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( 王王王 ) 111/06/23 OP Lab @ IM NTU 1
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Yi Cui, Member, IEEE, Yuan Xue, Member, IEEE, and Klara Nahrstedt, Member, IEEE A Utility-Based Distributed Maximum Lifetime Routing Algorithm for Wireless.
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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.
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.
OverviewCalculation of Marginal UtilitiesLoop-Free RoutingAlgorithmAnalysis
Simulation StudiesConclusion
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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)
18/78
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.
OverviewCalculation of Marginal UtilitiesLoop-Free RoutingAlgorithmAnalysis
Simulation StudiesConclusion
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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).
OverviewCalculation of Marginal UtilitiesLoop-Free RoutingAlgorithmAnalysis
Simulation StudiesConclusion
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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
OverviewCalculation of Marginal UtilitiesLoop-Free RoutingAlgorithmAnalysis
Simulation StudiesConclusion
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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