Yi Cui, Member, IEEE, Yuan Xue, Member, IEEE, and Klara Nahrstedt, Member, IEEE A Utility-Based Distributed Maximum Lifetime Routing Algorithm for Wireless

Yi Cui, Member, IEEE, Yuan Xue, Member, IEEE, and Klara Nahrstedt,

Member, IEEE

A Utility-Based Distributed Maximum Lifetime Routing

Algorithm for Wireless Networks

IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL.

55, NO. 3, MAY 2006

Presented by Yu-Shun Wang( 王猷順 ) 112/04/19OP Lab @ IM NTU1

AuthorYi Cui (M’05) received the B.S. and M.S.

degrees in computer science from Tsinghua University, Beijing, China, in 1997 and 1999, respectively, and the Ph.D. degree from the Department of Computer Science, University of Illinois at Urbana-Champaign in 2005.

Since 2005, he has been with the Department of Electrical Engineering and Computer Science, Vanderbilt University, Nashville, TN, where he is currently an Assistant Professor.

His research interests include overlay network, peer-to-peer system, multimedia system, and wireless sensor network.

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AuthorYuan Xue (M’04) received the B.S. degree in

computer science from the Harbin Institute of Technology, Harbin, China, in 1994 and the M.S. and Ph.D. degrees in computer science from the University of Illinois at Urbana-Champaign in 2002 and 2005, respectively.

She is currently an Assistant Professor at the Department of Electrical Engineering and Computer Science, Vanderbilt University, Nashville, TN.

Her research interests include wireless and sensor networks, mobile systems, and network security.

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Author Klara Nahrstedt (S’93–M’95) received the A.B. and

M.Sc. degrees in mathematics from Humboldt University, Berlin, Germany, in 1984 and 1985, respectively, and the Ph.D. degree in computer science from the University of Pennsylvania, Philadelphia in 1995.

She is a Professor at the Computer Science Department, University of Illinois at Urbana- Champaign, where she does research on quality-of service- aware systems with emphasis on end-to-end resource management, routing, and middleware issues for distributed multimedia systems.

She is the coauthor of the widely used multimedia book Multimedia: Computing, Communications and Applications (Englewood Cliffs, NJ: Prentice Hall).

Dr. Nahrstedt was the recipient of the Early National Science Foundation Career Award, the Junior Xerox Award, and the IEEE Communication Society Leonard Abraham Award for Research Achievements, and the Ralph and Catherine Fisher Professorship Chair. Since June 2001, she has been the Editor in- Chief of the Association for Computing Machinery/Springer Multimedia System Journal.

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AgendaIntroductionModelProblem FormulationOptimality ConditionsDistributed Routing

OverviewCalculation of Marginal UtilitiesLoop-Free RoutingAlgorithmAnalysis

Simulation StudiesConclusion

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IntroductionOne important issue in wireless networks is

the energy constraint.Radio communication consumes a large

fraction of power supply.Therefore, there is a critical demand to

design energy-efficient packet routing algorithms.

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IntroductionIn order to scale to larger networks, such

algorithms need to be localized.The key design challenge is to derive the

desired global system properties in terms of energy efficiency from the localized algorithms.

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IntroductionDifferent approaches

Minimum Energy Routing: User OptimizationIt tries to optimize the performance of a single user

(an end-to-end connection), minimizing its energy consumption.

the typical approach is to use a shortest path algorithm.

However, this approach can cause unbalance consumption distribution.

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IntroductionDifferent approaches(cont.)

Maximum Lifetime Routing: System OptimizationThis tries to maximally prolong the duration in which the

entire network properly functions.global coordination is required.There are two ways to solve this problem

Linear Optimization Formulationo A centralized algorithm is proposed, but it is hard to be

deployed on realistic wireless network environments.Nonlinear Optimization Formulation

o design a fully distributed routing algorithm that achieves the goal of maximizing network lifetime.

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IntroductionComparison

Linear Optimization Formulation

Nonlinear Optimization Formulation

Main concept

Through global coordination to maximize network lifetime.

Algorithm Centralized algorithm

Distributed algorithm

Adaptable

Environment

Centralized architecture

Wireless networks

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IntroductionOur goal is to maximize the lifetime of the

node that has the minimum lifetime among all nodes.

If we regard lifetime as a “resource”, then this goal can be regarded as to “allocate lifetime” to each node so that the max–min fairness criterion is satisfied.

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Introductionwe further adopt the concept of “utility” to

the problem. By defining an appropriate utility function, the problem is converted into the aggregated utility maximization problem.

Based on this formulation, the key to the distributed algorithm is to consider the marginal utility at each node.

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ModelThere are two models proposed in this

paperRouting modelEnergy model

Each model conducted with corresponding constrain.

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ModelRouting model

Notationinput traffic rii(j) ≥ 0: the traffic (in bits per second)

generating at node i and destined for node j.node flow ti(j): the total traffic at node i destined for

node j.routing variable φik(j): the fraction of the node flow

ti(j) routed over link (i, k).

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ModelFor routing variable φik(j)

φik(j) = 0, if (i, k) L,∈ as no traffic can be routed through a nonexistent link.

φik(j) = 0, if i = j, because traffic that has reached its destination is not sent back into the network.

As node i must route its entire node flow ti(j) through all outgoing links

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Model

φ14(6) = 1/4

t1(4) = 2 kb/s

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= link(1,4) 上以 6 為目的地之 flow/t1(6)

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ModelOne important constraint of traffic routing

in a network is flow conservation. This constraint can be formally expressed as:

Node i 上前往Node j 之總流量

Node i 本身發出以 node j 為目的地之流量

網路中其他 node l 透過link(l,i) 傳輸至 node j 之流量

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ModelEnergy model

NotationLet pr

i(joules per /bit) be the power consumption at node i, when it receives one unit of data.

ptik(J/bit) be the power consumption when one unit of

data is sent from i over link (i, k).pr

i=α, α is a distance-independent constant.

ptik=α + βdm

ik, β is the coefficient of the distance-dependent term that represents the transmit amplifier. The exponent m is typically a constant between 2 and 4.

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Modelwireless node i’s power consumption rate

pi in joules per second as:

Node i 之能耗自 i 至 j 之所有流量

由 node i 發出以 node j為目的地的能量消耗比率

接收訊息所引發的能量消耗發送訊息所引發的能量消耗

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ModelSummary

Notation Set Name

Content

Input set r {ri(j)|i, j ∈ N}

Node flow set t {ti(j)|i, j ∈ N}

Routing variable set

φ {φik(j)|i, j, k ∈ N}

Power consumption set

p {pi|i ∈ N}112/04/19OP Lab @ IM NTU22/78

ModelThe relations among the input set and the node

flow set are constrained by flow conservation. We further have the following lemma.

Lemma 1:Given the input set r and routing variable set φ, the

set in flow conservation constrain has a unique solution for t.

Each element ti(j) is nonnegative and continuously differentiable as a function of r and φ.

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Problem FormulationMaximum network lifetime routing tries to

maximally prolong the duration in which the entire network properly functions.

Here we consider the network lifetime as the lifetime of the wireless node who dies first.

The problem asks the given traffic demand how to route the traffic so that the network lifetime can be maximized.

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Problem FormulationLet Ti denote the lifetime of wireless node i,

and T as the lifetime of the wireless network.The problem can be formulated as a linear

optimization problem:

Node i 之power consumption

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Problem Formulationone can acquire the optimal routing

solution φ to maximize T by solving the above linear optimization problem via a centralized algorithm.

But as mentioned before, centralized algorithm is hard to deployed on a real wireless network.

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Problem FormulationTo address this challenge, we propose a utility-

based nonlinear optimization formulation, which can lead to a fully distributed routing algorithm.

This formulation inspired by the max–min resource allocation problem, if we regard lifetime of a node as a certain “resource” of its own.

the goal can be regarded as to “allocate lifetime” to each node so that the max–min fairness criterion is satisfied.

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Problem FormulationFurthermore, by defining an appropriate

utility function, the problem of achieving max–min fairness can be converted into the problem of maximizing the aggregated utility.

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Problem Formulationreformulate the problem of maximum

network lifetime as to maximize the aggregate utility of all nodes within the network.

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Problem FormulationSummary


Maximum Lifetime Routing Problem

Distributed

Algorithm

Centralized

Algorithm

Defining an Appropriate

Utility Function

Satisfy Max–min Fairness

Criterion

By a Unified

Framework

Multicommodity Flow Problem

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Problem FormulationSummary


Centralized Algorithm

Distributed Algorithm

Inspired concept Transportation

problem Resource allocation

Transforming method

Model this problem as a Multicommodity

Flow Problem.

Satisfying max–min fairness criterion to

achieve optimal point.

Approach

Apply a unified framework for a host of multicommodity flow and packing

problems algorithm.

By defining anappropriate utility

function to convert this problem to maximizing the

aggregated utility.

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Optimality ConditionsFrom the nonlinear optimization theory ,

we consider the first-order conditions in problem U.

Since the utility Ui is a function of node lifetime Ti, which directly associates with pi based on relation Ti = Ei/pi. Thus, we can write Ui(pi) as a function of pi.

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Optimality ConditionsPower consumption pi in turn depends on

the input set r and the routing variable set φ.

Thus, we calculate the partial derivatives of the aggregate utility U with respect to the inputs r and the routing variables φ.

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Optimality ConditionsWe first consider ∂U/∂ri(j), the marginal

utility on node i with respect to commodity j.

Assume that there is a small increment ε on the input traffic ri(j). Then the portion φik(j) from this new incoming traffic will flow over the wireless link (i, k).

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Optimality ConditionsThis will cause an increment power

consumption εφik(j)ptik on node i in order to

send out the incremented traffic. the consequent utility change of node i is

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Optimality ConditionsOn the receiver side, this will cause an

increment power consumption εφik(j)prk on

node k in order to receive the incremented traffic.

The consequent utility change of node k is

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Optimality ConditionsIf node k is not the destination node, then

the increment εφik(j) of extra traffic at node k will cause the same utility change onward as a result of the increment εφik(j) of input traffic at node k.

To first order this utility change will be εφik(j)∂U/∂rk(j).

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Optimality ConditionsSumming over all adjacent nodes k, then,

we find thatmarginal utility on link (i, k)

marginal utility of link (i, k) with respect to commodity j

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Optimality ConditionsAbove equation asserts that the marginal

utility of a node is the convex sum of the marginal utilities of its outgoing links with respect to the same commodity.

By the definition of φ, we can see that ∂U/∂rj(j) = 0, since φjk(j) = 0, i.e., no traffic of commodity j needs to be routed anymore once it arrives at the destination.

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Optimality ConditionsNext we consider ∂U/∂φik(j). An increment

ε in φik(j) causes an increment εti(j) in the portion of ti(j) flowing on link (i, k).

If k = j, this causes an addition εti(j) to the traffic at k destined for j.

Thus, for (i, k) L, i = j.∈

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Optimality ConditionsTo summarize above discussions, we have

the following theorem.Theorem 1: Let a network have traffic

input set r and routing variables φ, and let each marginal utility U'i(pi) be continuous in pi, i N. Then we have the following:∈the set in above equation i = j has a unique

solution for ∂U/∂ri(j).both ∂U/∂rii(j) and ∂U/∂φik(j) (i = j, (i, k) L) ∈

are continuous in r and φ.

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Optimality ConditionsThe optimal solution of the maximum

lifetime routing problem are given in Theorem 2.

Theorem 2: Assume that Ui is concave and continuously differentiable for pi i. U is ∀maximized if and only if for i, j N.∀ ∈

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Optimality ConditionsTheorem 2 states that the aggregate utility

is maximized if any node i, for a given commodity j, all links (i, k) that have any portion of flow ti(j) routed through (φik(j) > 0) must achieve the same marginal utility with respect to j.

And this maximum marginal utility must be greater than or equal to the marginal utilities of the links with no flow routed (φik(j) = 0).

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Distributed Routing-OverviewThe algorithm works in an iterative fashion. In each iteration, for each node i and a given

commodity j, i must incrementally decrease the fraction of traffic on link (i, k) (by decreasing φik(j)) whose marginal utility δik(j) is large.

And do the reverse for those links whose marginal utility is small, until the marginal utilities of all links carrying traffic are equal.

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Distributed Routing-Overviewfor each node i, each iteration involves two

steps:1) the calculation of marginal utility U'ik for

each outgoing link (i, k) and each of its downstream neighbors k’s marginal utility ∂U/∂rk(j).

2) the adjustment of routing variables φik(j) based on the values of U'ik and ∂U/∂rk(j).

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Distributed Routing-Calculation of Marginal UtilitiesWe first introduce how to calculate the link

marginal utility U'ik = ptikU'i(pi) + pr

kU'k (pk).Sending data over a wireless link (i, k) requires

power consumption of both sending node i and receiving node k.

The calculation of U'ik depends on the cooperation of both nodes. Node i is responsible to calculate the term pt

ikU'i(pi). U'i(pi) can be derived if the energy reserve Ei and power consumption rate pi are known.

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Distributed Routing-Calculation of Marginal UtilitiesNode k is responsible for calculate the term

prkU'k(pk). U'k(pk) can be calculated in the

same way as U'i(pi).After calculation, k can send the value of

prkU'k(pk) to node i, which in turn acquires

U'ik.

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Distributed Routing-Calculation of Marginal UtilitiesNow we see how each node i calculates its

marginal utility ∂U/∂ri(j) with respect to commodity j.

In order to do so, i needs to know δik(j) = U'ik + ∂U/∂rk(j), the marginal utility of all its outgoing links regarding commodity j.

It is clear that ∂U/∂ri(j) should be calculated in a recursive way.

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Distributed Routing-Calculation of Marginal UtilitiesStarting from node j, the recipient of

commodity j, ∂U/∂rj(j) = 0 based on the definition.

j then sends the values of ∂U/∂rj(j) and prjU'j(pj)

to its upstream neighbor, say k.Upon receiving the updates, node k can

calculate U'ik as described above, then acquire ∂U/∂rk(j). Then, k repeats the same procedure to its upstream neighbor until node i is reached.

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Distributed Routing-Calculation of Marginal Utilities

I K J‧ ‧ ‧ ‧ ‧∂U/∂rj(j) = 0

prjU'j(pj)

∂U/∂rk(j) pr

kU‘k(pk)

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Distributed Routing-Loop-Free RoutingFrom the above calculation, we can see that

among all nodes carrying traffic of commodity j, their marginal utilities follow a partial ordering.

The recipient node of commodity j has the highest marginal utility, which is 0. Its upstream neighbors have lower marginal utilities.

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Distributed Routing-Loop-Free RoutingTherefore, the recursive procedure of node

marginal utility calculation is free of deadlock if and only if such a partial ordering is maintained, i.e., the routing variable set φ is loop free.

In order to achieve loop-free routing, for each node i, with respect to commodity j, we introduce a set Bi,φ(j).

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Distributed Routing-Loop-Free RoutingThe set Bi,φ(j) of blocked nodes k for which

φik(j) = 0 and the algorithm is not permitted to increase φik(j) from 0. k B∈ i,φ(j) if one of the following conditions is met.(i, k) L, i.e., k is not a neighbor of i.∈φ ik=0 and ∂U/∂ri(j) ≥ ∂U/∂rk(j), i.e., the marginal

utility of i is already greater than or equal to the marginal utility of k.

φiik= 0 and (l,m) L such that∃ ∈a) l = k or l is downstream of k with respect to commodity j.b) φlm(j) > 0 and ∂U/∂rl(j) ≥ ∂U/∂rm(j), i.e., (l,m) is an

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Distributed Routing-Loop-Free Routing


1

2

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Distributed Routing-AlgorithmWe use φ(k) to represent the routing variable

set at the iteration k. Δφ(k) represents the changes made to φ(k) during the iteration k.

Apparently, φ(k+1) = φ(k) +Δφ(k). Also, for node i, we have the following:φi(j) = (φi1(j), . . . , φim(j))T is the vector of its

routing variable regarding commodity j.Δφi(j) = (Δφi1(j), . . . ,Δφim(j))T is the vector of

changes to φi(j).

δi(j) = (δi1(j), . . . , δim(j))T is the vector of marginal utilities of all i’s neighbors. 112/04/19OP Lab @ IM NTU62/78

Distributed Routing-AlgorithmAt iteration k, node i operates according to

the following steps:1) Calculate the link marginal utility U'ik for each

of its going links (i, k), get updates of marginal utility ∂U/∂rk(j) from each of its downstream neighbors k, and then calculate δik(j) = U'ik+ ∂U/∂rk(j).

2) Calculate its own marginal utility ∂U/∂ri(j) and send it to all its upstream neighbors.

3) Calculate Δφ(k)i(j).

4) Adjust routing variables. 112/04/19OP Lab @ IM NTU63/78

Distributed Routing-AlgorithmΔφ(k)

i(j) is calculated in the following way:

where δmin(j) = min δim(j), and ρ > 0 is some

positive stepsize.

m B ∈ i,φ(k)

(j)

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Distributed Routing-Algorithmrouting variables are Adjusted in the

following way:

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Distributed Routing-AnalysisThe following lemma shows some of the

properties of our algorithm.Lemma 2:

If φ(k)(j) is loop free, then φ(k+1)(j) is loop free.If φ(k)(j) is loop free and Δφ(k)(j) = 0, then φ(k)(j)

is optimal.If φ(k)(j) is optimal, then φ(k+1)(j) is also optimal.If Δφ(k)(j) = 0 for some i for which ti(j) > 0, then

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Distributed Routing-AnalysisThe following theorem shows the main

convergence result.Theorem 3: Let the initial routing φ(0) be

loop free and satisfy U(φ(0)) ≥ U0, where U0 is some scalar, then

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Simulation StudiesSimulation Setup

We randomly create 100 nodes on a 100 × 100 m2 square.

The maximum transmission range of each node is 25 m.

we set α = 50 nJ/b, β = 0.0013 pJ/b/m4, and m = 4 for the power consumption model.

The energy reserve on each node is 50 kJ.

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Simulation StudiesSimulation Setup(con’t)

We study two types of networking scenariossensor network

a node is picked as the base station (data sink), while a subset of other nodes act as data sources sending traffic to the sink at 0.5 kb/s.

The rest of the nodes act as relaying nodes.ad hoc network

we randomly create several pairs of unicast connections. Besides the unicast pairs, other nodes are responsible to

relay traffic. The sending rate of each connection is 0.5 kb/s.

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Simulation StudiesWe compare the performance of following

algorithms. The MinEnergy algorithm

It tries to minimize the energy consumption for each data unit routed through the network.

For each data source, the algorithm finds its shortest path to the destination in terms of energy cost.

The MaxLife algorithmIt tries to maximize the network lifetime, which can be

implemented in centralized or distributed fashions.The centralized algorithm derives the maximum lifetime by

solving the linear programming problem.The distributed algorithm is the one presented before.

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Simulation Studies

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Simulation Studies

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Simulation Studies

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ConclusionThis paper studies the problem of distributed

maximum network lifetime routing for a multihop wireless network.

Inspired by the max–min fair resource allocation, this paper presents a novel utility-based nonlinear formulation of this problem.

Based on this formulation, this paper further presents a distributed routing algorithm that achieves the goal of maximizing network lifetime.

The presented algorithm has shown to be effective and efficient under various simulation environments.

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Documents

Yi Cui, Member, IEEE, Yuan Xue, Member, IEEE, and Klara Nahrstedt, Member, IEEE A Utility-Based Distributed Maximum Lifetime Routing Algorithm for Wireless