A Cost Minimization Algorithm for Mobile DataGathering in Wireless Sensor Networks

Miao Zhao, Dawei Gong and Yuanyuan YangDepartment of Electrical and Computer Engineering, Stony Brook University, Stony Brook, NY 11794, USA

Abstract—Recent studies have shown that significant benefitcan be achieved in wireless sensor networks (WSNs) by employingmobile collectors for data gathering via short-range communi-cations. A typical scenario for such a scheme is that a mobilecollector roams over the sensing field and pauses at some anchorpoints on its moving tour such that it can traverse the trans-mission range of all the sensors in the field and directly collectdata from each sensor. In this paper, we study the performanceoptimization of such mobile data gathering by formulating itinto a cost minimization problem constrained by the channelcapacity, the minimum amount of data gathered from each sensorand the bound of total sojourn time at all anchor points. Weassume that the cost of a sensor for a particular anchor pointis a function of the data amount a sensor uploads to the mobilecollector during its sojourn time at this anchor point. In orderto provide an efficient and distributed algorithm, we decomposethis global optimization problem into two subproblems to besolved by each sensor and the mobile collector, respectively.We show that such decomposition can be characterized as apricing mechanism, in which each sensor independently adjustsits payment for the data uploading opportunity to the mobilecollector based on the shadow prices of different anchor points.Correspondingly, we give an efficient algorithm to jointly solvethe two subproblems. Our theoretical analysis demonstrates thatthe proposed algorithm can achieve the optimal data controlfor each sensor and the optimal sojourn time allocation forthe mobile collector, which minimizes the overall network cost.Finally, extensive simulation results further validate that ouralgorithm achieves lower cost than the compared data gatheringstrategy.

Index Terms—Mobile data gathering, convex problem, decom-position, Karush-Kuhn-Tucker (KKT) conditions, duality.

I. INTRODUCTION

Recent years have witnessed the proliferation of wirelesssensor networks (WSNs) as a new information-gatheringparadigm for a wide-range of applications, such as fieldexploration, environmental monitoring, and security surveil-lance. Besides the active (via in-situ observation) or passive(via remote-sensing technologies) sensing on the interestedreal-word phenomena [1], the paramount task in a WSN ishow to efficiently gather sensing data from scattered sensors.Traditional approaches, also referred to as static data gathering,typically inherit the basic idea of dynamic routing, wheresensing data is routed to a static data sink via selectedrelay sensors [2]. In more complex schemes, some in-networkprocessing, such as data aggregation and compression byexploiting spatiotemporal correlation, may be incorporatedinto routing [3][4]. Though these approaches can performeffective data forwarding in some applications, the majorperformance bottleneck of these approaches is the increasedand non-uniform energy consumption among sensors, which isattributed to the inherent nature of multi-hop wireless commu-nications. Recent studies have proposed to utilize controlled

mobility as a promising approach to tackling these difficulties[5]-[11]. Specifically, a mobile collector is employed roamingover the sensing field by moving close enough to sensors so asto collect data from them via short-range communications. Asthe routing burden has been shifted from sensors themselvesto the mobile collector, energy consumption at sensors isthus reduced and becomes more uniform as less wirelesstransmissions are required in the network [7]. Consequently,a WSN may be operated for an extended period of time onlimited energy supply.

Among a variety of different mobile data gathering schemes,a typical scheme is the anchor-based range traversing datagathering [8]-[10]. Specifically, a set of locations in thesensing field is chosen as anchor points. The mobile collectorperiodically carries out a data gathering tour by visiting eachanchor point such that it can traverse the transmission rangeof all the sensors in the network. When the mobile collectorarrives at an anchor point, it would collect data from sensorsin the neighborhood. Thanks to the direct data transmissionsbetween sensors and the mobile collector, uniform energyconsumption can be achieved as each sensor would no longerrelay data for other sensors. In this paper, we will focus onanchor-based mobile data gathering and study how to achieveoptimal performance in such a scheme. We characterize datagathering performance by introducing network cost, which isa function quantifying the aggregated cost on gathering datafrom sensors at different anchor points. The “cost” here phys-ically implies the energy consumption or monetary expenseon gathering a certain amount of data from a sensor at aparticular anchor point. In this way, optimizing data gatheringperformance is equivalent to solving the corresponding costminimization problem. To find the optimal solution to thisproblem, we consider regulating two tunable parameters undergiven constraints. One parameter is the amount of data a sensoruploads to the mobile collector at a particular anchor point.Since it is expected to collect a sufficient amount of dataduring a data gathering tour, we require that the aggregateddata uploaded from a sensor to the mobile collector at allanchor points should be no less than a specified amount.Another parameter is the sojourn time of the mobile collectorat each anchor point. We require that the total sojourn time atall anchor points should be constrained within a limit so thatthe latency of a data gathering tour is bounded.

Since the cost minimization problem essentially answers thequestions that where and how sensors communicate with themobile collector, we can characterize it as a pricing mech-anism, where sensors independently adjust their paymentscompeting for the data uploading opportunity to the mobilecollector based on the shadow prices of different anchor points

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set by the mobile collector. Using this feature, we decomposethe cost minimization problem into two simpler subproblemsthat describe the behaviors of sensors and the mobile collector[27], respectively. In this way, instead of directly resolvingthe original global problem, alternatively we can jointly solvethese two subproblems. By iteratively adjusting the paymentand the shadow price between each sensor and the mobilecollector, an equilibrium [27][29] that reconciliates the twosubproblems can be reached, where the overall network costis minimized.

The contributions of our work can be summarized asfollows: (1) We characterize data gathering performance bynetwork cost and formulate the problem of optimizing datagathering performance as a convex problem. (2) We show thatthis problem can be described as a pricing mechanism so thatit can be correspondingly decomposed into two subproblems.(3) We provide a pricing-based algorithm to jointly solvethese subproblems in a distributed manner. (4) We present atheoretical analysis and extensive simulation results to validatethe convergence of the proposed algorithm and demonstratethat our algorithm can achieve lower network cost than thecompared data gathering strategy.

The remainder of the paper is organized as follows. SectionII reviews the related work. Section III introduces the systemmodel and provides the formulation of the cost minimizationproblem. Section IV decomposes the problem into two sub-problems and presents a pricing-based algorithm that jointlysolves the two subproblems. Section V addresses how to solvethe subproblem at each sensor. Finally, Section VI presents thesimulation results of the proposed algorithm and Section VIIconcludes the paper.

II. RELATED WORK

There has been some work in the literature on optimizingdata gathering performance in WSNs. Most of the workstudied static data gathering and focused on optimal routingfor maximum lifetime. For example, Madan and Lall [12]proposed distributed algorithms using a dual decompositionapproach to computing an optimal routing that maximizes thetime the first node in the network depletes its energy. In [13],Madan, et al. modeled the circuit energy consumption and thetraditional physical, MAC and routing layers. They consid-ered the optimization of individual layers as well as cross-layer optimization by computing a strategy that maximizesnetwork lifetime. Zhang, et al. [1] studied the joint problemof sensing rate control, data routing and energy allocationto maximize system utility. They simplified the problem byconverting it to an equivalent routing problem and presenteda distributed gradient-based algorithm that iteratively adjuststhe per-node amount of energy allocated between sensing andcommunication to reach system-wide optimum. In [14], Huaand Yum jointly considered optimal data aggregation basedon the correlation of sensors and maximum lifetime routing,which aims to reduce the traffic across the network and balancetraffic to avoid overwhelming bottleneck nodes.

There has also been limited work in the literature on theoptimization of mobile data gathering schemes. The recentwork by Gatzianas [15] is mostly related to our work inthis paper. The work in [15] assumed a static data rate and

focused on finding optimal routing from sensors to each anchorpoint that maximizes network lifetime. In contrast, our workis significantly different from the work in [15] in the sensethat we consider the data control on each sensor and thesojourn time allocation for the mobile collector, instead ofthe routing problem. Moreover, in our model, we imposeconstraints on the data amount gathered from each sensor andthe total sojourn time at all anchor points. These considerationsaddress important practical issues in mobile data gatheringapplications, which have not been considered in the existingwork.

In the meanwhile, pricing mechanisms [25]-[28] have re-ceived much attention in recent years. The mechanisms wereespecially proposed for resource allocation problems, where aresource provider establishes resource prices to charge users,in order to regulate the behavior of selfish users and achievesocial welfare maximization [29]. In particular, Kelly et al.[25][26] proposed a scheme in a wired network for elastictraffic that a network provider charges the users based on thetraffic load on individual links and the users choose their trans-mission rates as a function of prices. Qiu and Marbach [28]extended Kelly’s work to the bandwidth allocation problemin ad hoc networks, where users can charge other users aprice for relaying their data packets. In a more recent work,Hou and Kumar [27] studied the utility maximization problemwith delay-based Quality-of-Service (QoS) requirements in awireless local area network. They characterized the problemas a bidding game, where clients bid for service time from theaccess point, and the access point assigns delivery ratios to theclients according to their bids. In contrast, our work demandsinelastic data traffic uploaded from sensors to the mobilecollector at different anchor points at the lowest possiblecost. We attempt to use pricing as a means to regulate thecommunication between sensors and the mobile collector,where the mobile collector sets the shadow prices for anchorpoints and each sensor learns link prices from itself to theneighboring anchor points based on these shadow prices, thendetermines its payment for the data uploading opportunity tothe mobile collector at different anchor points.

III. SYSTEM MODEL AND PROBLEM FORMULATION

Consider a sensor network which consists of a set ofstatic sensors, denoted by N , and a set of anchor points,denoted by A. We study the anchor-based range traversingdata gathering scheme, where the mobile collector gathersdata directly from sensors by visiting each anchor point in aperiodic data gathering tour. There are several ways to decidethe locations of anchor points. One way is to consider thesensing field as a grid and anchor points can be uniformlydistributed on grid intersections [16]. An alternative way isto use the positions of a subset of sensors as the locationsof anchor points [7][8]. In this paper, we would follow thelatter option, which not only simplifies the setting of anchorpoints but also can facilitate the distributed implementation ofour algorithm that will be presented in subsequent sections.An example of this data gathering scheme is illustrated inFig. 1, where the positions of seven sensors are chosen asanchor points and the mobile collector starts its tour fromthe static data sink and sequentially visits each anchor point

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Sensor

Moving tour

Wireless link betweensensor and anchor point

Anchor point

Static sink

Fig. 1. An example of anchor-based range traversing data gathering schemein a WSN, where the positions of a subset of sensors are used as anchorpoints.

for data gathering. The mobile collector can be a robot or avehicle equipped with a powerful transceiver and battery. Forconvenience, we simply call it SenCar in the rest of this paper.

Since the SenCar moves over different anchor points, wenow define two sets that depict the relationship between theSenCar and sensors in the movement. One set is N a (a ∈ A),which represents the sensors in the coverage area of anchorpoint a. These sensors can directly upload data to the SenCarwhen it arrives at anchor point a. Another set is Ai (i ∈ N ),which contains the anchor points sensor i can reach in a singlehop. To ensure that each sensor has the opportunity to uploaddata to the SenCar, we assume that Ai is always non-empty.This can be guaranteed by choosing the anchor points throughfinding a set of neighbor sets of sensors such that the selectedsets contain all the sensors in the neighbor sets.

We assume that each sensor has enough buffered sensingdata and sensor i would upload an amount of data xa

i to theSenCar when it stops at anchor point a. In order to ensure thatthe SenCar can obtain a sufficient amount of data from eachsensor in a data gathering tour, we impose a minimum dataamount for each sensor, Mi, which indicates the minimumaggregated data uploaded from sensor i to the SenCar at allanchor points in a data gathering tour.

The SenCar would stay at anchor point a for a period ofsojourn time ta to gather data from nearby sensors. In sometime-sensitive applications, the data gathering task is expectedto be completed in a bounded period, which is equivalent toconstraining the total sojourn time at all anchor points withina limit. We denote such a limit by T and call it the boundof total sojourn time. Moreover, considering the prevalence ofunreliable channels in WSNs [21], we assume that the datatransmission between sensor i and the SenCar at anchor pointa experiences a lossy link with a successful delivery ratio ofpa

i . Thus, in order to ensure the SenCar receives xai amount

of data, sensor i needs to send out xai

pai

amount of packets.In order to characterize the impact of data uploading from a

sensor to the SenCar at a particular anchor point on the overalldata gathering performance, we introduce a cost function,Ca

i (·), as a strictly convex, increasing and twice-differentiablefunction with respect to the amount of data uploaded fromsensor i to the SenCar at anchor point a (i.e., xa

i ). In practice,the “cost” can be evaluate in terms of energy consumption,monetary cost or other metrics modeling user applicationneeds. Cost function Ca

i (·) implicitly quantifies the suitabilityfor sensor i to upload data towards the SenCar at anchor pointa. Correspondingly, the network cost is defined as the sum of

TABLE ILIST OF NOTATIONS USED IN PROBLEM FORMULATION.

Notation DefinitionN Set of sensorsA Set of anchor pointsN a Set of sensors in the coverage of anchor point a, N a ⊆ NAi Set of neighboring anchor points of sensor i, Ai ⊆ Ata Sojourn time of SenCar at anchor pint aT Bound of total sojourn time at all anchor points in a data

gathering tourxa

i Data amount sensor i uploads to the SenCar at anchor point ain a data gathering tour

pai Stochastic successful delivery ratio of a link from sensor

i to the SenCar at anchor point aB Channel bandwidth of the systemMi Minimum amount of data sensor i needs to upload to the

SenCar in a data gathering tour

data gathering costs of all sensors at all anchor points. Ourwork in this paper is to minimize the network cost by means ofproperly scheduling communications between sensors and theSenCar and dynamically adjusting the sojourn time at differentanchor points.

The network cost minimization problem can be formalizedas follows:

Definition 1: (NCM: Network Cost Minimization Problemfor Mobile Data Gathering in WSNs.) Given a set of sensors,N , a set of anchor points, A, the minimum data amount ofsensor i (i ∈ N ), Mi, and the bound of total sojourn time atall anchor points, T , find: (1) the data amount xa

i uploadedfrom sensor i to the SenCar at anchor point a; (2) the sojourntime ta of the SenCar at anchor point a, such that networkcost is minimized.

Using the notations listed in Table I, the NCM problem canbe formulated into the following convex optimization problem.NCM:

Minimize∑

a∈A

∑i∈Na

Cai (xa

i ) (1)

Subject to∑

a∈Ai

xai ≥Mi, ∀i ∈ N (2)

∑i∈Na

xai

pai≤ B · ta, ∀a ∈ A (3)∑

a∈Ata ≤ T, (4)

Over xai , ta ≥ 0, ∀i ∈ N a, ∀a ∈ A (5)

The constraints in the NCM problem can be explained asfollows.• Data constraint (2) shows that for each sensor, its aggre-

gated uploaded data at all anchor points should be no lessthan the specified minimum amount.

• Link capacity constraint (3) enforces that when the Sen-Car is located at anchor point a, the total transmitted dataamount from the sensors in the neighborhood is restrictedby the product of channel bandwidth B and sojourn timeta.

• Total sojourn time constraint (4) ensures that the totalsojourn time of the SenCar at all anchor points is boundedby T .

IV. PROBLEM DECOMPOSITION AND PRICING-BASED

ALGORITHM

In the previous section, we provided the formulation ofthe NCM problem. Since the problem has a strictly convex

324

function with respect to xai (a ∈ A, i ∈ N a) and is over a

convex feasible region, the NCM problem is mathematicallytractable. However, there exist some difficulties to directlysolve it: (1) Cost functions, Ca

i (·), for all a ∈ A, i ∈ N a,are typically the knowledge of sensors and are unlikely to beknown by the network provider or a central controller; (2) Dueto the asymmetry of wireless channels, the successful deliveryratio pa

i that indicates the uplink channel quality from a sensorto the SenCar at an anchor point may not be easily obtainedby sensors, which are the senders in the transmissions. Thisinformation can be available at the SenCar by performing areceiving estimation; (3) The adjustable variables xa

i and ta

in the formulation in fact reveal the behaviors of differententities, where xa

i reflects the schedule on data uploading ateach sensor, and in contrast, ta characterizes the movementof the SenCar; (4) Finally, it would be difficult to implementa solution in any centralized way in a WSN. To circumventthese difficulties, next we decompose the NCM problem intotwo simpler subproblems [25][28].

Suppose sensor i chooses to pay qai for the data uploading

opportunity when the SenCar stops at anchor point a in a datagathering tour, and in return is permitted to upload xa

i amountof data proportional to qa

i , i.e., qai = λa

i xai , where λa

i can beconsidered as the price for uploading a unit amount of dataover the link from sensor i to the SenCar at anchor point a.In the following, we simply call λa

i link price. Then, the localcost minimization problem for sensor i can be expressed asfollows.

SENSOR−i:

Minimize∑

a∈Ai

Cai

(qa

i

λai

)+

∑a∈Ai

qai

Subject to∑

a∈Ai

qai

λai≥Mi,

Over qai ≥ 0, ∀a ∈ Ai

(6)

In the above we consider two parts of costs for sensor i:∑a∈Ai

Cai

(qa

i

λai

)represents the sum of data uploading cost to

all the neighboring anchor points of sensor i, and∑

a∈Ai

qai is

the total payment used in competing for the data uploadingopportunity. In SENSOR−i problem, given link prices λa

i ’s,sensor i independently minimizes its overall cost under theconstraint that its aggregated uploaded data is no less than Mi.Note that to solve this problem, there is no need for sensor ito have the knowledge of link condition pa

i for all a ∈ Ai.On the other hand, given the payments from all sensors, the

SenCar tries to maximize function∑

a∈A

∑i∈Na

qai log(xa

i ) under

the constraints of channel capacity and total sojourn timebound. In other words, the SenCar needs to solve the followingoptimization problem.

SENCAR:

Maximize∑

a∈A

∑i∈Na

qai log(xa

i )

Subject to∑

i∈Na

xai

pai≤ B · ta, ∀a ∈ A∑

a∈Ata ≤ T,

Over xai , ta ≥ 0, ∀i ∈ N a, ∀a ∈ A

(7)

Clearly, the above maximization problem does not require theSenCar to know the cost functions Ca

i (·) for all a ∈ A andi ∈ N a.

The following theorem shows that by solving SENSOR−iand SENCAR problems, optimal data control and sojourn timeallocation can be achieved as the global cost minimization (i.e.NCM) problem.

Theorem 1: There exist non-negative matrices x = {xai |a ∈

A, i ∈ N a}, q = {qai |a ∈ A, i ∈ N a} and λ = {λa

i |a ∈A, i ∈ N a}, and non-negative vector t = {ta|a ∈ A} withqai = λa

i xai , ∀i ∈ N , a ∈ Ai such that

(a) For i ∈ N , with λai > 0 for all a ∈ Ai, qi = {qa

i |a ∈ Ai}is the solution to the SENSOR−i problem;(b) Given that each sensor is charged qa

i for uploading data tothe SenCar when it is located at anchor point a, (x, t) is thesolution to the SENCAR problem;In addition, given such x, λ, t � 0, matrix x and vector t solvethe NCM problem.

Proof: We first show the existence of x, q and λ thatsatisfy (a) and (b), and then prove that the corresponding (x, t)is the solution to the NCM problem.

We assume that with proper settings of parameters Mi

and T , there always exist feasible variable matrices x andq, and variable vector t that satisfy the constraints in NCM,SENSOR−i and SENCAR problems with strict inequality,which means that they are interior points in the feasible regionof the respective problem. Thus, the Slater’s condition forconstraint qualification is satisfied [19][12]. Since SENSOR−i,SENCAR and NCM problems are all convex problems, the so-lution to each problem that satisfies the corresponding Karush-Kuhn-Tucker (KKT) conditions is sufficient to be optimal forthe respective problem [19].

For the global cost minimization problem (i.e., NCM), weintroduce non-negative Lagrangian multipliers σa, μi and γ forthe constraints in (2)-(4), respectively. Then, the Lagrangianof NCM can be obtained as

Lsys(x, t, σ, μ, γ)=

∑a∈A

∑i∈Na

Cai (xa

i ) +∑

a∈Aσa(

∑i∈Na

xai

pai−Bta)

−∑i∈N

μi(∑

a∈Axa

i −Mi) + γ(∑

a∈Ata − T ).

Assuming x∗ = {xai∗|a ∈ A, i ∈ N a} and t∗ = {ta∗|a ∈ A}

are the optimal solution to the NCM problem, we obtain thefollowing KKT conditions.

∂Lsys∂xa

i

= Cai

′(xa

i∗) +

σa∗

pai

− μ∗i = 0, (8)

∀a ∈ A, i ∈ N a,∂Lsys∂ta

= −σa∗B + γ∗ = 0, ∀a ∈ A, (9)

σa∗(∑

i∈Na

xai∗

pai

−Bta∗) = 0, ∀a ∈ A, (10)

μ∗i (∑a∈A

xai∗ −Mi) = 0, ∀i ∈ N , (11)

γ∗(∑a∈A

ta∗ − T ) = 0, (12)

xai∗ ≥ 0, ta∗ ≥ 0, ∀a ∈ A, i ∈ N a, (13)

σa∗, μi∗, γ∗ ≥ 0, ∀a ∈ A, i ∈ N . (14)

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Introducing νi as the Lagrangian multipliers for the dataconstraint of SENSOR−i, its Lagrangian is given by

Lsen−i(q, ν, ε)=

∑a∈Ai

Cai

(qa

i

λai

)+

∑a∈Ai

qai − νi(

∑a∈A

qai

λai−Mi).

By the KKT conditions, q∗i = {qai∗|a ∈ Ai} is the optimal

solution to SENSOR−i problem if and only if there exists ν∗ithat satisfies

∂Lsen−i

∂qai

=1λa

i

· Cai

′(

qai∗

λai

)+ 1− ν∗i

λai

= 0, ∀a ∈ Ai,(15)

ν∗i (∑a∈A

qai∗

λai

−Mi) = 0, (16)

qai∗ ≥ 0, ∀a ∈ Ai, (17)

ν∗i ≥ 0. (18)

Similarly, introducing multipliers αa and β for the constraintsof SENCAR problem, the Lagrangian of SENCAR can beexpressed as follows.

Lcar(x, t, α, β, ρ, η)= −∑

a∈A

∑i∈Na

qai log xa

i +∑

a∈Aαa(

∑i∈Na

xai

pai−Bta)

+β(∑

a∈Ata − T )

For a given q, by the KKT conditions, we have that matrix x∗

and vector t∗ are the optimal solutions to SENCAR problemif and only if there exist α∗ = {αa∗|a ∈ A} and β∗ such that

∂Lcar∂xa

i

= − qai

xai∗ +

αa∗

pai

= 0, ∀a ∈ A, i ∈ N a, (19)

∂Lcar∂ta

= −αa∗B + β∗ = 0, ∀a ∈ A, (20)

αa∗(∑

i∈Na

xai∗

pai

−Bta∗) = 0, ∀a ∈ A, (21)

β∗(∑a∈A

ta∗ − T ) = 0, (22)

xai∗ ≥ 0, ta∗ ≥ 0, ∀a ∈ A, i ∈ N a, (23)

αa∗ ≥ 0, β∗ ≥ 0, ∀a ∈ A. (24)

Let (x∗, t∗) be the optimal solution to the NCM problem andσ∗, μ∗ and γ∗ be the corresponding multipliers that satisfy theKKT conditions in (8)-(14). Let xa

i = xai∗, ta = ta∗, λa

i =σa∗pa

iand qa

i = σa∗pa

ixa

i∗. It is clear that xa

i , ta, λai and qa

i are allnon-negative. By defining αa = σa∗ and β = γ∗, we find thatx, t, α and β satisfy the KKT conditions for SENCAR problemin (19)-(24). Thus, (x, t) solves SENCAR, which implies thatthe solution satisfying KKT conditions of NCM also identifiesa solution to SENCAR. Defining νi = μi

∗ together with λai =

σa∗pa

i, the KKT conditions of SENSOR−i problem are satisfied

such that qai = σa∗

pai

xai∗ is the solution to SENSOR−i. This

analysis establishes the existence of x, λ and t.On the other hand, suppose we are given x, t and λ

satisfying conditions (a) and (b) of the theorem. We showthat (x, t) is the solution to NCM. It is clear that by (19) andthe definition of λa

i , we have λai = αa

pai

. Letting σa = αa and

μi = νi, we have that λai = σa

pai

and condition (8) of NCM

holds by (15). Furthermore, letting γ = β, conditions in (9)-(14) are equivalent to (16)-(18) and (20)-(24). Thus, x, t, σ, μand γ satisfy the KKT conditions in (8)-(14). Therefore, weconclude that (x, t) solves NCM.

Theorem 1 implies that instead of directly solving the NCMproblem, alternatively we can jointly solve the SENSOR−iand SENCAR subproblems, which have less complexity andfacilitate a distributed implementation for the solution. Systemoptimum can be achieved when sensors’ payment q andSenCar’s data control x and link price λ reach equilibrium,i.e., qa

i = λai xa

i for all a ∈ A, i ∈ N a.The SENCAR problem requires the payment information

from all the sensors. It may incur high communication over-head if the SENCAR problem is solved in a centralized way[26]. Thus, we consider its dual problem to decompose itinto a set of subproblems with respect to each anchor point[20][23]. By taking advantage of the fact that there is a sensorat each anchor point, the subproblems can be solved with theaid of these sensors. For clarity, we call them help nodes inthe following. This way, to announce payment qa

i , sensor ionly needs to locally inform the help node at anchor point a.

We form the dual problem of SENCAR by introducingLagrangian multipliers αa’s (a ∈ A) for channel capacityconstraints. This results in the partial Lagrangian as

L′car(x, t, α) = −∑a∈A

∑i∈Na

qai log xa

i +∑

a∈Aαa(

∑i∈Na

xai

pai−Bta)

=∑

a∈A

∑i∈Na

(−qa

i log xai + αa xa

i

pai

)− ∑

a∈AαaBta,

where αa is also referred to as shadow price of anchor pointa.

Given price λ, the minimum of L′car occurs when xai =

qai /λa

i . Thus, the dual function is defined as

g(α) = inft

{L′car

(qλ , t, α

) ∣∣∣∣ ∑a∈A

ta ≤ T

}.

Correspondingly, the dual problem is to find a shadow pricevector α∗ that maximizes dual function g(α).

Moreover, based on Lagrangian L′car, we have

∂L′car∂xa

i= − qa

i

xai

+ αa

pai

= −λai + αa

pai.

For the optimum solution to the SENCAR problem, we have∂L′car∂xa

i= 0, i.e., λa

i = αa

pai

. This can be interpreted as that linkprice λa

i from sensor i to anchor point a is actually determinedby the shadow price of this anchor point and the quality ofthe link between them.

From the above analysis, we can see that data matrix x∗

is the optimal solution to the NCM problem if and only ifthere exists shadow price vector α∗ solving the dual problemof SENCAR such that for each a ∈ A, i ∈ N a, we have thatxa

i∗ = qa

i

λai

, where λai = αa∗

pai

and qai is the solution to the

SENSOR−i problem for a given λai . Based on this result, to

find the optimal solution, we will gradually vary shadow priceα of anchor points, derive link price λ accordingly and givedata amount x as a function of link price λ. When shadowprice vector α iteratively converges to its optimum α∗, theoptimal solution to the NCM problem can be achieved. Notethat the shadow price is associated with an anchor point.

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Therefore, the task of finding optimal vector α∗ can be doneby the help node at each anchor point in a distributed manner.Next, we propose a pricing-based algorithm to jointly solvethe dual problem of SENCAR and SENSOR−i.

Pricing-based Algorithm:For all a ∈ A, the help node independently initializes the

shadow price αa for anchor point a to a positive value.Repeat the following iteration until the shadow price vector

α converges to α∗.At iteration n,

• For all a ∈ A, the help node at anchor point a determineslink price λa

i (n) for all i ∈ N a by setting

λai (n) = αa(n)

pai

,

and then sends this information to sensors in its neigh-borhood.

• For all i ∈ N , after learning link price λai (n) for

all a ∈ Ai, sensor i decides its payments qai (n)’s for

its neighboring anchor points by solving SENSOR−iproblem to minimize the local cost, i.e.,

qai (n) = arg min

qai ≥0

{ ∑a∈Ai

Cai

(qa

i

λai (n)

)+

∑a∈Ai

qai∣∣∣∣∣ ∑

a∈Ai

qai

λai (n) ≥ Mi

}, λa

i (n) > 0,

and then announces these payments to the correspondinghelp nodes at neighboring anchor points.

• Help nodes exchange the information of shadow prices.In the cases that the help nodes are not connected, weassume that they can use slightly higher transmissionpower to ensure a minimum degree of connectivity amongthemselves. In order to minimize L′car, each help nodesets the sojourn time for its located anchor point byfollowing rule

ta(n) =

{T, If a = arg max

a∈Aαa(n)

0, Otherwise.(25)

• Upon receiving the payment information from all sensorsin its neighborhood and having identified the sojourntime, each help node updates the shadow price for itslocated anchor point according to

αa(n + 1) =[αa(n) + θ(n)

( ∑i∈Na

xai (n)pa

i−Bta(n)

)]+

with xai (n) = qa

i (n)/λai (n),

(26)where [·]+ denotes the projection onto the positive orthantand θ(n) is a properly chosen scalar stepsize for iterationn. In our algorithm, we choose the diminishing stepsize,i.e., θ(n) = d/(b + cn), ∀n, c, d > 0, b ≥ 0, where b, cand d are adjustable parameters that regulate the conver-gence speed. The diminishing stepsize can guarantee theconvergence regardless of the initial value of αa [19].

• Note that SENCAR problem is not strictly concave withrespect to sojourn time ta, which implies that the valuesof ta in the optimal solution to the Lagrangian dual can-not be directly applied to the primal SENCAR problem.

In view of this, we recover the solutions by applying themethod introduced in [22]. For iteration n, we composea primal feasible ta(n) as follows.

ta(n) = 1n

n∑h=1

ta(h)

={

ta(1) n = 1n−1

n ta(n− 1) + 1n ta(n) n > 1

(27)

It was proved in [22] that when the diminishing stepsizeis used, any accumulation point of sequence {ta(n)}generated by (27) is feasible to the primal problem and{ta(n)} can converge to a primal optimal solution.

V. LOCAL COST MINIMIZATION AT SENSORS

In this section, we consider the second step of the pricing-based algorithm: how to solve SENSOR−i problem by eachsensor under given link price vector λi = {λa

i |a ∈ Ai}.As aforementioned, Ca

i (·) is a monotonic increasing function.Thus, the minimum of the objective function in (6) should beachieved when

∑a∈Ai

qai

λai

= Mi. Considering this fact, theSENSOR−i problem can be rewritten as follows.

SENSOR−i:

Minimize∑

a∈Ai

Cai

(qa

i

λai

)+

∑a∈Ai

qai

Subject to∑

a∈Ai

qai

λai

= Mi,

Over qai ≥ 0, ∀a ∈ Ai

(28)

Let fi denote the objective function of SENSOR−i. Since fi =∑a∈Ai

Cai ( qa

i

λai) +

∑a∈Ai

qai =

∑a∈Ai

Cai ( qa

i

λai) +

∑a∈Ai

λai ( qa

i

λai), fi is a

function with respect to variable vector xi = {xai = qa

i

λai|a ∈

Ai}, where xi can be considered as the demand of uploadingdata vector at sensor i.

For each sensor i, let ai be the index of the minimum-marginal-cost anchor point for sensor i. That is,

ai = arg mina∈Ai

{∂fi(xi)

∂xai

}= arg min

a∈Ai

{Ca

i

′(

qai

λai

)+ λa

i

}.

If there are multiple minimum-marginal-cost anchor points,we can randomly choose one. Since SENSOR−i is a convexproblem, we can characterize solution q∗i by the followingoptimality condition [18][30][31].∑

a∈Ai

∂fi(x∗i )

∂xai

(xai − xa

i∗)

=∑

a∈Ai

(Ca

i

′ ( qai∗

λai

)+ λa

i

)(qa

i −qai∗

λai

)≥ 0.

This optimality condition can be equivalently expressed as

qai∗ > 0, only if

[∂fi(xi

∗)∂xa′

i

≥ ∂fi(xi∗)

∂xai

, ∀a′ ∈ Ai

].

That is, for each anchor point a ∈ Ai, sensor i only pays forthe data uploading opportunity to the SenCar at those anchorpoints that incur the minimum marginal cost. This intuitivelysuggests that sensor i should gradually shift the payment to theminimum-marginal-cost anchor point from other neighboringanchor points and finally reach an equilibrium, where theaggregated marginal cost of anchor points selected for datauploading is less than or equal to that of unselected anchor

327

points [24]. In the following, we present an adaptation algo-rithm that strikes for such equilibrium.

Adaptation algorithm:1. Case I: If |Ai| = 1, then qa

i = λai Mi;

2. Case II: If |Ai| > 1, sensor i first initializes its pay-ment vector qi(0) = {qa

i (0) ≥ 0|a ∈ Ai} that satisfies∑a∈Ai

qai (0)λa

i= Mi. For example, we can let qa

i (0) = Miλai

|Ai| ,where |Ai| represents the cardinality of set Ai. Then, ititeratively updates vector qi(k) according to

qai (k + 1) = ϕ(k)qa

i (k) + [1− ϕ(k)]qai (k), ∀a ∈ Ai (29)

qai (k) =

⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩

[qai (k)− δ(k)λa

i

(∂fi(xi(k))

∂xai

− ∂fi(xi(k))

∂xaii

)]+

if a ∈ Ai, a �= ai and qai (k) ≥ 0,

λai

i ·(

Mi −∑

a∈Ai,a �=ai

qai (k)λa

i

), if a = ai,

(30)where [·]+ denotes the projection onto the non-negative or-thant, k stands for the iteration index, δ(k) is a small positivescalar stepsize, and ϕ(k) is a scalar on [a, 1] with 0 < a ≤ 1.In other words, the new payment for each neighboring anchorpoint is a weighted average of the amount in the previousiteration and currently derived optimal value.

The adaptation algorithm can be explained as follows. Ifanchor point a is not chosen as the minimum-marginal-costanchor point by sensor i (i.e., a �= ai) and there still existspositive payment for it, this payment should be reduced. Onthe contrary, if a is chosen as the minimum-marginal-costanchor point (i.e., a = ai), the payment for it should beincreased and the increased amount is proportional to thelinear combination of the aggregated payment shifted from allother neighboring anchor points of sensor i in order to ensure∑

a∈Ai

qai

λai

= Mi.We have the following theorem regarding the convergence

of the adaptation algorithm.Theorem 2: When stepsize δ(k) is small enough, the adap-

tation algorithm converges to a unique optimal solution q∗i tothe SENSOR−i problem.

Proof: We first show that when δ(k) is no more than acertain value, adjusting payment vector qi by the adaptationalgorithm in (29)-(30) always results in the decrease of localcost at sensor i, i.e., fi(xi(k + 1)) ≤ f(xi(k)). Then weshow that such adaptation would finally reach an equilibriumto achieve the unique optimal solution q∗i = {qa

i∗|a ∈ Ai}.

From the adaptation algorithm, it is straightforward to verify∑a∈Ai

qai (k)−qa

i

λai

= 0, and (31)

(q

aii −q

aii

λaii

)2

=

( ∑a∈Ai,a �=ai

qai −qa

i

λai

)2

≤ (|Ai| − 1) · ∑a∈Ai,a �=ai

( qai −qa

i

λai

)2.(32)

It is clear that fi(xi) is defined on the compact set χ ={xa

i ∈ xi|∑

a∈Ai

xai = Mi, x

ai ≥ 0}. As ∇2fi is continuous

on χ, we assume that its norm is bounded by some scalarL > 0 [32]. Denoting the cost difference between two

consecutive iterations as Δxi and applying the mean valuetheorem [18][32], we have

Δxi = fi(xi(k + 1))− fi(xi(k))≤ < ∇fi(xi(k), xi(k + 1)− xi(k) >

+L2 |xi(k + 1)− xi(k)|2

(33)

Based on (33) and qai (k + 1)− qa

i (k) = ϕ(k)(qai (k)− qa

i (k))by (29), Δxi can be rewritten as

Δxi ≤∑

a∈Ai

∂fi(xi(k))∂xa

iϕ(k)

(qa

i (k)−qai (k)

λai

)+L

2 ϕ2(k)∑

a∈Ai

(qa

i (k)−qai (k)

λai

)2

=∑

a∈Ai

(∂fi(xi(k))

∂xai

− ∂fi(xi(k))

∂xaii

)ϕ(k)

(qa

i (k)−qai (k)

λai

)+∂fi(xi(k))

∂xaii

ϕ(k)∑

a∈Ai

(qa

i (k)−qai (k)

λai

)+L

2 ϕ2(k)∑

a∈Ai

(qa

i (k)−qai (k)

λai

)2

=∑

a∈Ai,a �=ai

(∂fi(xi(k))

∂xai

− ∂fi(xi(k))

∂xaii

)ϕ(k)

(qa

i (k)−qai (k)

λai

)+L

2 ϕ2(k)∑

a∈Ai

(qa

i (k)−qai (k)

λai

)2

≤ − ∑a∈Ai,a �=ai

aδ(k)

(qa

i (k)−qai (k)

λai

)2

+ L2 ·[(

qaii (k)−q

aii (k)

λaii

)2

+∑

a∈Ai,a �=ai

(qa

i (k)−qai (k)

λai

)2]

≤ ∑a∈Ai,a �=ai

−(

aδ(k) − L

2 |Ai|)(

qai −qa

i

λai

)2

,

(34)where the first equality follows from adding and subtractingthe same term ∂fi(xi(k))

∂xaii

ϕ(k)∑

a∈Ai

(qa

i (k)−qai (k)

λai

), the second

equality holds by (31), the second inequality follows fromthe fact that a ≤ ϕ ≤ 1 and the observation by (30) thatfor all a �= ai,

∂fi(xi(k))∂xa

i− ∂fi(xi(k))

∂xaii

≥ qai (k)−qa

i (k)δ(k)λa

i, and the

third inequality holds by (32). Therefore, when δ(k) ≤ 2aL|Ai| ,

the right side of (34) is non-positive so that fi(xi(k + 1)) ≤fi(xi(k)) always holds. This implies that the updating on{qa

i (k)} by the adaptation algorithm always reduces the localcost at sensor i.

From the KKT conditions of the SENSOR−i problem listedin (15)-(18), we can obtain Ca

i

′( qa

i∗

λai

)+λai = ν∗i . Since Ca

i (·) isstrictly convex, increasing and twice differentiable, the inversefunction of Ca

i

′(·), i.e., Ca

i

′−1(·), exists and is continuous.Thus, over the orthant qa

i∗ ≥ 0 for all a ∈ Ai, we have

qai∗ =

{0, if ν∗i < Ca

i

′(0) + λa

i

λai · Ca

i

′−1(ν∗i − λai ), if ν∗i ≥ Ca

i

′(0) + λa

i .(35)

In the adaptation algorithm, in order to obtain the optimalsolution, we always increase the minimum marginal cost, i.e.,

C ai

i

′ (q

aii

λaii

)+ λai

i , by increasing the corresponding payment

qai

i for anchor point ai, and decrease other marginal costs

Cai

′ ( qai

λai

)+ λa

i for all a ∈ Ai and a �= ai by reducingthe payments for them. In this way, we stipulate the marginalcosts towards to the same value ν∗i for all anchor points with

328

8

11

12

12

Anchor Point 3 Anchor Point 2

Anchor Point 1

3 4

5

6

7

9

10

Fig. 2. An example network with 12 sensors and 3 anchor points.

TABLE IIPARAMETER SETTINGS

Notation Value Notation Valueωa

i ranging from 0.01 to 0.08 T 42 secondsB 250kbps θ(n) 1

1+20npa

i ranging from 0.7 to 1 δ(k) 0.03Mi 800Kb ϕ(k) 0.8

positive qai∗’s (a ∈ Ai). Therefore, at the equilibrium, the

unique optimal solution can be achieved by (35).

VI. SIMULATION RESULTS

In this section, we provide simulation results to demonstratethe usage and efficiency of the proposed algorithm and com-pare its performance with another data gathering strategy.

A. Convergence

In this subsection, we illustrate the convergence of thepricing-based algorithm via a numerical case study. We con-sider a WSN with a total of 12 sensors as shown in Fig. 2. Thelocations of sensors 3, 4 and 5 are chosen as anchor pointsand each of these sensors would act as the helping node incomputing for the respective anchor point. In the figure, thereis a link between an anchor point and each of its neighboringsensors. We define the cost function as Ca

i (xai ) = ωa

i xai2,

where ωai is the weight of cost for sensor i to upload data

to the SenCar at anchor point a. Clearly, a larger weight ωai

would have more impact on the entire network cost. For clarity,we list all the parameter settings in Table II.

Fig. 3 shows the evolution of network cost, shadow price αa,recovered sojourn time variable ta, and data variable xa

i versusthe number of iterations in the pricing-based algorithm. It canbe seen from Fig. 3(a) that network cost first drops sharplyin the first few iterations and then slightly decreases until itreaches optimum. It falls within 2% of its optimum after only40 iterations. Fig. 3(b) shows that the shadow prices of threeanchor points converge very fast and they finally reach almostthe same value in the equilibrium. Since all shadow prices aremuch larger than zero, it indicates that the communicationopportunity between sensors and the SenCar at all anchorpoints is fully utilized. By the adjustment policy on shadowprices in the pricing-based algorithm, when T is large enoughto satisfy all the data uploading demands from sensors toeach anchor point, the corresponding shadow prices can bereduced to almost zero. Fig. 3(c) shows the convergence ofrecovered sojourn time at different anchor points. It furthervalidates that at any iteration step, the recovered sojourn timeis feasible to the primal SENCAR problem, i.e, satisfying thetotal sojourn time constraint, and when diminishing stepsizeis used, the recovery process guarantees its convergence tooptimum. In Fig. 3(d)-(f), we investigate the evolution of thedata amount uploaded from selected sensors 1, 6 and 10 to

their neighboring anchor points. We can see that they allapproach the stable state after 200 iterations. For a particularsensor, say, sensor 1, as its weight of the cost for anchorpoint 1 is smaller than those of other two anchor points,more data would destine to the SenCar at anchor point 1 soas to minimize the cost. In Fig. 4, we plot two instances todemonstrate the convergence of the adaptation algorithm forsolving the SENSOR−i problem with stepsize δ(k) = 0.03.We focus on sensor 1 in two cases where link price vectorsare λ1 = {1.11, 2, 3} and λ1 = {124.6, 112.3, 111.97},respectively. In both cases, we find that the payment for eachneighboring anchor point can be determined in about 1000iterations. It is clear that the smaller the stepsize is, the slowerthe convergence is, however, the smoother the adaptationtowards optimum. In practice, besides using the constantstepsize for the adaptation algorithm as in our simulations,each sensor can dynamically set its stepsize by first choosinga larger value to ensure faster convergence, and subsequentlyreducing the stepsize once there is an oscillating around somevalues.

B. Network Cost

In this subsection, we conduct a suite of simulations to eval-uate the network cost achieved by the pricing-based algorithmand compare the results with another data gathering strategycalled cluster-based algorithm, where sensors are virtuallyclustered, i.e., each sensor is randomly associated with ananchor point and uploads data to the SenCar only whenit arrives at this anchor point. This algorithm is commonlyconsidered as a simple and effective strategy for the anchor-based range traversing data gathering scheme in the existingliterature [8][9]. We consider a generic sensor network with|N | sensors randomly distributed over the sensing field. Westill assume that there are three anchor points that cover all thesensors. The cost functions are defined as Ca

i (xai ) = ωa

i xai2

for all a ∈ A, i ∈ N a and the weights of the cost ωai ’s are

generated as discrete uniform random numbers ranging from0.01 ∼ 0.10. The minimum data amount Mi for each sensoris equally set to 800Kb and channel bandwidth B equals250Kbps. If not specified otherwise, the successful deliveryratio pa

i of each link between a sensor and an anchor pointranges from 0.7 ∼ 1. Considering the randomness of thenetwork topology, each performance point in the figures belowis the average of the results in 100 simulation experiments.

Fig. 5 plots the network cost of the pricing-based algorithmwhen the bound of total sojourn time T is varied from 175seconds to 220 seconds. The number of sensors |N | is set to50. We introduce p to denote the average successful deliveryratio of all links and use it to characterize the physicalcondition of the network. We investigate network cost in fourcases, where p equals 0.85, 0.9, 0.95 and 1, respectively. Itcan be seen from the figure that in most cases, network costdecreases as T increases. This result is reasonable and can beexplained as follows. Since cost function Ca

i (·) is convex, it isexpected that each sensor sends parts of its data to the SenCarat different anchor points so as to minimize the aggregatedcost. When the restriction on the total sojourn time becomesloose, each sensor can send the preferred amount of data to

329

0 200 400 600 800 10002.6

2.8

3

3.2

3.4

3.6

3.8

4

4.2x 105

Number of iterations

Net

wor

k co

st

(a) Network cost vs. n.

0 200 400 600 800 10000

50

100

150

200

250

Number of iterations

Lagr

angi

an m

ultip

lier

α1, α2,α3

(b) αa vs. n.

200 400 600 800 10000

5

10

15

20

25

30

Number of iterations

Rec

over

ed s

ojou

rn ti

me

at a

ncho

r po

ints

(s) t1

t2t3

(c) ta vs. n.

200 400 600 800 10000

200

400

600

800

Number of iterations

Dat

a fr

om s

enso

r 1

to

neig

borin

g an

chor

poi

nts

(Kbi

ts)

x11

x12 x

13

(d) xa1 vs. n.

200 400 600 800 10000

200

400

600

800

Number of iterations

Dat

a fr

om s

enso

r 6

to

neig

borin

g an

chor

poi

nts

(Kbi

ts)

x62 x

63 x

61

(e) xa6 vs. n.

200 400 600 800 10000

200

400

600

800

Number of iterations

Dat

a fr

om s

enso

r 10

to

neig

hbor

ing

anch

or p

oint

s (K

bits

)

x101

x102

(f) xa10 vs. n.

Fig. 3. The evolution of network cost, shadow prices of different anchor points, recovered sojourn time for SenCar stopping at different anchor points, anduploading data from sensors 1, 6 and 10 versus the number of iterations in the pricing-based algorithm.

the SenCar more freely at different anchor points, otherwise,in order to ensure the bound of total sojourn time, sensors arerestricted to send more data to the SenCar at some particularanchor points in order to complete the data uploading in ashorter time. We also notice that for a given T , the caseswith a larger p always achieve lower network cost than thecases with a smaller p. For instance, when T = 180s, thecase of p = 0.95 results in 22% improvement on the networkcost with respect to the case of p = 0.85. When T becomeslarge enough, such as T > 205s, all the cases reach the sameminimum network cost, which implies that T no longer affectsthe network performance and the benefit of data control thatsmartly schedules the communication between sensors and theSenCar at different anchor points can be fully extracted by allcases.

Fig. 6 shows the network cost comparison between thepricing-based algorithm and the cluster-based algorithm whenthe number of sensors is varied from 10 to 200. The boundof sojourn time T is set to 4.5|N | seconds, which canaccommodate the data uploading from all sensors. From thefigure, we can draw some observations. First, network costincreases in both algorithms investigated as the number ofsensors increases. This is intuitive. As each sensor needs toupload 800Kb data to the SenCar, more cost is incurred bythe increase of sensors. Second, the pricing-based algorithmalways achieves lower network cost. For example, when|N | = 100, the pricing-based algorithm results in 32% lessnetwork cost with respect to the cluster-based algorithm. Theunderlying reason for such superiority of the pricing-basedalgorithm is that each sensor can adaptively split its data andsend the data to the SenCar at different neighboring anchorpoints such that the cost is minimized.

0 500 1000 1500 2000 2500 3000200

300

400

500

600

700

800

Number of iterations in adaptation algorithm

Pay

men

t fro

m s

enso

r 1

for

neig

hbor

ing

anch

or p

oint

sλ

1 = {1.11,2,3}

q11

q12

q13

(a) qa1 vs. k.

0 500 1000 1500 2000 2500 30001.5

2

2.5

3

3.5

4

4.5

5x 10

4

Number of iterations in adaptation algorithm

Pay

men

t fro

m s

enso

r 1

for

neig

hbor

ing

anch

or p

oint

s

λ1 = {124.6,112.3,111.97}

q11 q

12

q13

(b) qa1 vs. k.

Fig. 4. The evolution of the payment from sensor 1 for different anchorpoints versus the number of iterations in the adaptation algorithm.

VII. CONCLUSIONS

In this paper, we have studied performance optimization ofmobile data gathering in WSNs. We formalized the problem asa cost minimization problem constrained by channel capacity,the minimum amount of data uploaded from each sensor

330

180 190 200 210 2201.1

1.2

1.3

1.4

1.5

x 106

Bound of total sojourn time (s)

Net

wor

k co

st

|N| = 50

p = 1p = 0.95p = 0.9p = 0.85

Fig. 5. Network cost of the pricing-based algorithm as a function of thebound of total sojourn time T .

0 50 100 150 2000

2

4

6

8x 106

Number of sensors

Net

wor

k co

st

T = 4.5|N|

Pricing−based algorithmCluster−based algorithm

Fig. 6. Network cost of the pricing-based algorithm and the cluster-basedalgorithm as a function of the number of sensors.

and the bound of total sojourn time at all anchor points.We characterized this problem as a pricing mechanism anddecomposed it into two simpler subproblems, i.e., SENSOR−iand SENCAR subproblems. We have proved that network costcan be minimized by jointly solving the two subproblems.Correspondingly, we described a pricing-based algorithm thatiteratively solves SENSOR−i and the dual problem of SEN-CAR. In each iteration, the help node sets the shadow pricefor its local anchor point and derives link prices betweenneighboring sensors and the anchor point. Each neighboringsensor then determines the payments to minimize its local cost.The minimum network cost can be achieved when reachingthe equilibrium that reconciliates the two subproblems. Wealso proposed an efficient adaptation algorithm for solvingthe SENSOR−i subproblem at each sensor. Finally, we gaveextensive simulation results to validate the efficiency of theproposed algorithm and compare the performance with anotherdata gathering strategy.

ACKNOWLEDGMENTS

This research work was supported in part by the U.S.National Science Foundation under grant number ECCS-0801438, and U.S. Army Research Office under grant numberW911NF-09-1-0154.

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