The Impact of Transmit Rate Control on Energy-Efficient Estimation in Wireless Sensor Networks

IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 11, NO. 9, SEPTEMBER 2012 3261

The Impact of Transmit Rate Control onEnergy-Efficient Estimation in

Wireless Sensor NetworksIordanis Koutsopoulos, Member, IEEE, and Sławomir Stanczak, Senior Member, IEEE

Abstract—We study the impact of physical layer (PHY) trans-mit rate control on energy efficient estimation in wireless sensornetworks. A sensor network collects measurements about anunknown evolving process. Each sensor controls its sampling rateand its PHY transmit rate to the next hop or to the Fusion Center(FC). The FC performs estimation of the unknown process basedon sensor measurements and needs to adhere to an estimationaccuracy constraint. The objective is to maximize sensor networklifetime. The tradeoff is that, high PHY transmit rates consumemore energy per transmitted bit, but they increase the amountof transmitted sensor measurement data per unit time, and thusthey aid in improving estimation quality and in satisfying theestimation error constraint. First, we study a single-hop networkwhere sensors transmit directly to the FC. In this case, sensorsampling rates are directly mapped onto PHY transmit rates.We identify fundamental structural properties of the optimalsolution, and we propose a distributed, iterative sensor PHYrate adaptation algorithm for reaching a solution, based on light-weight feedback from the FC. Next, we consider the multi-hopversion of the problem, where the sensor measurement (sampling)rates, PHY transmit rates and data flows to the FC are controlled.We extend the distributed optimization framework above toinclude all controllable parameters, and we devise an iterativealgorithm for maximizing network lifetime.

Index Terms—Wireless sensor networks, distributed optimiza-tion, PHY transmit rate adaptation, estimation, energy efficiency.

I. INTRODUCTION

RECENT advances in hardware and signal processingmodules of miniature sensor devices have enabled dy-

namic adaptation of physical layer parameters such as mod-ulation level or channel coding rate, in response to varyinglink conditions. High transmission rates, realized through highmodulation levels and/or coding rates are used in favorablelink conditions so as to convey more bits to the receiver. Lowtransmission rates are used when link conditions deteriorateso as to make transmission more robust to link errors.

Manuscript received July 24, 2011; revised December 19, 2011 and March24, 2012; accepted April 23, 2012. The associate editor coordinating thereview of this paper and approving it for publication was S. Cui.

I. Koutsopoulos is with the Department of Computer and CommunicationsEngineering, University of Thessaly, Volos, Greece, and the Centre forResearch and Technology Hellas (CERTH), Greece (e-mail: [email protected]).

S. Stanczak is with the Fraunhofer Heinrich Hertz Institute, Berlin, and theDepartment of Electrical Engineering and Computer Science, Technical Uni-versity of Berlin, Germany (e-mail: [email protected]).

Part of the material in the paper appeared in [1] at the GLOBECOM 2010Conference, Miami, FL.

Digital Object Identifier 10.1109/TWC.2012.062012.111392

We study the impact of physical layer (PHY) transmit ratecontrol on the lifetime of a wireless sensor network. A sensornetwork collects measurements about an unknown evolvingprocess. Each sensor controls its sampling rate and its PHYtransmit rate to the next hop or to the Fusion Center (FC).The FC performs estimation of the unknown process based onsensor measurements, and it needs to adhere to an estimationaccuracy constraint. Although high transmit rates require alarger amount of energy per bit in order to be supported undergiven link conditions, they turn out to be beneficial in termsof estimation quality and could be employed when needed.With higher PHY transmit rates, a larger amount of sensormeasurement data per unit time is conveyed to the destina-tion, since the packet transmission duration is reduced. Byopportunistically exploiting good channel conditions, a largeamount of data is transmitted to the FC, and better estimationperformance is achieved, namely a smaller estimation error.

The problem we address in this paper is precisely therelation between energy efficiency (which is signified bymaximum operational lifetime of the sensor network) andestimation accuracy, and how this relation is shaped by PHYtransmit rate control.

A. Related work

Although transmit rate control has been around in cellularwireless systems for quite some time (see e.g. [2] whichstudies the throughput advantage of adaptive modulation incellular systems, and references therein), its employment inwireless sensor networks started later [3] due to the needfor advanced signal processing modules to support adaptationtechniques in sensor devices. The work [4] is among the firstones to consider energy consumption for both the transmit andthe electronic circuitry, towards identifying the most energy-efficient modulation scheme in the sense of minimizing energyconsumption to transmit a given number of bits.

A seminal work on lifetime maximization in wireless sensornetworks is [5], where routing of sensor data flows with givensampling rates is addressed, albeit with no consideration ofestimation constraints. Network lifetime is defined as the timeelapsed until the first sensor battery empties. The key ideais that routing flows should be such that the sensor batteryenergy depletion rate is balanced across sensors. A cross-layer design framework for lifetime maximization is proposedin [6] with optimization of routing flows, link schedules andtransmit powers. In [7], the same authors formulate and solve

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3262 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 11, NO. 9, SEPTEMBER 2012

a total energy minimization problem. Along with link androuting layer decisions, they use the framework of [4] toperform transmit rate control to shape the tradeoff betweendelay and energy consumption. Routing and sensor placementfor maximizing network lifetime were studied in [8], wherethe distortion depended on sensor positions relative to cluster-heads. Rate distortion theory was used for formulating a typeof lifetime-distortion tradeoff in [9].

Various aspects of estimation objectives are considered inthe literature [10]-[12]. In [10], the authors target energyminimization through finding the number of quantizationlevels for given estimation error. The work in [11] studiesthe problem of power allocation for minimum estimationerror subject to a power constraint and the problem of powerminimization subject to an estimation error constraint. Theseproblems possess the water-filling structure, i.e, at the optimalsolution, only a subset of sensors transmit, and the rest areoff. In [12], a definition of functional lifetime is used, i.e.the cycles elapsed until the network satisfies the estimationconstraint. A common assumption in these works is that eachsensor transmits at most one sample to the fusion center (FC)which then performs the estimation. The work [13] studiesthe impact of spatial correlation on sensor sampling rate androuting in order to maximize lifetime subject to an estimationerror constraint. The recent work [14] studies total utilitymaximization in a sensor network with rechargeable batteries,where the utility is a concave function of sampling rate ofeach sensor. A dual decomposition algorithm is proposed forcomputing the sampling rates and routes, subject to a flowconservation and an energy conservation constraint for eachsensor.

B. Our contribution

To the best of our knowledge, this is the first work thatbrings together the PHY layer transmit rate control and theapplication layer aspect of estimation quality, as well astheir joint impact on sensor network lifetime. The effect ofPHY rate control on energy consumption is modeled throughthe energy consumed for transmitting a message through thepower amplifier and the energy for keeping the hardwarecircuitry on. A higher PHY transmit rate needs more transmitenergy per bit, but less energy to keep the circuitry on,since the packet duration is smaller. The problem is studiedfirst for an one-hop network where sensors transmit theirmeasurements directly to the FC [1], and then for a multi-hop network where measurements reach the FC in multi-hopfashion.

Our contribution is as follows: (i) we relate PHY layertransmit rate adaptation to estimation error performance, andwe formulate the problem of sensor lifetime maximizationsubject to an estimation error constraint, (ii) we solve theproblem precisely for two sensors and one-hop transmissionand show that it is an instance of a max-min optimizationproblem, (iii) for the one-hop network, where sensor samplingrates and PHY transmit rates to the FC are controlled, we showthat sampling rates are mapped onto PHY transmit rates andwe provide a lightweight decentralized algorithm for reachinga solution, (iv) for the multi-hop network, we extend the

i

Fusion Center

and control messages

measurementsSensor Transmission

im

TT’ss

bi1 N

Fig. 1. One-hop transmission from the m sensors to the FC. At each epoch,each sensor i takes Ni measurements during the measurement and controlinterval of duration T ′

s. Control messaging between each sensor and the FCtakes place at that time. Then, during the transmission interval of durationTs, each sensor transmits its measurements to the FC with PHY rate bi.

framework above and devise a decentralized algorithm forcontrolling sensor sampling rates, PHY transmit rates androuting flows. The rest of the paper is organized as follows.First, we study the one-hop network problem. In section IIwe present the model, and in section III we solve the specialcase for two sensors and present our algorithm. In section IVwe consider the multi-hop network problem. In section V weprovide numerical results, and in section VI we conclude.

II. THE ONE-HOP NETWORK SCENARIO

In order to gain insight on the problem and the structure ofthe solution, we first consider the one-hop network problem,where a set N of m sensors transmit directly to a fusion center(FC) in single-hop. Each sensor has initial energy reserve Ai.

A. Sensor Measurement Model

Time is divided in intervals of duration (Ts + T ′s) each,

which we call epochs. Within each epoch, sensors submitmeasurements to the FC about a slowly time-varying, un-known, spatially homogeneous phenomenon process. This isa parameter sequence {θt}t=1,2,..., where θt is the unknownvalue of the process at epoch t, which is assumed to remainfixed for the epoch duration.

Each epoch consists of a measurement / control interval ofduration T ′

s and a transmission interval of duration Ts (Fig.1). In the former, the FC broadcasts control information tosensors. The FC transmit power is large enough so that itreaches all sensors. Within the same interval, each sensor icollects measurements at time instants τ , given by xi(τ) =θτ+ni(τ), where xi(τ) is the measurement at time τ and θτ =θt, ni(τ) = ni(t), if time instant τ is in epoch t. The noiseprocess ni(t) captures uncertainty of sensor i measurementdue to different perception of the phenomenon process and dueto residual measurement errors. For each i, ni(t) is Gaussian,zero mean, wide-sense stationary and uncorrelated in time.The variance of ni(t), σ2

i = E[n2i (t)], is fixed for all epochs t

and captures the magnitude of measurement inaccuracy. Thenoise processes of any two sensors i and j are assumed to bespatially and temporally uncorrelated.

During the transmission interval, sensors transmit measure-ments to the FC. We assume no quantization or compression

KOUTSOPOULOS and STANCZAK: THE IMPACT OF TRANSMIT RATE CONTROL ON ENERGY-EFFICIENT ESTIMATION IN WIRELESS SENSOR . . . 3263

at bit level. The epoch is large enough so that measurementsof sensors reach the FC. At the end of the epoch, the FCmakes the estimation. Coordination of sensor transmissions togather the data to the FC is realized through an underlyingchannelized access protocol over a set of orthogonal channels(frequencies). A channel allocation scheme can ensure thatdifferent channels are used for sensors that are within wirelessinterference range of each other. A channel can be reused bysensors that are located far away from each other.

B. Transmission Model

Each sensor i can adapt its PHY layer transmission rate bi(in bits/symbol) at each epoch. We treat PHY transmissionrate as a continuous variable, and we do not place additionalrestrictions on its value, so as to gain insight on propertiesof the solution. Note that in reality, the transmission rateis chosen from a finite set B = {b(1), . . . , b(K)} of Kdifferent modulation levels, where b(�) denotes the numberof bits/symbol for the �-th modulation level, and b(1) <. . . < b(K). Define s > 0 to be the (fixed) symbol rate (insymbols/sec), common for all sensors. Let L be the numberof bits of each measurement packet which is assumed to befixed. The PHY transmission rate (in bits/sec) for sensor i isbis and the required time to transmit one packet is L/(bis)sec.

We consider the following types of energy:

• The energy to feed the power amplifier for transmitting ameasurement packet to the FC under certain bit error rate(BER) specification ε. This is referred to as the transmitenergy (per packet) and is denoted by et,i for sensor i.

• The energy required for the transmit circuitry to be onduring transmission of one packet. This energy is calledon-energy and is denoted by eo.

Fix attention to sensor i. Let Pi be the required transmitpower to transmit a packet to the FC so that BER ≤ ε.Clearly, Pi depends on the propagation environment betweensensor i and the FC receiver, which includes path loss,shadowing and fading. All the above are captured by linkgain G

(t)i from sensor i to the FC, which is assumed invariant

during epoch t. We drop the epoch index t for now. TheSNR at the FC receiver for a received packet by sensor i isSNRi =

GiPi

w , where w is the average Gaussian noise powerat the receiver.

For an M -QAM modulation level of b bits/symbol withM = 2b, the minimum SNR that guarantees BER ≤ ε isγ(b) = d(ε)(2b − 1), where d(·) : (0, 1/2] �→ [0,∞) is a non-increasing function of the BER requirement ε > 0 that dependson physical layer realization, with d(ε) → 0 as ε → 1/2 andd(ε)→∞ as ε→ 0. If ε < 1/5, a widely used assumption isd(ε) = − ln(5ε)/1.5 [2]. By combining the SNR expressionabove with the requirement SNRi ≥ γ(bi), we get that Pi ≥wGi

γ(bi) must hold, in order to have BER ≤ ε. A necessarycondition for achieving a maximum network lifetime is thatPi =

wGi

γ(bi) for each i ∈ N , which is assumed to be truein all that follows. Indeed, there is no reason to exceed thepower in the equality above, since that would lead to higherenergy consumption.

The transmit energy of sensor i for one packet is et,i =Pi

Lbis

= PiLbis

(2bi − 1), where Pi = wGi

d(ε) (in Watts)depends on the link gain from sensor i to the FC, the receivernoise power and the BER specification. Each sensor knows Pi

through feedback from the FC.The on-power required to transmit a packet is constant, call

it Po, and it depends on the circuit hardware implementation,semiconductor material and other related factors. The on-energy to transmit a packet depends on the packet duration.For sensor i with transmission rate bi bits/symbol, the on-energy per packet is eo,i = LPo/(bis). Thus, the total amountof energy per packet for sensor i is

ei = et,i + eo,i =L

sbi[Pi(2

bi − 1) + Po] . (1)

C. Maximum Likelihood Estimation

Within each epoch, the FC obtains sensor measurementsand computes an estimate θ of the unknown parameter valueθ at that epoch. First, assume that each sensor sends onemeasurement to the FC, so that a vector of m measurementsx = (x1, . . . , xm)

T is available. Since measurements ofsensors are uncorrelated, the joint p.d.f. of vector x is pθ(x) =∏m

i=1 pθ(xi), where p.d.f. pθ(xi) is Gaussian, with mean θand variance σ2

i . We consider estimation in the MaximumLikelihood (ML) sense; this is a special case of MaximumA Posteriori estimation in case there is no prior statisticalinformation about the unknown θ. The ML estimate of θ isθML = argmaxθ pθ(x) = argmaxθ log pθ(x), and it is

θML =

(m∑i=1

1

σ2i

)−1

·(

m∑i=1

xi

σ2i

). (2)

The criterion for estimation quality is the mean squarederror (MSE), E[(θ − θML)

2], where the expectation is withrespect to the randomness of observations. To make estimationquality independent of θ, we consider unbiased estimators,for which E(θML) = θ. Then, the MSE equals var(θML).The minimum variance unbiased (MVU) estimator minimizes

var(θML), and we get var(θML) =(∑m

i=11σ2i

)−1

. We require

that var(θML) ≤ ε where ε > 0 specifies a given estimationaccuracy.

Assume now that at a given epoch, each sensor i =1, . . . ,m, takes Ni measurements, denoted by vector x(i) =

(x(i)1 , . . . , x

(i)Ni

), and it transmits them to the FC. Let ni,i = 1, . . . ,m be indices with 1 ≤ ni ≤ Ni. Measurements ofeach sensor are uncorrelated among themselves. Since sensorsare also spatially uncorrelated, all sensor measurements areuncorrelated, and under Gaussian assumption, they are alsoindependent. The ML estimate of θ is

θML =

(m∑i=1

Ni∑ni=1

x(i)ni

σ2i

)·(

m∑i=1

Ni

σ2i

)−1

, (3)

and the variance of estimation error is

var(θML) =

(m∑i=1

Ni

σ2i

)−1

. (4)


III. PHY TRANSMIT RATE CONTROL IN A ONE-HOP

NETWORK

A. Problem statement

A high PHY transmission rate implies larger consumedenergy per packet, as observed by (1). On the other hand,with high transmission rate, the packet duration decreases,and thus more measurement packets can be packed and sentto the FC during the transmission interval Ts of an epoch.Therefore, the performance in terms of estimation error isimproved, as implied by (4). That is, with more measurementdata, the variance of estimation error decreases, and thusestimation becomes more accurate. The problem of optimizingthe energy-accuracy tradeoff above becomes interesting, sincedifferent sensors have different link gain to the FC due todifferent physical locations and fading. Therefore, they differin the amount of consumed energy per transmitted packet.Sensors may also have different initial energy budgets andmeasurement accuracies. For instance, among sensors with thesame energy budget and measurement accuracy, it is betterto use higher PHY transmit rate (and thus transmit moremeasurements) for the sensor that has higher link gain, sincein that case, the consumed amount of energy per packet issmaller.

For sensor i that takes Ni measurements and transmits Ni

measurement packets per epoch, the energy depletion rate (inJoules per epoch) is eiNi. Since each sensor uses one channelto transmit its packets, the total time needed for sensor i totransmit the Ni measurement packets should not exceed Ts.The time duration to transmit one packet is L/(sbi), and thuswe require that Ni

Lsbi≤ Ts. Therefore, packets should be

transmitted with PHY rate bi ≥ (LNi)/(sTs). Since energyconsumption ei is an increasing function of PHY transmit rate,the PHY rate employed must be equal to the lower bound, i.e.

bi =LNi

sTs. (5)

The number of measurement packets per epoch is thus pro-portional to PHY transmit rate, i.e. Ni = sTsbi/L. Due to theequality in (5), the entire transmission interval of duration Ts

is utilized. The energy depletion rate of sensor i in Joules perepoch, as a function of its transmission rate bi is

zi(bi) = Niei = Ts[Pi(2bi−1)+Po] = Ei(2

bi−1)+Eo , (6)

where Ei = PiTs and Eo = PoTs are the energy depletionrates due to energy consumed for transmission and for keepingthe circuitry on.

Remark: A remark about the multiple access of the msensors to the FC is in place. Since the channels of sensorsare parallel orthogonal channels in the frequency domain,the multiple access requirement reduces to that of simplyhaving each sensor utilize its entire measurement intervalTs to transmit its Ni measurements to the FC. A differentassumption about the channel configuration would lead to adifferent multiple access constraint. If, for instance, all sensorswere using the same channel, time sharing would need tobe employed. The channels would be orthogonal in the timedomain and the constraint

∑mi=1 Ni

Lsbi≤ Ts would be needed.

B. Problem formulation

The lifetime Li(·) of sensor i is the number of epochs untilits battery empties. As a function of the instantaneous PHYrate bi at an epoch, it is

Li(bi) =Ai

zi(bi)=

Ai

Ei(2bi − 1) + Eo. (7)

As will be seen later, the PHY rate changes from epoch toepoch, and so is the anticipated lifetime Li(·). The sensornetwork lifetime is defined as the number of epochs elapseduntil the battery of the first sensor empties, namely L(b) =mini=1,...,m Li(bi) [5]. This is depends on the number ofmeasurement packets from each sensor and thus on the sensortransmit rate vector b = (b1, . . . , bm).

The problem of PHY transmit rate control with the objectiveto maximize lifetime subject to a maximum tolerable varianceof estimation error, var(θML) ≤ ε is formulated as:

supb≥0

ω(b), ω(b) := mini=1,...,m

Ai

Ei(2bi − 1) + Eo(8)

subject to:m∑i=1

biσ2i

≥ L

sεTs. (9)

Inequality (9) that relates the variance of estimation error toPHY transmit rates, emerges from the estimation constraintand the fact that Ni = sTs

L bi. Note that since ω(b) isstrictly decreasing in each bi, we must have supb≥0 ω(b) =

supb∈Π ω(b), where Π = {b ≥ 0 :∑m

i=1biσ2i

= LsεTs}.

Moreover, as Π is a compact set and ω(·) is continuous on Π,the supremum in (8) is attained for some b ≥ 0. Thus, at theoptimal solution, constraint (9) is satisfied with equality.

It is worth pointing out that the problem always has asolution, since we have neglected any constraints on transmitpowers, and with it on sensor transmit rates. In contrast, ifthere were individual power constraints Pi ≤ Pi on eachsensor i, it can be verified that the problem has a solutionif and only if

m∑i=1

log2(1 +PiGi

wd(ε))

σ2i

≥ L

sεTs.

If the inequality above is not satisfied, the estimation qualityrequirement cannot be met.

C. Solution for m = 2 sensors

For m = 2 sensors, the lifetime maximization problembecomes:

maxb1,b2

min{ A1

E1(2b1 − 1) + Eo,

A2

E2(2b2 − 1) + Eo

}(10)

subject to:b1σ21

+b2σ22

=L

sεTs, (11)

and b1, b2 ≥ 0. Since an optimal solution is positive, wecan define xi = 2bi , so that bi = log2 xi, for i = 1, 2 andtransform the problem to the following equivalent one:

minx1,x2

max{Eo + E1(x1 − 1)

A1,Eo + E2(x2 − 1)

A2

}(12)


subject to:

x1/σ2

11 x

1/σ22

2 = 2L/sεTs , (13)

and x1, x2 > 0. From (13), variables x1 and x2 should satisfy:

x2 =2Lσ2

2/sεTs

xσ22/σ

21

1

=β

xα1

, (14)

with α = σ22/σ

21 > 0 and β = 2Lσ2

2/sεTs > 0. The solution tothe problem is given as the solution of

minx1

max{Eo + E1(x1 − 1)

A1,Eo + E2(βx

−α1 − 1)

A2

}(15)

with x1 > 0. Functions f1(x) = [Eo + E1(x − 1)]/A1 andf2(x) = [Eo + E2(βx

−α − 1)]/A2 are positive. Functionf1(x) is affine and strictly increasing with x > 0. Functionf2(x), x > 0, is convex and strictly decreasing for any α > 0.Since both functions are also bijective, they have exactly onepoint of intersection, which is given by the solution to equation

A2E1xα+11 + γxα − βA1E2 = 0 ,

with γ = A2(Eo − E1) − A1(Eo − E2). For equal initialenergy reserves A1 = A2 and variances σ2

1 = σ22 , so that

α = 1, the positive solution to the second-order polynomialequation E1x

21 + (E2 − E1)x1 − βE2 = 0 is,

x∗1 =−(E2 − E1) +

√(E2 − E1)

2 + 4βE1E2

2E1. (16)

The optimal transmit rates of sensors are b∗1 = log2 x∗1 and

b∗2 = log2 β − αb∗1. In Fig. 2 we depict functions f1(·) andf2(·) and their intersection min-max point x∗

1.

D. Distributed algorithm for sensor PHY rate adaptation

We provide an iterative decentralized algorithm for the caseof m sensors. First, we convert the formulation above to anequivalent one, similarly in spirit to [5], by defining a newvariable

ω = mini=1,...,m

Ai

Ei(2bi − 1) + Eo.

The equivalent problem becomes

maxω>0,b≥0

ω (17)

subject to:

ω[Ei(2

bi − 1) + Eo

] ≤ Ai , for i = 1, . . . ,m , (18)

and subject to the estimation error constraint:

m∑i=1

biσ2i

=L

sεTs. (19)

We relax the problem constraints and form the Lagrangian,

L(ω,b,λ, μ) = −ω +

m∑i=1

λi

[ω(Ei(2

bi − 1) + Eo

)−Ai

]

+ μ( L

sεTs−

m∑i=1

biσ2i

), (20)

x*

f (x)

f (x)

1

2

min−max point

1 − E / E 0 1

x=x

Fig. 2. Graphical representation of the solution for 2 sensors. Functionsf1(x) = [Eo +E1(x1 − 1)]/A1 and f2(x) = [Eo +E2(βx

−α1 − 1)]/A2

represent the inverse of lifetime of sensors 1 and 2 respectively, as functionsof x = 2b1 , where b1 is the transmission rate of sensor 1, and α and β aredefined in (14). The thick part of the plot shows function max{f1(x), f2(x)}.The min-max point x∗ is the solution, and b∗1 = log2 x

∗.

where λ = (λ1, . . . , λm) ≥ 0 and μ are the dual variables forconstraints (18) and (19) respectively. For given λ, μ, considerfirst the primal problem,

minω>0,b≥0

L(ω,b,λ, μ) . (21)

A gradient descent step, ω(t) = ω(t−1) − st∂L(·)∂ω , is executed

by the FC to adjust ω(t), where st > 0 is the step size atiteration t. The optimization with respect to bi is performedby each sensor separately, by taking ∂L(·)/∂bi = 0. The dualproblem is:

maxλ,μ

D(λ, μ) = maxλ,μ

minω,b

L(ω,b,λ, μ) , (22)

where D(λ, μ) is the dual function. Note that due to the min-operator in the definition of ω, the dual function is not ingeneral differentiable with respect to λ, μ, since it is piecewiselinear. Hence, gradient update methods cannot be used for thedual problem, and we resort to sub-gradient ones [15, Sec.6.3.1]. It should be noted that there is duality gap since theproblem is not convex.

For concave function f(·), vector d is called a sub-gradientat point x0 if f(x) ≤ f(x0) + (x− x0)

Td. As the dual

function D(λ, μ) is always concave in λ, μ, a sub-gradientfor D(λ, μ) with respect to λ at point λ(t−1) is the vectorwhose i-th component is ω(t−1)

(Ei(2

b(t−1)i − 1) +Eo

)−Ai.

Each sensor i knows its PHY transmission rate b(t−1)i and

the value of ω(t−1) that is broadcast by the FC at the end ofthe previous iteration. It updates its Lagrange multiplier λ

(t)i

with sub-gradient ascent,

λ(t)i =

[λ(t−1)i + st

(ω(t−1)

(Ei(2

b(t−1)

i − 1) + Eo

)−Ai

)]+(23)

where x+ = max{x, 0}. Similarly, a sub-gradient of D(λ, μ)with respect to μ at point μ(t−1) is found, and we obtain the


sub-gradient update for Lagrange multiplier μ at the fusioncenter,

μ(t) = μ(t−1) + st

(L

sεTs−

m∑i=1

b(t−1)i

σ2i

). (24)

The channel gain G(t)i , and thus the energy E

(t)i =

wTsd(ε)/G(t)i for sensor i changes at each epoch t. Here, we

assume that the inverse channel gain process {1/G(t)i }t=1,2,...

is stationary and its mean exists and is 1/gi. Then, the process{E(t)

i }t=1,2,..., is also stationary with mean Ei = wTsd(ε)/gi.This mean energy Ei is used in the proposed primal-dualalgorithm which goes as follows:

• STEP 0: Initialization. At epoch t = 0, the FC initializesμ(0) ≥ 0 and ω(0) > 0 and broadcasts them to thenetwork. Each sensor i initializes λ(0)

i and b(0)i and sends

them to FC via a feedback control channel. Set t = 1.Go to Step 1.

• STEP 1: Each sensor i updates its multiplier λ(t)i based

on (23) and sends it to the FC.• STEP 2: The FC updates multiplier μ(t) using (24).• STEP 3: The FC updates parameter ω(t) according to

ω(t) = [ω(t−1)−st(−1+ m∑

i=1

λ(t)i (Eo+Ei(2

b(t−1)i −1))]+ .

(25)It then broadcasts μ(t) and ω(t) to the sensors.

• STEP 4: Each sensor, say sensor i adapts its PHYtransmit rate b

(t)i based on

b(t)i =

⎧⎪⎨⎪⎩

log2( 1

σ2iEi(ln 2)

μ(t)

λ(t)i ω(t)

), if μ(t)> 0,

[b(t−1)i − stλ

(t)i ω(t)Ei(ln 2)2

b(t−1)i ]

+

, if μ(t)= 0.(26)

• STEP 5: Each sensor transmits its measurements duringepoch t to the FC. The FC performs the estimation.

• STEP 6: t ← t+ 1. Go to Step 1. Continue until sometermination condition is satisfied.

Steps 1-4 take place during the measurement and controlinterval of each epoch t, while step 5 takes place in thetransmission interval. In step 4, either the optimization is one-shot with ∂L(·)

∂bi= 0, or a gradient descent step is performed

when the one-shot optimization is not feasible.During the iterations, if constraint (19) is violated and the

variance of estimation error exceeds ε, the update rule (24)increases μ and thus, by (26) the transmission rates (and thus,the numbers of measurements) Ni for each sensor i increase,so as to reduce the estimation error. If constraint (18) isviolated for some sensor i, this implies that for sensor i theenergy depletion rate is larger than the minimum depletionrate across sensors. Then, multiplier λi is reduced due to (23)and variable ω is reduced as in (25) so as to make the energydepletion rate as balanced as possible.

Note that, in addition to its Ni measurements, a sensor isubmits a signaling parameter λ

(t)i to the FC in step 1 of

each epoch. The additional energy consumption to transmitthis information can be taken into account by defining theenergy depletion rate in (6) as zi(bi) = (Ni + 1)ei.

kiFC

bij

i

Ni

T’ss

T

reception measurements and controlmessages

transmission

sT

j

kb

Fig. 3. Multi-hop transmission from the m sensors to the FC. At eachepoch, each sensor receives measurement data from its in-neighbors duringthe reception interval of duration Ts, with rate bki from in-neighbor k. Next,it takes Ni measurements during the measurement and control interval ofduration T ′

s. Control messaging between neighboring sensors and betweensensors and the FC takes place at that time. Finally, each sensor forwardsmeasurement data to its out-neighbors during the transmission interval ofduration Ts, with rate bij for out-neighbor j.

If the sequence of steps {st} satisfies limt→+∞ st = 0 and∑t st = +∞,

∑t s

2t < +∞, the algorithm converges at least

to a local optimum of the original problem [15].

IV. THE PROBLEM IN A MULTI-HOP SENSOR NETWORK

A. Model and notation

We now consider a sensor network where measurement datais to be transported to the FC in multi-hop fashion (Fig. 3).The multi-hop version of the problem is a nontrivial extensionof the one-hop problem and warrants separate investigation.

For each sensor i, we define N outi to be the set of out-

neighbors of i, namely the set of sensors that can be directlyreached by sensor i with a certain transmit power level. LetN in

i be the set of in-neighbors of i, i.e. those sensors whichcan directly reach i. The transmit power level of i is assumedto be adjusted to the minimum level needed for an intendedreceiver j ∈ N out

i . Using the rationale of section II-B, thetransmit energy per packet from sensor i to j ∈ N out

i is,

et,ij = PijL

sbij(2bij − 1) , (27)

where bij is the PHY transmit rate used from sensor i to j,and Pij =

wGij

d(ε), where Gij is link gain between sensors iand j.

The epoch for each sensor is divided into (i) a receiveinterval of duration Ts, within which data from neighbors arereceived, (ii) a control and measurement interval of durationT ′s, where control packets are exchanged among sensors and

the FC, and (iii) a transmit interval of duration Ts, wheredata are transmitted to each neighbor. For each sensor i, thefollowing types of energy are considered:

• The transmit energy per packet, et,ij for transmission toeach out-neighbor of i, j ∈ N out

i , given by (27).• The on-energy per packet, eo,ij = LPo

sbijwhich is the

required energy for the transmit circuitry for transmissionfrom i to j ∈ N out

i ,


• The receive energy per packet, er,ki = LP ′o/(sbki) that

captures the required energy of the receive circuitry forreception from in-neighbor k ∈ N in

i .

In the expressions above, Po, P ′o are the on-powers to

transmit and receive a packet respectively. A sensor i may alsotransmit directly to the FC, and thus the FC can be neighborof i.

The channels used for transmission from sensor i to eachneighbor j ∈ N out

i are orthogonal to each other in thefrequency domain, and a different portion of the transmitcircuitry needs to be powered to transmit to each orthogonalchannel. Let Nij be the number of measurement packets thatare forwarded from sensor i to j in an epoch. The durationof a packet is L/(sbij). Since transmissions to each neighborj ∈ N out

i take place in orthogonal channels, the time neededfor sensor i to transmit its measurements to each j shouldnot exceed Ts, that is, Nij

Lsbij≤ Ts. Thus, the PHY rate bij

should satisfy bij ≥ LNij

sTs, and since energy consumption is an

increasing function of PHY rate, we have bij = LNij/(sTs),for i = 1, . . . ,m, j ∈ N out

i .Let bi = (bij : j ∈ N out

i ) be the vector of PHY transmitrates of sensor i to its out-neighbors. The energy depletionrate of sensor i, in Joules per epoch, is

zi(bi) =∑

j∈Nouti

(Eij(2bij − 1) + Eo) +

∑k∈N in

i

E′o =

∑j∈Nout

i

Eij(2bij − 1) + |N out

i |Eo + |N ini |E′

o, (28)

with Eij = PijTs, Eo = PoTs and E′o = P ′

oTs. In the sequel,we set ei = |N out

i |Eo + |N ini |E′

o.

B. Problem formulation

Let b = (b1, . . . ,bm) be the ensemble of PHY ratevectors of all sensors. Denote again by Ni the numberof measurements per epoch obtained by sensor i, and letN = (N1, . . . , Nm). The problem of maximizing networklifetime is formulated as:

supb≥0

ω(b), ω(b) := mini=1,...,m

Ai

ei +∑

j∈Nouti

Eij(2bij − 1)

(29)subject to a flow conservation constraint (in bits per epoch)for each sensor i,

∑k∈N in

iNki +Ni =

∑j∈Nout

iNij , which

is written in terms of PHY rates as

sTs

L

∑k∈N in

i

bki +Ni =sTs

L

∑j∈Nout

i

bij , i = 1, . . . ,m, (30)

and subject to estimation error constraint,

m∑i=1

Ni

σ2i

≥ 1

ε. (31)

Note that the flow conservation constraint (30) brings to-gether the sensor sampling rates and PHY transmit rates andmakes the problem and the solution procedure different fromthe previous one.

Again, we define the new variable ω that appears in (29),and we get the equivalent problem:

maxω>0,b≥0,N≥0

ω (32)

subject to:

ω(ei +

∑j∈Nout

i

Eij(2bij − 1)

) ≤ Ai , i = 1, . . . ,m , (33)

and constraints (30) and (31). The Lagrangian is

Lo(ω,b,N,λ,μ, ν) = −ω + ν(1ε−

m∑i=1

Ni

σ2i

)

+m∑i=1

λi

[ω(ei +

∑j∈Nout

i

Eij(2bij − 1)

)−Ai

]

+

m∑i=1

μi

(sTs

L

∑k∈N in

i

bki +Ni − sTs

L

∑j∈Nout

i

bij),

(34)

where λ ≥ 0, μ, and ν ≥ 0 are the real-valued Lagrangemultipliers for constraints (33),(30) and (31) respectively. Wehave that

m∑i=1

∑k∈N in

i

μibki =

m∑k=1

∑i∈Nout

k

μibki =

m∑i=1

∑j∈Nout

i

μjbij ,

(35)so that the Lagrangian yields

Lo(ω,b,N,λ,μ, ν) = −ω + ν(1ε−

m∑i=1

Ni

σ2i

)

+

m∑i=1

λi

[ω(ei +

∑j∈Nout

i

Eij(2bij − 1)

)−Ai

]

+sTs

L

m∑i=1

∑j∈Nout

i

(μj − μi)bij +

m∑i=1

μiNi .

(36)

The primal problem is

minω>0,b>0,N>0

Lo(ω,b,N,λ,μ, ν) , (37)

while the dual one yields

maxλ,μ,ν

minω,b,N

Lo(ω,b,N,λ,μ, ν) . (38)

The primal-dual algorithm goes as follows:• STEP 0: Initialization. At epoch t = 0, the FC initializes

ν(0) ≥ 0 and ω(0) > 0 and broadcasts them to thenetwork. Each sensor i initializes λ

(0)i > 0, μ(0)

i , N (0)i

and b(0)i and sends them to the FC via a feedback control

channel. Set t = 1. Go to Step 1.• STEP 1: Each sensor i updates its multiplier λ

(t)i with

the sub-gradient ascent rule,

λ(t)i =

[λ(t−1)i + st

(ω(t−1)(ei + ∑

j∈Nouti

Eij(2b(t−1)ij − 1)

)− Ai

)]+

(39)and sends it to the FC. Each sensor also updates its

multiplier μ(t)i with the sub-gradient ascent rule,

μ(t)i = μ

t−1)i +st

(sTs

L(∑

k∈N ini

b(t−1)ki −

∑j∈Nout

i

b(t−1)ij )+N

(t−1)i

)

(40)


and sends it to its in-neighbors.• STEP 2: The FC updates multiplier ν(t) with

ν(t) =[ν(t−1) + st

(1ε−

m∑i=1

N(t−1)i

σ2i

)]+. (41)

• STEP 3: The FC updates parameter ω(t) as

ω(t) =[ω(t−1)−st

(−1+

m∑i=1

λ(t)i (ei+

∑j∈Nout

i

Eij(2b(t−1)ij −1))

)]+.

(42)• STEP 4: Each sensor i updates the number of measure-

ments per epoch (sampling rate) N (t)i ,

N(t)i = [N

(t−1)i − st(μ

(t)i −

ν(t)

σ2i

)]

+

. (43)

Each sensor i adapts its PHY transmit rate vector to itsout-neighbors, b(t)

i based on

b(t)ij = log2

μ(t)i − μ

(t)j

Eijλ(t)i ω(t) ln 2

(μ(t)i −μ(t)

j ), if μ(t)i −μ(t)

j > 0,

(44)otherwise,

b(t)ij = [b

(t−1)ij −st

((μ

(t)j − μ

(t)i ) + ω(t)λ

(t)i Eij2

b(t)ij ln 2

)]+

.(45)

• STEP 5: Each sensor transmits its measurements duringepoch t.

• STEP 6: All sensor measurements during epoch t reachthe FC. The FC performs the estimation.

• STEP 7: t ← t + 1. Go to Step 1. Continue until atermination condition holds.

Steps 1-4 again take place during the measurement andcontrol interval at each epoch t, while step 5 takes placeduring the transmission interval. In step 4, the optimizationis either one-shot with ∂L(·)

∂bij= 0, or a gradient descent step

is performed when the one-shot optimization is not feasible.As in the single-hop case, if the estimation error constraint

(31) is violated and the variance of estimation error exceeds ε,the update rule (41) increases ν and thus, by (43) the samplingrates for each sensor i increase so as to reduce the estimationerror. If constraint (33) is violated, this implies that for sensor ithe energy depletion rate is larger than the minimum depletionrate across sensors. Then, multiplier λi is reduced in (39), andvariable ω, which denotes the minimum energy depletion rateis reduced as in (42) so as to make the energy depletion rateas balanced as possible.

If the flow conservation constraint (30) is violated forsome sensor i, then μi increases or decreases, depending onwhether the in-flow is larger than the out-flow or not. In theformer case, the increase of μi makes it easier for i to pushdata towards out-neighbors j. Indeed, observe that a centralquantity in deciding the update rule for bij is μi − μj . Recallthan for a sensor i, multiplier μi corresponds to the flowconservation constraint, and a larger value for μi implies alarger penalty per unit of violation of the constraint for sensori. The update rule for bij favors the increase in the amountof forwarded measurement data towards out-neighbors j of ifor which μi − μj > 0, i.e the corresponding penalty per unit

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

100

200

300

400

500

600

700

800

Range of channel means, D

Lifetime in number of epochs

Lifetime versus range of channel means for channel variance=0.1

MaxLT =0.1MinE =0.1MaxLT =0.5MinE =0.5

Fig. 4. Single-hop transmission: Lifetime vs. range of channel means, Dfor the MaxLT and MinE approaches for low channel variance, σ2

G = 0.1.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

100

200

300

400

500

600

700

800

900

1000



Lifetime versus range of channel means for channel variance=0.5


Fig. 5. Single-hop transmission: Lifetime vs. range of channel means, Dfor the MaxLT and MinE approaches for high channel variance, σ2

G = 0.5.

of violation of the constraint is smaller, so as to facilitateminimization of the value of the Lagrangian. The secondupdate rule for bij is along the same spirit. On the otherhand, if the out-flow is larger than the in-flow at sensor i,μi decreases and this makes it easier in-neighbors k of i toincrease their transmitted data to i, provided than μ−

k μi > 0.The same convergence result as that of the single-hop primal-dual algorithm holds.

V. NUMERICAL RESULTS

We consider a sensor network with m = 16 sensors locatedat the vertices of a 4× 4 square grid. The energy reserve foreach sensor i is Ai = A = 0.1, and the measurement varianceis σ2

i = σ2 = 1. At each slot, link quality between sensor i andthe FC is log-normally distributed with mean Gi and varianceσ2G. The mean of all channel means is G = 1

m

∑mi=1 Gi = 1.

We define D as the range of channel means. Thus, channelmeans of sensors are uniform in [G− D

2 , G+ D2 ].

First, sensors transmit directly to a FC located at the centerof the grid. We consider the following policies for comparison:


4 4.2 4.4 4.6 4.8 5 5.2 5.4 5.6 5.8 60

500

1000

1500

2000

2500

3000

3500

4000

4500Wireless network lifetime as a function of the mean channel gain


Sum rates (bits/symbol)

MaxLT =0.3MaxLT =0.5MaxLT =0.1

Fig. 6. Single-hop transmission: Lifetime vs. total transmit rate of sensors.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

100

200

300

400

500

600

700

800

900Lifetime vs range of channel means for channel variance=0.1



MaxLT =0.1MinE =0.1MinE =0.5MaxLT =0.5

Fig. 7. Multi-hop transmission: Lifetime vs. range of channel means, D forMaxLT and MinE approaches for low channel variance, σ2

G = 0.1.

• Maximum lifetime (MaxLT) policy. MaxLT is proposedin this paper and attempts to balance energy depletionrate of sensor batteries subject to the estimation errorconstraint.

• Minimum total energy (MinE) policy. This policy findsthe PHY transmit rates that minimize the total amountof consumed energy, subject to the estimation errorconstraint.

In Fig. 4, we assume small channel variance, σ2G = 0.1.

We plot lifetime (measured in number of epochs until thefirst battery empties) versus the range of channel means D,for estimation error tolerance ε = 0.1 and ε = 0.5. Forthe MinE policy, we use the PHY rates that minimize totalenergy and compute lifetime based on them. First, as expected,both MaxLT and MinE lifetimes increase with increasing ε.For channel mean range D > 0, MaxLT always outperformsMinE. This is due to the fact that MaxLT empties batteriessimultaneously. On the other hand, MinE minimizes the energyconsumption of the entire network at each epoch, with the riskthat some sensors may consume higher energy than others inorder to achieve their required rates. For channel mean rangeD = 0, all sensors have the same channel quality on average,

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

100

200

300

400

500

600

700

800

900

1000Lifetime vs range of channel means for channel variance=0.5




Fig. 8. Multi-hop transmission: Lifetime vs. range of channel means, D forMaxLT and MinE approaches for high channel variance, σ2

G = 0.5.

and thus MaxLT and MinE have very similar performance,with a very slight advantage for MaxLT. Note also that thelifetime of the MaxLT policy slightly increases with increasingD.

For strongly varying channel with σ2G = 0.5 (Fig. 5), we

observe that for large enough values of D, MaxLT is still betterthan MinE for given ε. The performance difference increaseswith increasing D and ε. However, for small values of D,i.e. when sensors have similar channel quality on average,MInE outperforms MaxLT for given ε. This can be explainedas follows. MaxLT tries to empty batteries simultaneously atevery epoch, while MinE tries to minimize total consump-tion by attempting to adapt to channel conditions. However,since on average sensors have similar channel quality, MinEis able to minimize total consumption and empty batteriessimultaneously, thus leading to a higher lifetime. That is, MinEtries to adapt to channel conditions by allocating low rates tosensors having a bad channel and high rates to the ones witha good channel. MaxLT however allocates rates in a way thatall sensors are emptied equally at each epoch. The risk takenin MinE strategy pays off in this case, since sensors havesimilar channel quality on average, and thus a higher lifetimeis achieved by MinE. Note that for σ2

G = 0.1 and σ2G = 0.5,

the performance of MinE changes considerably with differentvalues of D, while that of MaxLT changes very little.

In Fig. 6 we depict lifetime as a function of total rate.Parameter D = 0 and the average channel gain G is varied inthe range [2, 3]. Since the transmit rate is mapped onto channelquality, for each value of the channel mean, we calculatetotal transmit rate of sensors. Lifetime is a convex increasingfunction of transmit rate and, for given total rate, its derivativeincreases as ε increases.

In the multi-hop case, sensors in the 4 × 4 communicatewith the FC via their neighbors in the grid. In Figures 7 and 8we study multi-hop transmission from sensors to the FC in the4×4 sensor grid. Similar trends as those in the multi-hop casecan be observed as well, but the main difference here is thatMaxLT always outperforms MinE. Furthermore, by comparingthe multi-hop and single-hop transmission scenario, it can be


−1000 0 1000 2000 3000 4000 5000 6000 7000 8000 9000 100000

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

number of iterations

Rates (bits/symbol)

Convergence of the distributed algorithm

Sensor 1

Sensor 2

Fig. 9. Convergence of the iterative primal-dual algorithm for a simplesingle-hop scenario with two sensors.

seen that, for given values of D, σ2G and ε, the former always

leads to higher lifetime than the latter. For instance, for ε = 0.5and σ2

G = 0.1, network lifetime is at the range 800 − 850epochs for the multi-hop scenario and 600−700 for the single-hop one. The higher lifetime in the multi-hop case is due tothe fact that most sensors do not need to transmit directlyto the FC but only to their neighbors. Finally, in Fig. 9 weverify the convergence of the algorithm for a simple single-hop scenario with two sensors, channel variance σ2

G = 0.5,ε = 0.5 and channel mean difference 0.5. After some transient,convergence is achieved with fewer than 100 iterations. Fig.10 shows convergence to final transmit rates for the multi-hopcase, for three sample sensors out of the 16 ones. Convergenceoccurs at about 1500 iterations.

VI. CONCLUSION

We addressed the problem of PHY transmit rate adaptationfor maximizing sensor network lifetime subject to an estima-tion error constraint in a single-hop and a multi-hop setting,and we presented convergent iterative algorithms towardsreaching a solution. Our algorithms maximize network lifetimeby alleviating different channel qualities across sensors (i.elarge ranges of channel means, D) and large variances σ2

G

in individual sensor channel quality. Lifetime is a convexincreasing function of total transmission rate, and the rate ofimprovement in lifetime per unit of transmit rate increase islarger for higher tolerance in estimation error.

VII. ACKNOWLEDGMENTS

The authors wish to thank Mr. Rudi Mullens for hishelp with the numerical study of the paper. The work ofIordanis Koutsopoulos was supported in part by the EuropeanCommission through projects CONECT (FP7-ICT-257616)and FORLAB (FP7-SEC-285052). The work of SławomirStanczak was supported in part by the German Ministry forEducation and Research (BMBF) under grant 01BN0712C,and by the Deutsche Forschungsgemeinschaft under GrantSTA 864/3-2.

−1000 0 1000 2000 3000 4000 5000 6000 7000 8000 9000 100000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9Convergence of the distributed algorithm in the multi−hop case

number of iterations

Rates (bits/symbol)

Sensor 1

Sensor 2

Sensor 3

Fig. 10. Convergence of the iterative primal-dual algorithm for the multi-hopscenario for three indicative sensors in the 4× 4 sensor grid.

REFERENCES

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[4] S. Cui, A. J. Goldsmith, and A. Bahai, “Energy-constrained modulationoptimization,” IEEE Trans. Wireless Commun., vol. 4, no. 5, pp. 2349–2360, Sep. 2005.

[5] J.-H. Chang and L. Tassiulas, “Maximum lifetime routing in wirelesssensor networks,” IEEE/ACM Trans. Networking, vol. 12, no. 4, pp. 609–619, Aug. 2004.

[6] R. Madan, S. Cui, S. Lall, and A. J. Goldsmith, “Modeling and opti-mization of transmission schemes in energy-constrained wireless sensornetworks,” IEEE/ACM Trans. Networking, vol. 15, no. 6, pp. 1359–1372,Dec. 2007.

[7] S. Cui, R. Madan, A. J. Goldsmith, and S. Lall, “Cross-layer energy anddelay optimization in small-scale sensor networks,” IEEE Trans. WirelessCommun., vol. 6, no. 10, pp. 3688–3699, Oct. 2007.

[8] V. Sharma, E. Frazzoli, and P. G. Voulgaris, “Improving lifetime datagathering and distortion for mobile sensing networks,” in Proc. 2004IEEE SECON.

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Iordanis Koutsopoulos (S ’99, M ’03) is AssistantProfessor in the Dept. of Computer and Communi-cations Engineering, University of Thessaly, Greecesince 2010. He was Lecturer in the same departmentfrom 2005 to 2010. He is also affiliated with the In-stitute for Telematics and Informatics of the Centrefor Research and Technology Hellas (CERTH). Hewas born on December 17, 1974 in Athens, Greece.He obtained the Diploma in Electrical and ComputerEngineering from the National Technical Universityof Athens, Athens, Greece in 1997 and the M.Sc.

and Ph.D degrees in Electrical and Computer Engineering from the Universityof Maryland, College Park (UMCP) in 1999 and 2002 respectively. From 1997to 2002 he was a Fulbright Fellow and a Research Assistant with the Institutefor Systems Research (ISR) of UMCP. He has held internship positions withHughes Network Systems, Germantown, MD, Hughes Research LaboratoriesLLC, Malibu, CA, and Aperto Networks Inc., Milpitas, CA, in 1998, 1999 and2000 respectively. For the summer period of 2005 he was a visiting scholarwith University of Washington, Seattle, WA. He was awarded a EuropeanResearch Council (ERC) runner-up Grant (funded by national funds) for theperiod 2012-2015, and a Marie Curie International Reintegration Grant (IRG)for the period 2005-2007. His research interests are in the general area ofnetwork control, optimization and performance analysis.

Sławomir Stanczak studied control systems engi-neering at the Wroclaw University of Technology,Poland, and at the Technical University of Berlin(TU Berlin), Germany. He received his Dipl.-Ing.degree and Dr.-Ing. degree with distinction (summacum laude) in Electrical Engineering from the TU-Berlin in 1998 and 2003, respectively. Since 2006 healso holds a habilitation degree (venialegendi) andis an associate professor (privatdozent) at the TU-Berlin. Dr. Stanczak has been involved in researchand development activities in wireless communica-

tions since 1997. Since 2003 he leads a research at the Fraunhofer HeinrichHertz Institute and since 2010 Dr. Stanczak has been the acting director of theHeinrich-Hertz-Lehrstuhl at the TU Berlin. Dr. Stanczak is a co-author of twobooks and more than 100 peer-reviewed journal articles and conference papersin the area of information theory, wireless communications and networking.In 2008, he was a visiting scientist at the Stanford University, CA, USA(Prof. Nick Bambos). He is a recipient of research fellowships from theGerman research foundation. He was a co-chair of the 2009 InternationalITG Workshop on Smart Antennas (WSA 2009), the general chair of the2010 Workshop on Resource Allocation in Wireless Networks (RAWNET2010) and the co-chair of 14th IEEE International workshop on signalprocessing advances in wireless communications (SPAWC 2013). From 2009until 2011, Dr. Stanczak was Associate Editor for European Transactions forTelecommunications (information theory). More information is available at:http://www.mk.tu-berlin.de/mitarbeiter/tub/prof/Stanczak.

Documents

The Impact of Transmit Rate Control on Energy-Efficient Estimation in Wireless Sensor Networks