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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

								
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