Optimum Deployment in Heterogeneous Sensor Networks
390 likes | 485 Vues
Study on node deployment parameters in sensor networks to maximize data gathering cycles. Focus on minimizing overall network cost and reducing energy waste.
Optimum Deployment in Heterogeneous Sensor Networks
E N D
Presentation Transcript
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
Outline • Introduction • Previous work • Problem • Solution to random deployment scenario • Solution to grid deployment scenario • Numerical Results • Conclusions
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
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
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
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
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
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
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
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
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
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
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
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
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
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)
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
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
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
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
From substitutions we have • Assuming that a feasible solutions exists, i.e. λ0, λ1, E0, E1 > 0 we get μ0 and μ2
From KKT • Using μ0 • Reformulate optimization problem as • We obtain as function of single variable λ1 • Rewriting a from previous equations
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]
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
Since t=e-λ1πr2 • If µrk >> l • We eliminate a from previous equation to obtain • For r << 1 • For sufficiently small r
H is fixed and have scenario where nodes have very small sensing/coverage radius r • First order approximations for λ1 is obtained as
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
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
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
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
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
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
Worthwhile having more type 1 nodes • For smaller values of H, λ1 is higher • λ0 >> λ1
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
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
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?
Q/A? Thanks!
