε such that probability of connectivity and coverage be at least 1-ε. We require • Minimizing above equation as function of θ under constraint γ+2θ=1 when εr2 < 1 • The constraint in above equation reduces to λ0+λ1≥ u(θ0) = a where a is dependent on ε, p, r • In the grid case, required number of nodes is λ0 + λ1,and connectivity coverage requirement for a unit area takes the simple form

15. Lifetime of the system is the number of cycles until all the cluster heads as well as all the critical nodes are active • Can not ensure sharp cutoff due to inherent non-uniform nature of energy drainage in cluster • type 0 nodes near periphery of cluster have little relaying to do • best is try to ensure cluster heads and critical nodes expire at same time • P0 is the average energy spent by a typical critical node and P1 by typical cluster head in each cycle and E0/P0 is average number of cycles that critical nodes can sustain, • to ensure lifetime of at least T cycles, we require

16. P0 consists of • relaying packets to other nodes that are in the same cell (P0r per packet) and • transmitting ones own data (E0t per packet) • P1 consists of energy spent on • receiving data from other nodes in the cell (Er0 per packet), • processing and compressing the received data (Ef per packet) and • transmitting the compressed data to the aircraft (Et1 per packet) • Assume radio model wherein the energy required to transmit a packet over distance x is l+μxk • μxk is the energy spent in the RF amplifier to counter propagation loss • Cluster heads coordinate MAC and routing in cluster • E[Nv] expected number of type 0 nodes in a typical cluster

17. Like to determine parameters of the minimum cost network • Guaranteed lifetime of T cycles • Ensuring connectivity and coverage with probability 1-ε • Have fallowing optimization problem for random deployment scenario • For grid deployment, the problem formulation is similar to

18. Solution for Random Deployment Scenario • First determine an expression of P0r • Find expected number of critical nodes in a typical Voronoi cell • Find expected number of type 0 nodes outside circle of radius r around type 1 node • E[Nv] is the expected number of type 0 nodes in cell C0 (using Campbell's theorem and Slivnyak’s theorem)

19. Using equivalence with event of a point of type 0 in a small area xdxdθ located at (x, θ) and there is no other point of type 1 in a circle or radius x gives • Expected number of type 0 nodes located within distance r from type 1 node • Average relaying load on a typical critical node (Pr0) is • From E[Nv] we derive values of P0 and P1

20. Combining equations we have E1P0-E0P1=0 which gives us • Rewriting coverage constraint • We also have E1≥TP1 and inequality constraint on T • From cost equation

21. We get optimization problem • This is a standard optimization equation and can be solved by Karush-Kuhn-Tucker Theorem • Exact solution can be obtained by numerically solving equation for λ1

22. Solution to minimization problem • Solve the minimization problem by solving with μ0, μ1 and μ2 constants of the KKT theorem • From previous equations we have

23. From substitutions we have • Assuming that a feasible solutions exists, i.e. λ0, λ1, E0, E1 > 0 we get μ0 and μ2

24. From KKT • Using μ0 • Reformulate optimization problem as • We obtain as function of single variable λ1 • Rewriting a from previous equations

25. Since λ0=a-λ1 • Since we get E1 • Since we get E0 and f(λ1) • Local extremum of f(.) is attained when df/dλ=0 • Solve equation numerically to get exact solution for λ1 • Equation implies f(λ1) is convex on (0,a]

26. Making approximations λ0>>λ1 and with a-λ1 ≈ a, λ0 ≈ a • With simplification, and c given by • Since distance of node from aircraft is much larger than r • Since μHk>>l+μrk, we get c>>1 • The only feasible (t<1) solution is

27. Since t=e-λ1πr2 • If µrk >> l • We eliminate a from previous equation to obtain • For r << 1 • For sufficiently small r

28. H is fixed and have scenario where nodes have very small sensing/coverage radius r • First order approximations for λ1 is obtained as

29. Random Deployment contd... • For closed form expression for λ1, we note that H >> r and λ0 >> λ1 • We get following closed form expression for required cluster head intensity with given c • We get further simplification for typical transceiver radio parameters when μrk >> l with propagation loss index k equals 2 • λ1 scales approximately as √λ0 • Exact solution of λ1 obtained by numerically solving previous equation

30. The optimum number of cluster heads required when N sensor nodes are uniformly distributed over a unit area and nodes used single hop communication to reach cluster head • Cluster head are periodically rotated for efficient load balancing • Assuming line of sight communication between cluster head and base station and • propagation loss model of ε1x2 between node and cluster head and • ε2x4 between cluster head and base station

31. An approximate solution for λ0 is a • We can determine E0 and E1 • Assume its possible to equip nodes with as much energy as required for T data gathering cycles • Cluster heads can serve purely as fusion centers as their intensity is lower and λ0 >> λ1

32. Solution for Grid Deployment Scenario • Consider a simple grid of nodes placed along grid points with distance r between them • Connectivity and coverage condition take form shown with a’ being minimum number of nodes required • P0r is calculated by noting that in a grid there are only four critical nodes

33. Same minimization problem as random case based on KKT • For local minimum • c2 dominates over other ci • For k=2 and simplifying λ1 • Striking similarity between form of λ1 for random deployment and grid deployment • Work can be generalized to the case of unreliable nodes • Coverage-connectivity constraint still has logλ/λ form

34. Numerical Results • Provide justifications for approximations by using some typical transceiver radio parameters • Consider an area A of 10kmx10km to be covered by sensor nodes • Sensing radius of nodes varies from 10m to 100m and distance of nodes from aircraft varies from 1km to 10km • Compare approximate solution for λ1 with exact solution obtained numerically • Approximation works quiet well for settings of normal interest

35. Worthwhile having more type 1 nodes • For smaller values of H, λ1 is higher • λ0 >> λ1

36. Conclusions • Provide results that guarantee a minimum lifetime, i.e. T successful data gathering trips of the sensor networks • Ensure conditions for connectivity and coverage of area • Cluster heads and nodes within one hop of cluster heads have maximum relaying burden • Minimize overall costs within constraints

37. Compare results for random deployment with those of grid deployment • In both deployment scenarios, the required cluster head intensity λ1 scales as √λ0 • Analysis can be easily extended to scenarios where unreliable nodes are deployed randomly or along grid points

38. Critique • The analysis complex with many assumptions made along the way • Hard to understand if the problem still fits general case • Unreliable sensor case mentioned frequently yet not clear how analysis will be impacted • Lack of experimental results in real world scenarios glaring • Is energy efficiency this critical, i.e. is not better to do more pessimistic deployment and not bother with minimizing residual energy?

39. Q/A? Thanks!