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A Theory for Maximizing the Lifetime of Sensor Networks

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					A Theory for Maximizing the
Lifetime of Sensor Networks

   Joseph C. Dagher, Michael W. Marcellin, and
   Mark A. Neifeld. The University of Arizona.

   IEEE Transactions on Communications, VOL.
   55, NO. 2, February 2007.


                                                 1
Outline

   Introduction
   Problem Statement
   An Optimal Algorithm
   Experimental Results
   Conclusion




                           2
Introduction
   Maximize the lifetime of sensor networks
       The limited energy and bandwidth resources in wireless sensor networks
   In a unicast network routing model, Chang and Tassiulas [18] considered the
    problem of maximizing the network lifetime,
       Chang defined to be the time until the first node runs out of battery power.
       In fact, the authors in [18] were the first to treat this problem as a linear programming (LP)
        problem.
       [18] J. H. Chang and L. Tassiulas, “Routing for maximum system lifetime in wireless ad
        hoc networks,” in Proc. 37th Annu. Allerton Conf. Commun., Control, Comput., Sep. 1999,
        CD-ROM.
   The lifetime of the network is equal to the lifetime of that “weak” sensor.
       “weak” sensor : a sensor that has a much higher energy consumption per bit, as
        compared with the rest of the sensors
       There exist multiple solutions that will yield the same network lifetime. However, each of
        those solutions will result in different lifetimes for the remaining sensors.
   In this paper, we consider the problem of maximizing the lifetime of a unicast
    network
       Develop a theory and a corresponding algorithm that maximize the n-th minimum sensor
        lifetime in the network, subject to the (n-1)th minimum lifetime being maximum,… ,subject
        to the minimum lifetime being maximum,
           n is an arbitrary positive integer less than or equal to the total number of sensors in the network.
           The choice of n depends on the application of interest.
                                                                                                               3
4
Problem Statement
   Network model
       N sensors are deployed some distance away from a central Base Station
        (BS).
           BS is a unique receiver.
       Bi : the remaining energy (in Joules) of the i-th sensor prior to a
        transmission event.
       Xi,j : the number of bits that will be routed from sensor i to sensor j .
           Xi,0 : the number of bits that will be directly transmitted from sensor I to the
            base station.
           Xi,i : the initial load Xi,i (bits) that sensor i needs to transmit to the base station
            via some route.
       ti,j : the communication cost for sensor i to transmit one bit to sensor j
       rj,i : the communication cost for sensor i to receive one bit from sensor j
       αi : the lifetime of sensor i
       data is acquired at a fixed rate
           e.g., images acquired at a fixed coding rate


                                                                                                      5
6
Problem Statement
   The communication cost (in Joules/bit) along a given path
    depends on
     the path loss of the channel

     the length of the path

     the cost of processing a bit

     the cost of receiver electronics

   Different sensors could have different transmission capabilities,
    depending on their communication costs.
     ti,j , rj,i…

   Assume that the operation of every sensor is equally important to
    the network.
     It would be desirable to maximize the lifetime of every sensor in
       this network.
     However, this is not always possible to achieve, because the
       lifetime functions are not independent of one another.

                                                                      7
 Problem Statement
The total amount of energy spent by a given sensor for a given transmission event




  The lifetime of sensor will be a function of the data transmission strategy




                                                                                8
Problem Statement
   Definition: Define C to be the constraint set. That is, X  C if
    the elements of X are nonnegative and satisfy (5).
   “conservation of load” constraint
       At every sensor node i, the total transmitted load equals the sum
        of sensor i’s initial load and the total load received from other
        sensors.




                                                                            9
Problem Statement
   MultiObjective (MO) optimization problem in which
    the objectives are the respective sensor lifetimes.
     One notion of optimality is that of Pareto-optimal
       (PO) solutions here.
   A vector x*  C is said to be PO if it is not possible
    to improve one objective without making at least
    one other objective worse.
   Definition: A vector x*  C is said to be PO if there
    exists no x’  C such that αi(x’) ≥αi(x*)   i=1,…,N ,
    with at least one strict inequality.
   Definition: Let P be the subset of solutions in C that
    are PO.

                                                             10
柏拉圖最佳解(Pareto-optimal solution)

   在多目標最佳化(multi-objective optimization)問題
    中,我們所要找的最佳解並不 是所有子目標的最佳
    解,而是所謂的「Pareto 最佳解(Pareto-optimal
    solution) 。
   如果要在現有的設計點上要改進其中的某一個子目標
    (sub-objective),勢必要在其他子目標上有所犧牲,
    而現有的設計點在某種意義上,卻是一個最佳解,即
    稱為 『Pareto 最佳解』。
   很顯然的,Pareto 最佳解不只一個,事實上在一般多
    目標最佳化問題中, Pareto 最佳解常是連續的而有
    無限多個。

                                          11
Problem Statement
   In this paper, we want to find the PO solution that maximizes
    the n-th minimum lifetime s.t. the (n-1)th minimum lifetime is
    maximized,…, s.t. the second minimum lifetime is maximized,
    s.t. the minimum lifetime is maximum,
         n is any integer between 1 and N
         The desired solution is the solution that maximizes the
          “ n-th conditional lifetime”




                                                              = 1,…,N
      k
min   i   {αi(x)} : the k-th minimum lifetime among {αi(x)}   i


                                                                         12
                                                            It is impossible here to
                                                            simultaneously
                                                            maximize the lifetimes of
                                                            both sensors.




N=2 sensors with equal initial batteries B1=B2=1J, and with initial loads x1,1=
103 bits, x2,2=102 bits. The transmit costs chosen here are t1,0=210-3,
t2,0=710-4 (J/bit), and t1,2=t2,1=0 (no intersensor communication cost). The
cost of receiving data is also chosen to be zero here.                            13
An Optimal Algorithm

   Theorem 1: If  x  P s.t. α1(x) = α2(x) = … = αN(x) = α, then
    x maximizes the N-th conditional lifetime.
   If there exists a PO solution in the constraint set that makes
    all lifetimes equal, then this solution maximizes the N-th
    conditional lifetime.
       Two curves cross; or equivalently, when both lifetimes are equal.
   Unfortunately, we are not always guaranteed to find P a
    solution in that equalizes all lifetimes.
       If the solution does not exist, the algorithm then proceeds in an
        iterative fashion.
       The algorithm eventually attains the desired objective of
        maximizing the n-th conditional lifetime.
       The number of steps required for convergence is upper bounded
        by n.

                                                                        14
An Optimal Algorithm

   The algorithm finds the maximum value to which all lifetimes
    can be) set equal in the first iteration.
            (1
   Let xe be a solution that makes all lifetimes equal to a
    maximum value α(1)
   If α(1) is not PO,
       We find the maximum value α(2) to which we can set equal all
        lifetimes of those sensors not in E(1).
   If α(2) is not PO,
       …...
    Iterate the same steps until we find the first xe  P, where n
                                                      (n)

    is the iteration index.
   Xe, te, and re denote the “extended version” of vectors x, t, and
    r, respectively.

                                                                       15
Algorithm




            16
An Optimal Algorithm

   Remarks about the algorithm :
       The maximum number of iterations required to
        find the solution is N.
           N is the number of sensors in the network.
           N is not a tight upper bound.
       The algorithm is guaranteed to find a PO solution.
       Depending on the application of interest, we can
        stop the algorithm at any iteration m and then
        obtain a solution that maximizes the m-th
        conditional lifetime.

                                                         17
• Step 3 of (7) determines whether a given solution X(n)e is PO.
• Check each sensor k to determine if it is possible to find a better lifetime X’e.
• Each k in (8) can be tested with one LP instance.




                                                                                  18
Experimental Results

   N sensors are deployed randomly in 100m 100 m
    square area.
   The location of the Base Station(BS) is also random
    within this square.
   We only account for communication costs and
    neglect all processing costs.
   di,j : the distance (in meters) between node i and
    node j.
   The cost for transmitting one bit along a path of
    length :
   The cost of receiving a bit :

                                                      19
Assume all sensors possess the same initial energy of Bi = 1J.




 Using (4), we get the sensor lifetimes:α1=64.74, α2=64.74, α3=64.74, and
 α4=64.74. In this case, the final result was obtained with just one iteration.
                                                                                  20
We get the sensor lifetimes:α1=95.33, α2=95.33, α3=95.33, and α4=505.25. In
                                                                                  21
this case, the final result was obtained with four iterations of our algorithm.
Experimental Results

   An interesting question :
     How often does our algorithm perform better than methods that

        only aim at maximizing the minimum lifetime?
   Compare with an algorithm from the literature. Chang and
    Tassiulas [19]
     [19] J. H. Chang and L. Tassiulas, “Maximum lifetime routing in

        wireless sensor networks,” IEEE Trans. on Networking, vol. 12,
        no. 4, pp. 609–619, Aug. 2004.
   Compute and compare the percentage gain in the n-th minimum
    lifetime
     αi and α’I are the lifetimes for sensor i , obtained using our

        algorithm and the algorithm of [19].


                                                                         22
n = 1. On average, g1:4=0.5%. Our algorithm is able to attain higher
first minimum lifetimes than the LP algorithm described in [19].
                                                                       23
n = 2. The probability of obtaining a gain in the second minimum lifetime in the
interval [10%–20%) is about 0.05. On average, the expected gain in the second
minimum lifetime due to our algorithm is around 7.9%.
                                                                              24
n = 3. The probability of obtaining a gain in the third minimum lifetime in the
interval [10%–20%) lifetime is about 0.07. The expected gain in the third minimum
lifetime is 10.75%.
                                                                             25
n=4   26
Table I. N=4, the high
value of average
percentage gain for the
fourth minimum lifetime.




                       27
                                                         The probability of
                                                         obtaining a gain in
                                                         any of the lifetimes
                                                         for seven sensors
                                                         is around 0.57.




pgN : the rate at which our algorithm succeeds in finding a
                                                                         28
larger value for any of the th minimum lifetimes.
Conclusion

   Developed a theory and algorithm for
    maximizing the lifetimes of unicast multihop
    wireless sensor networks.
   Presented the optimal solution in the form of
    an iterative algorithm.
   Future work
       The implementation of a distributed algorithm to
        achieve the desired solution.
       The formulation could be extended to allow
        sensor loads (rates) to become dynamic.

                                                           29
References
   [18] J. H. Chang and L. Tassiulas, “Routing for maximum system lifetime in wireless
    ad hoc networks,” in Proc. 37th Annu. Allerton Conf. Commun., Control, Comput.,
    Sep. 1999, CD-ROM.
   [19] J. H. Chang and L. Tassiulas, “Maximum lifetime routing in wireless sensor
    networks,” IEEE Trans. on Networking., vol. 12, no. 4, pp. 609–619, Aug. 2004.
   [32] R. K. Sundaram, A First Course in Optimization Theory. New York: Cambridge
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   [33] D. Schmeidler, “The nucleus of a characteristic function game,” SIAM J. Appl.
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   [34] D. Bertsekas and R. Gallager, Data Networks. Englewood Cliffs, NJ: Prentice-
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   [35] J. Jaffe, “Bottleneck flow control,” IEEE Trans. Commun., vol. COM-29, no. 7, pp.
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   [36] D. Nace and M. Pioro, “A tutorial on max-min fairness and its applications to
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    http://www.hds.utc.fr/dnace/recherche/Publication/tutorial-MMF.pdf.
   [37] Y. Bejerano, S. Han, and L. Li, “Fairness and load balancing in wireless LAN’s
    using association control,” in Proc. 10th Annu. Int. Conf. Mobile Comput. Netw., Sep.
    2004, pp. 315–329.
   [40] J. Deng, Y. Han, P. Chen, and P. Varshney, “Optimum transmission range for
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    vol. 2, pp. 1024–1029.
                                                                                        30

				
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