# A Theory for Maximizing the Lifetime of Sensor Networks by ewghwehws

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```									A Theory for Maximizing the

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

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

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Outline

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

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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.
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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

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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.

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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

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

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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.

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   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。

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Problem Statement
   In this paper, we want to find the PO solution that maximizes
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

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

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It is impossible here to
simultaneously
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
   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
   The number of steps required for convergence is upper bounded
by n.

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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.

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Algorithm

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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

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• 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.

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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 :

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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.
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We get the sensor lifetimes:α1=95.33, α2=95.33, α3=95.33, and α4=505.25. In
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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
 αi and α’I are the lifetimes for sensor i , obtained using our

algorithm and the algorithm of [19].

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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].
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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%.
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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
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n=4   26
Table I. N=4, the high
value of average
percentage gain for the

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The probability of
obtaining a gain in
for seven sensors
is around 0.57.

pgN : the rate at which our algorithm succeeds in finding a
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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.

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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.
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   [33] D. Schmeidler, “The nucleus of a characteristic function game,” SIAM J. Appl.
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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
routing, load-balancing and network design,” [Online]. Available:
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
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vol. 2, pp. 1024–1029.
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