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 ?0 and
  battery energy E0 and type 1 with intensity ?1
  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

5
(No Transcript)
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 ?0, E0, ?1 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 ?1 equally spaced
  type 1 nodes and ?0 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 CiaißEi
  where a and ß 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 ?0 and ?1 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 ?r(?) which are
  shifted by ?r(?) where ?2? 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(?)
• For n nodes deployed along grid points

14
Problem Contd

• In a network dimensioning problem, designers
  provide parameter e such that probability of
  connectivity and coverage be at least 1-e. We
  require
• Minimizing above equation as function of ? under
  constraint ?2?1 when er2 lt 1
• The constraint in above equation reduces to
  ?0?1 u(?0) a where a is dependent on e, p, r
• In the grid case, required number of nodes is ?0
  ?1, 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 lµxk
• µxk is the energy spent in the RF amplifier to
  counter propagation loss
• Cluster heads coordinate MAC and routing in
  cluster
• ENv 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-e
• 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
• ENv is the expected number of type 0 nodes in
  cell C0 (using Campbell's theorem and Slivnyaks
  theorem)

19

• Using equivalence with event of a point of type 0
  in a small area xdxd? located at (x, ?) 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 ENv we derive values of P0 and P1

20

• Combining equations we have E1P0-E0P10 which
  gives us
• Rewriting coverage constraint
• We also have E1TP1 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 ?1

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.
  ?0, ?1, E0, E1 gt 0 we get µ0 and µ2

24

• From KKT
• Using µ0
• Reformulate optimization problem as
• We obtain as function of single variable ?1
• Rewriting a from previous equations

25

• Since ?0a-?1
• Since we get E1
• Since we get E0 and f(?1)
• Local extremum of f(.) is attained when df/d?0
• Solve equation numerically to get exact solution
  for ?1
• Equation implies f(?1) is convex on (0,a

26

• Making approximations ?0gtgt?1 and with a-?1 a,
  ?0 a
• With simplification, and c given by
• Since distance of node from aircraft is much
  larger than r
• Since µHkgtgtlµrk, we get cgtgt1
• The only feasible (tlt1) solution is

27

• Since te-?1pr2
• If µrk gtgt l
• We eliminate a from previous equation to obtain
• For r ltlt 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 ?1 is obtained as

29
Random Deployment contd...

• For closed form expression for ?1, we note that H
  gtgt r and ?0 gtgt ?1
• 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 gtgt l with
  propagation loss index k equals 2
• ?1 scales approximately as v?0
• Exact solution of ?1 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 e1x2 between node and
  cluster head and
• e2x4 between cluster head and base station

31

• An approximate solution for ?0 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 ?0 gtgt ?1

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 k2 and simplifying ?1
• Striking similarity between form of ?1 for random
  deployment and grid deployment
• Work can be generalized to the case of unreliable
  nodes
• Coverage-connectivity constraint still has log?/?
  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 ?1 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, ?1 is higher
• ?0 gtgt ?1

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 ?1 scales as v?0
• 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!