hithesh nama et al. 1 optimal utility-lifetime trade-off in ...chiangm/utilitylife.pdfhithesh nama...

HITHESH NAMA ET AL. 1

Optimal Utility-Lifetime Trade-off

in Wireless Sensor Networks:

Characterization and Distributed Algorithms

Hithesh Nama,Student Member, IEEE,Mung Chiang,Member, IEEE,Narayan

Mandayam,Member, IEEE

Abstract

The performance of wireless sensor network applications typically increases with the amount of

data collected by the individual sensors and delivered to a set of sinks through multi-hop routing within

the network. However, the energy-constrained nature of thenodes limits the operational lifetime of the

network since energy is dissipated both in sensing and in communicating data across the network. There

is thus an inherent trade-off in simultaneously maximizingthe application performance (characterized

here by utility functions) and the network lifetime. In thispaper, we characterize this trade-off by

considering a cross-layer design problem that jointly maximizes the network utility and lifetime. The

solution to the problem results in optimal source rates and also optimal routes through the network

between each source and the sink. Through the framework of “layering as optimization decomposition”,

we show that the cross-layer design problem decomposes bothvertically (across different layers of the

protocol stack) and horizontally (across nodes in the network) into simpler subproblems allowing a fully

distributed solution. Our formulation leverages the presence of a common sink node in the network to

design a distributed algorithm that minimizes the energy overhead in its implementation. We apply our

generic model to study the utility-lifetime trade-off in networks with proportionally fair rate allocation

and discuss practical design implications through analysis and numerical simulations.

Index Terms

Cross-layer, distributed algorithm, optimization, sensor networks, network lifetime, network utility

maximization.

The work in this paper is funded in part by the NJ Commission on Science and Technology under the MUSE project and

NSF grants.


I. I NTRODUCTION

In typical sensor network applications, the application performance improves with the amount

of data gathered by asink from each node in the network. For instance, more data from a video

sensor could mean a better quality image while high-precision samples of a temperature field

result in a more accurate characterization of the underlying physical process. However, higher

data rates entail greater energy dissipation in the nodes which could result in a shorter operational

lifetime for the network. There is thus an inherent trade-off between application performance

and the lifetime of the network. Further, this trade-off is affected by actions taken at different

layers of the communication protocol stack, each of which have a direct impact on the energy

dissipated in the network. Therefore, it is important to consider a cross-layer design methodology

in order to understand the trade-off between application performance and network lifetime.

In this work, we use the framework of “layering as optimization decomposition” [1] to design

distributed protocols for the transport and network layersthat allow the network to be operated at

any point on theoptimal trade-off curve between application performance and network lifetime.

Such a framework allows the systematic analysis and design of network protocols as distributed

solutions to a global optimization problem - a view that treats the network itself as an optimizer

trying to maximize its service objectives subject to resource constraints. Specifically, we consider

the maximization of a combination of application performance and lifetime, in a network with

capacitated links and energy-limited nodes. Our formulation leverages the presence of a common

sink in the network to design distributed algorithms that minimize the energy expended in

their implementation. We apply our framework to study the optimal utility-lifetime trade-off in

networks with proportionally fair rate allocation and illustrate the performance of the proposed

distributed algorithms through numerical simulations.

While there are several ways to characterize application performance, in this paper, we

characterize it using a network utility function which is the sum of individual node utility

functions [2], [3]. The utility of each node is assumed to be an increasing and strictly concave

function of its source rate, thus reflecting the applicationperformance. Distributed algorithms for

solving the network utility maximization problem in the context of wired networks are studied

in [4], [5]. The network utility formulation has been used in[6], [7], [8], [9] in the cross-layer

design of transport, network, and radio resource layers formaximizing the throughput or network


utility of ad hoc wireless networks. See [1] for an extensivelist of references.

We define network lifetime as the time until the death of the first node as in [10]. Using

this definition, distributed routing algorithms and cross-layer design approaches to maximize

network lifetime have been studied in [11], [12], and [13]. However, the models in these papers

assume fixed source rates which need not be optimal from a utility maximization perspective

or might even be infeasible for a given set of resource constraints. In this paper, we consider

a self-regulating wireless sensor network in which nodes can adapt their source rates to jointly

maximize the network utility and lifetime. Furthermore, byusing the sink to coordinate the

distributed lifetime maximization algorithm, we reduce the energy expended in its implementation

since fewer protocol packets are exchanged between nodes compared to algorithms in existing

literature.

In [14], a utility-based formulation to maximize the allocated rates in anad hoc wireless

network while achieving a desired lifetime is proposed. However, given an arbitrary network, it

is difficult to verify a priori if a desired network lifetime is indeed feasible. In this context, we

introduce a system design parameter called the lifetime-control factor and argue its advantage in

bootstrapping and in operating networks in practice. Further, the penalty function-based approach

of [14] results in optimal solutions only in a doubly-asymptotic regime involving two model

parameters. In our work, we use dual decomposition techniques that guarantee optimal solutions

in a non-asymptotic regime and the optimal dual variables allow an intuitive understanding

of bottleneck resources through apricing interpretation. We show that our cross-layer design

problem decomposesvertically into separate subproblems corresponding to the transport layer

and the network layer, and a network lifetime maximization problem, all of which interact

through dual prices corresponding to bottleneck resources. Further, these “layered” subproblems

are also shown to decomposehorizontallyacross nodes in the network allowing fully distributed

algorithms to compute the optimal source rates and optimal routes.

We design distributed algorithms for two kinds of sensor networks - those with unique routes

from each source to a sink and more generally those with potentially multiple routes between

each source and a sink. While the proposed algorithms for these two kinds of networks are

similar in flavor, their convergence properties are considerably different as illustrated analytically

and through numerical simulations in subsequent sections.Further, such a distinction between

networks could arise in practice due to the presence or absence of a network layer routing protocol


that chooses unique routes between sources and sinks. Thus,the distinction between these two

kinds of networks is of interest both from an analytical as well as a practical perspective.

Section II describes the system model and discusses the joint utility-lifetime maximization

problem. A dual-based distributed algorithm for sensor networks with single-path routing is

presented in Section III. In Section IV we propose a distributed algorithm for networks with

potentially multiple routes between each source and the sink. In Section V, we apply our

system model to networks with proportionally fair rate allocation and use numerical simulations

to illustrate the optimal utility-lifetime trade-off as well as the convergence of the proposed

distributed algorithms.

II. SYSTEM MODEL

Consider a wireless sensor network consisting of a set1 of sensor nodes, indexed1 to N , and

a single sink with indexN + 1 that collects data from these nodes. LetR denote the set of

feasible routes, obtained using a network layer routing protocol ( [15], [16], [17]). Each router

is assumed to carry data from auniquesource noden(r) to the sink.

Let a matrix2 H ∈ {0, 1}N×R be defined such thatHnr = 1 if and only if n = n(r). We

associate a flowyr ≥ 0 with each router ∈ R. The source ratexn corresponding to noden is

defined as the sum of flows over all routes from noden to the sink. Thusx = Hy, wherex ∈ RN+

andy ∈ RR+ denote the vector of source rates and the vector of route flows, respectively3. Let

Iy = {y |y ≥ 0} be the set of admissible flow vectors, where an inequality between vectors

denotes component-wise inequality. Similarly, we defineIx = {x |xmin ≤ x ≤ xmax} to be the

set of admissible source rates.

Let L denote the set of all links that are a part of at least one route. We define a matrix

A ∈ {0, 1}L×R such thatAlr = 1 if and only if link l is a part of router. The capacitycl of

link l is defined by the underlying physical and MAC layers (for example based on the IEEE

1With a slight abuse of notation, we use the same symbol to denote a set as well as its cardinality.

2We denote vectors using bold lower-case letters and matrices using bold upper-case letters. Further,xn and(x)n both denote

the nth component of vectorx.

3R

N

+ denotes the set of vectors inRN with non-negative components andRN

++ denotes the set of vectors inRN with positive

components.


802.15.4 standard [18]) and is assumed to be a known constantin our model. Letc ∈ RL+ denote

the vector of link capacities.

We defineX = {x |x = Hy, Ay ≤ c, x ∈ Ix, y ∈ Iy} to be the set offeasiblesource rates

and assumeAymin ≤ c so that the setX is non-empty. In most applications of sensor networks,

the performance of the application improves with the amountof data gathered from the nodes

and thus it might be desirable to operate the network at a Pareto optimal point [19] of the set

X . In the next section, we introduce the network utility maximization problem that computes

such an efficient operating point for the network.

A. Network Utility Maximization

We assume a self-regulating network in that the sensor nodescan adjust their source rates

depending on the application requirement. With each sourcen we associate a utility function

Un(xn), which is continuously differentiable, increasing, and strictly concave inxn. The utilities

are assumed to be additive so that the network utility is defined as the sum of utilities of

individual nodes. The network utility maximization (NUM) problem is then stated as follows:

maxx∈X

N∑

n=1

Un(xn). (1)

The polyhedral constraint setX together with the strictly concave objective function results in

a unique maximizerx∗ in (1) that is Pareto optimal [19].

In the context of sensor networks, however, nodes are energyconstrained and higher data rates

result in greater energy dissipation in sensing, transmitting, and receiving data. Maximizing the

network utility could thus result in widely varying power dissipation levels across the nodes and

could potentially result in a disconnected network within ashort time. In the next section we

present a model for power dissipation in the nodes and introduce the notion of lifetime of a

network.

B. Power Dissipation Model and Network Lifetime

Let Es, Etx, andErx denote the energy consumed per bit in the hardware in sensing, trans-

mitting, and receiving data, respectively. Note thatEtx also includes the radiated energy per bit

for reliable communication. In this work, we assume that nodes transmit at a fixed power as

is common in most practical deployments of wireless sensor networks. However, using power


control to transmit at the smallest power that guarantees successful reception can result in energy

savings. Our cross-layer design framework can be easily extended to include transmit power at

the physical layer as an additional resource optimization parameter [20].

We assume that all nodes have identical power dissipation characteristics and thus define a

matrix E ∈ RN×R+ such that

Enr =

Es + Etx if node n is the origin of router,

Erx + Etx if node n is an intermediate node of router,

0 if node n is not a part of router.

(2)

For a particular network flowy, the average power dissipation at noden is given byP avgn =

∑

r∈R Enr yr. We assume that the sink is not energy constrained unlike thesensor nodes. We

therefore consider only the energy dissipated in the sensornodes in our subsequent analysis. Let

en denote the initial energy of noden ande ∈ RN++ the corresponding vector of initial energies.

Definition 2.1: The lifetime of noden is defined astn = en/P avgn . The network lifetime

corresponding to flowy is then defined astnwk = min{tn |n = 1, . . . , N} i.e., the time until the

death of the first node in the network.

Let v = 1/tnwk be the inverse-lifetime of the network so that, from the definition of network

lifetime, tn ≥ tnwk = 1/v. We defineIv = {v | vmin ≤ v ≤ vmax}, wherevmin and vmax denote

lower and upper bounds onv, respectively. Combining the above definitions, the setXv of

feasible source rates and network inverse-lifetime can be stated as follows:

Xv = {(x, v) |x = Hy, Ay ≤ c, Ey ≤ e v, x ∈ Ix, y ∈ Iy, v ∈ Iv}. (3)

Definition 2.2: The Pareto boundary of the setXv, consisting of its maximal elements, is

defined asX ∗v = {(x, v) ∈ Xv | (x, u) ≤ (z, v) for some(z, u) ∈ Xv ⇒ (z, u) = (x, v)}.

Note that every(x, v) ∈ X ∗v is said to be Pareto optimal. The Pareto optimal setX ∗

v is

important in practice since it characterizes the entire setof efficient operating points of an

energy-constrained network. We now consider a joint utility-lifetime maximization problem that

results in a Pareto optimal operating point for the network.We associate a lifetime-penalty

function F (v), which is continuously differentiable, strictly convex, and increasing inv ≥ 0.

The network utility-lifetime maximization problem is stated as follows:

max(x,v)∈Xv

N∑

n=1

Un(xn) − F (v). (4)


The convex constraint setXv together with the strictly concave objective function results in a

unique maximizer(x∗, v∗) in (4) that is a Pareto optimal allocation. In the next section, we

consider networks with a single-path routing and propose distributed algorithms to operate the

network at a desired point on the optimal utility-lifetime trade-off curve. The case of networks

with multipath routing is discussed in Section IV.

III. N ETWORKS WITH SINGLE-PATH ROUTING

In this section, we consider a sensor network consisting ofN sources and a sink with each

source having a unique route to the sink possibly involving multiple hops through the network.

Let r(n) denote the unique route from source noden to the sink. Such a network could be

obtained when the underlying routing protocol chooses a unique route between a source and the

sink based on a particular routing policy. The unique route assumption implies that there are as

many routes as there are nodes (R = N ) and the matrixH defined in Section II is an identity

matrix. Therefore,x = Hy ⇒ x = y. The utility-lifetime maximization problem of (4), also

called the primal problem henceforth in this section, can berestated as

maxx∈Ix, v∈Iv

N∑

n=1

Un(xn) − F (v)

subject to,Ax ≤ c, Ex ≤ e v (5)

and can be used to modify the source rates depending on the application requirements so as to

maximize the lifetime of the network. However, solving (5) directly might be infeasible in a large

network and it might be desirable to have a distributed algorithm to obtain the optimal primal

solutions. Since (5) is a convex optimization problem without any non-linear constraints, strong

duality holds [19] i.e., the optimal values of the primal anddual objective functions are equal.

We can thus obtain the primal solutions indirectly by first solving the dual problem. Moreover

the dual-based approach leads to an efficient distributed algorithm as shown below.

We introduce Lagrange multipliersλ ∈ RL+ and µ ∈ RN

+ to formulate the Lagrangian dual

function corresponding to primal problem (5) as below

D(λ,µ) = maxx∈Ix, v∈Iv

N∑

n=1

Un(xn) − F (v) − λT (Ax − c) − µT (Ex − e v). (6)


The dual problem corresponding to the primal problem in (5) is then given by

minλ≥0, µ≥0

D(λ,µ). (7)

The dual function in (6) can be decomposed into the followingtwo subproblems, which are

evaluated separately in the source ratesx and the network inverse-lifetimev.

D(λ,µ) = λTc + D1(λ,µ) + D2(λ,µ) where,

D1(λ,µ) = maxx∈Ix

N∑

n=1

Un(xn) − (λTA + µTE)x (8)

D2(λ,µ) = maxv∈Iv

µTe v − F (v). (9)

The objective functions in the above two subproblems are separable, strictly concave functions

of the individual variables and admit a unique analytical solution for the optimalxn andv as

xn =[

U′

n

−1 ((λTA + µTE)n

)

]xmaxn

xminn

, n = 1, . . . , N (10)

v =[

F′−1 (

µTe)

]vmax

vmin(11)

where [x]Mm = max{min{x,M},m}, i.e., the projection ofx on to the interval[m,M ]. Since

the objective function in the primal problem (5) isstrictly concave in the primal variablesx

andv, the dual function in (6) is continuously differentiable. Therefore we can use the iterative

gradient projection algorithm [21] to solve the dual problem in (7). The partial gradients of the

dual function are given as (see Prop A.43 in [21])∇λD = c−Ax and∇µD = e v −Ex. Note

that the primal variablesx andv are implicitly a function of the dual variables through (10)and

(11). The dual variable update equations at thekth iteration in the direction of negative gradient

are given below

λ(k + 1) = [λ(k) − β(c − Ax(k))]+ and µ(k + 1) = [µ(k) − β(e v(k) − Ex(k))]+ (12)

where,β > 0 is a constant scalar step size and[x]+ denotes projection of each component ofx

on to R+.

The dual variables admit an intuitive resource-pricing interpretation withλl denoting thelink

congestion priceand µn denoting thenode lifetime price. From (12), the link-price decreases

during thekth iteration if the total traffic through a link is less than its capacity. From (10), the

reduced link-price could mean a greater source rate for all sources using this link, thus leading


to an efficient utilization of the link. Similarly, the node-price decreases during thekth iteration

if the maximum allowed average power dissipationen v is greater than the actual average power

dissipation(Ex(k))n. The reduced node-price could mean a greater source rate forall sources

whose routes pass through this node (see (10)). Combining (10), (11), and (12), the distributed

algorithm that solves the primal problem in (5) is given as follows:

Algorithm 1: At the kth iteration,

1) Source rate update: Each noden computes its source rate based on theaggregateprice of

links and nodes along its router to the sink

xn(k + 1) =

U′

n

−1

∑

l∈L(r)

λl(k) +∑

m∈N(r)

µm(k)Emr

xmaxn

xminn

whereL(r) andN(r) denote the set of links and set of nodes, respectively, alongrouter.

2) Network inverse-lifetime update: The sinkN +1 adjusts the network inverse-lifetime based

on theaggregatenode-prices weighted by the node energies

v(k + 1) =

[

F′−1

(

N∑

n=1

µn(k)en

)]vmax

vmin

3) Link and node prices update: Based on the aggregate traffic passing through it, each node

n updates its node-priceµn and link-priceλl, wherel is the outgoing link from the node

along its route to the sink

λl(k + 1) =

λl(k) − β(cl −∑

n∈N(l)

xn(k))

+

,

µn(k + 1) =

µn(k) − β(env(k) −∑

m∈N(n)

Enmxm(k))

+

4) Message passing toward the sink: The updated source rates(xn(k + 1)) are propagated

toward the sink by aggregating them at intermediate nodes. Similarly, the node-prices

weighted by the node energies (µn(k + 1)en) are aggregated at each node and propagated

to the sink.

5) Message passing away from the sink: The network inverse-lifetime (v(k + 1)) computed

by the sink is broadcast to all the nodes. Starting from the nodes closest to the sink, the

updated link and node-prices (λl(k+1) andµn(k+1)) are propagated through the network

along individual routes by aggregating them at intermediate nodes.


An important feature of the above algorithm is that only aggregate variables are passed through

the network and not the individual variables correspondingto each node. A total ofN messages

are passed in the network toward the sink and anotherN messages away from the sink at the

end of each iteration. Further, these updates can be piggy-backed on data destined to the sink

and broadcast from the sink.

In prior work related to distributed routing algorithms [12] or cross-layer design approaches

[20] to maximize network lifetime, each node updates its ownestimate of the network lifetime

based on the estimates of all its neighbors. However, in the above algorithm only the sink

coordinates the network lifetime by aggregating the weighted node lifetime prices. Since each

node need not receive the estimates of all its neighbors at each iteration, the energy dissipated

in implementing Algorithm 1 is significantly lesser than prior algorithms in literature.

The source rate update and link-price update in the above algorithm are similar to that in

standard transport layer congestion control algorithms [4]. However, the source rate update here

depends not just on the link congestion prices but also on thenode lifetime prices, arising from

the additional network lifetime maximization objective. In [22], an alternative implementation

of a primal-dual algorithm for transport layer congestion control in the Internet is proposed that

does not involve explicit message passing between sources and links. The source rate update

and the link-price update of Algorithm 1 above could also be adapted to avoid explicit message

passing and instead use only local measurements. The node lifetime price update that is unique

to this algorithm could be performed using local power dissipation measurements (to compute∑

m∈N(n) Enmxm(k)) by monitoring the battery drain in a unit interval of time. However, the

algorithm requires the network inverse-lifetime updatev(k) to be explicitly passed from the

sink. The performance of the algorithm without explicit message passing and using only local

measurements needs to be studied in further detail and is outof the scope of this work.

We now state the following theorem that guarantees convergence of the above algorithm to

the optimal primal and dual solutions.

Theorem 3.1:Starting from any initial pointsx(0) ∈ Ix, v(0) ∈ Iv, λ(0) ≥ 0, andµ(0) ≥ 0,

every limit point of the sequence(x(k), v(k),λ(k),µ(k)) generated by Algorithm 1 is primal-

dual optimal, provided the step sizeβ is chosen small enough and the following conditions

hold:

1) For all source nodesn and for everyx ∈ Ix, the utility function Un(xn) is increasing,


strictly concave, and twice continuously differentiable,U′

n(xn) < ∞, and there is anαn

such that−U′′

n (xn) ≥ 1/αn > 0.

2) For everyv ∈ Iv, the lifetime penalty functionF (v) is increasing, strictly convex, and

twice continuously differentiable,F′

(v) < ∞, and there is aτ such thatF′′

(v) ≥ 1/τ > 0.

The proof of the theorem is derived in [23] and it also gives anupper bound on the range

of values ofβ that guarantees convergence of Algorithm 1. We illustrate the performance of

Algorithm 1 using numerical simulations in Section V-B.

IV. N ETWORKS WITH MULTIPATH ROUTING

In this section, we present a distributed algorithm to solvethe utility-lifetime maximization

problem (4) when each sensor node can possibly have multiplepaths to the sink (R ≥ N ). For

convenience, we restate the problem below in its entirety:

maxx∈Ix,y∈Iy, v∈Iv

N∑

n=1

Un(xn) − F (v)

subject to,Hy = x, Ay ≤ c, Ey ≤ e v. (13)

Since (13) represents a convex optimization problem with only linear constraints, strong duality

holds and we can obtain the primal optimal solutions indirectly by first solving the dual problem.

We introduce Lagrange multipliersη ∈ RN , λ ∈ RL+, andµ ∈ RN

+ to formulate the Lagrangian

dual function corresponding to primal problem (13) as below

D(η,λ,µ) = maxx∈Ix,y∈Iy, v∈Iv

N∑

n=1

Un(xn) − F (v) + ηT (Hy − x)

− λT (Ay − c) − µT (Ey − ev). (14)

The dual function can be decomposed into the following threesubproblems

D(η,λ,µ) = λTc + D1(η,λ,µ) + D2(η,λ,µ) + D3(η,λ,µ) where,

D1(η,λ,µ) = maxx∈Ix

N∑

n=1

Un(xn) − ηTx (15)

D2(η,λ,µ) = maxy∈Iy

(ηTH − λTA − µTE)y =

0, if ηTH ≤ λTA + µTE

∞, otherwise(16)

D3(η,λ,µ) = maxv∈Iv

µTe v − F (v). (17)


The dual problem corresponding to the primal problem in (13)is then given by

min D(η,λ,µ)

subject to,λ ≥ 0, µ ≥ 0, ηTH ≤ λTA + µTE. (18)

Each of the three subproblems (15), (16), and (17) evaluatesvariables corresponding to different

layers of the protocol stack and thus they represent a vertical decomposition of the primal

problem (13). The cross-layer interaction between the different layers is coordinated through

the dual variables and the distributed algorithm presentedbelow represents a further horizontal

decomposition. Fig. 1 illustrates the vertical and horizontal decomposition of the primal problem.

Unlike the single-path routing case (6), the dual objectivefunction in (14) is not differen-

tiable since the objective function in (13) is not strictly concave in all the primal variables, in

particular in the route flowsy. We therefore use a subgradient4 based descent approach [24]

to solve the dual problem (18). Using Danskin’s Theorem (seepp. 717 of [24]), the set of all

subgradients∂D(η,λ,µ) of the dual objective function at(η,λ,µ) can be obtained from the

set of maximizersZ = {(x,y, v)} of (14). It is easy to verify that the partial subgradients with

respect toλ, µ, andη are given by∂λD = c − Ay, ∂µD = e v − Ey, and∂ηD = Hy − x,

respectively.

The constraintηTH ≤ λTA + µTE in (18) is equivalent toηn ≤ minr∈R(n)

(

λTA + µTE)

r,

n = 1, . . . , N . Now, given feasible dual variablesλ and µ, considerη′

and η′′

such that

η′

m = η′′

m,m = 1, . . . , N,m 6= n and η′

n > η′′

n. If η′

n < minr∈R(n)

(

λTA + µTE)

r, then yr =

0, r ∈ R(n) from (16) and the partial subgradient(Hy − x)n ≤ 0. From the definition of

subgradient,D(η′

,λ,µ) ≤ D(η′′

,λ,µ). Therefore, in solving the dual problem (18), we only

need to considerηn = minr∈R(n)

(

λTA + µTE)

r, n = 1, . . . , N for any feasible dual variables

λ andµ.

Subproblems (15) and (17) admit unique solutions forx andv, respectively, as given below

xn =[

U′

n

−1(ηn)

]xmaxn

xminn

, n = 1, . . . , N and v =[

F′−1 (

µTe)

]vmax

vmin. (19)

From (16), if yr∗ > 0 for somer∗ ∈ R(n), then for all other routesr′ ∈ R(n),

ηn =(

λTA + µTE)

r∗≤(

λTA + µTE)

r′. (20)

4Given a convex functionf : Rn→ R, a vectord ∈ R

n is a subgradient off at a pointu ∈ Rn if f(v) ≥ f(u) + (v −

u)Td, v ∈ R

n.


Interpretingλ andµ as link congestion price and node lifetime price, respectively, the route-price

of route r is given by(

λTA + µTE)

r. From (20), all routes from a node with non-zero flows

have equal price, which is the least route-price compared toother routes from the same node.

The source rate update and the routing algorithm thus involve each node choosing its source rate

according to the least route-price (and arbitrarily partitioning this flow among routes with the

least price). We summarize the distributed algorithm for networks with multipath routing below.

Algorithm 2: At the kth iteration,

1) Source rate and route flow update: Each noden computes the minimum route-price among

all its routes and updates its source rate as below

xn(k + 1) =[

U′

n

−1 ((λTA + µTE)r∗

)

]xmaxn

xminn

wherer∗ is a route with minimum price. The source rate is then arbitrarily divided among

all its routes with minimum price. Routes with a higher price do not carry any flow.

2) Network inverse-lifetime update: Same as network inverse-lifetime update of Algorithm 1.

3) Link and node prices update: Using subgradient-based descent, each noden updates its

node-priceµn and link-pricesλl, wherel is an outgoing link from the node along any of

its routes to the sink

λl(k + 1) =

λl(k) − β(cl −∑

r∈R(l)

yr(k))

+

,

µn(k + 1) =

[

µn(k) − β(env(k) −∑

r

Enryr(k))

]+

whereβ > 0 is a constant scalar step size.

The message passing involved in the above algorithm is similar to that in Algorithm 1 of

previous section. Several results regarding the convergence of subgradient-based descent methods

can be found in [24] and [25]. For constant step size as in the above algorithm, [6] show that

the (arithmetic) averages of the primal and dual iterates approach the optimal solutions for large

number of iterations, provided the subgradients are bounded and the step size is chosen small

enough. Since the primal variables are assumed to be boundedin our model, the subgradients are

bounded and simulation results presented in Section V-C show that the primal and dual variables

approach the optimal values within a finite number of iterations.


V. NETWORKS WITH PROPORTIONALLY FAIR RATE ALLOCATION

In this section, we apply our system model to study utility-lifetime trade-off in networks with

proportionally fair rate allocation and illustrate the performance of the distributed algorithms

through numerical simulations. We consider the network utility to be a convex combination of

logarithmic node utility functions, which corresponds to aweighted proportionally fair source

rate allocation [3] and considerF (v) = γ v2/2, whereγ ≥ 0 is a lifetime-control factor. The

network utility-lifetime maximization problem (4) can then be stated as

maxx∈Ix,y∈Iy, v∈Iv

N∑

n=1

wn log xn − γ v2/2

subject to,Hy = x, Ay ≤ c, Ey ≤ e v (21)

wherewn ≥ 0, n = 1, . . . , N ,∑N

n=1 wn = 1, and we assumexmin > 0 andvmin = 0. By varying

the weightswn and the lifetime-control factorγ, different points on the Pareto boundary (see

Definition 2.2) can be obtained as unique solutions to the above convex optimization problem.

The optimal primal and dual solutions of (21) must satisfy

γ v = µTe;N∑

n=1

wn

xn

Hnr ≤ (λTA + µTE)r, w.e. if yr > 0, r ∈ R andx > xmin (22)

λT (Ay − c) = 0; µT (Ey − e v) = 0 (23)

where (22) and (23) represent Lagrangian stationarity and complementary slackness conditions

[19], respectively. Using (22) and (23), we get

γ v2 = µTEy ≥ 1 − λTAy = 1 − λTc (24)

with equality if x > xmin. When the links are unsaturated, the link congestion pricesλ are zero

from (23) and the network lifetime depends only on the lifetime-control factor and is independent

of othe

The lifetime-control factor has important implications inpractice for bootstrapping sensor

networks. Given an arbitrary network, it might be difficult to verify if a desired lifetime is

indeed feasible. However, initializing the distributed algorithms withany positive value of the

lifetime control factorγ results in a feasible network lifetime. From then on, the network lifetime

can be steered to any feasible value as desired by appropriately varying the lifetime control factor.


A. Utility-Lifetime Trade-off: Numerical Results

Consider a network consisting of six sensor nodes (N=6) and a sink as shown in Fig. 2.

The network consists of a total of seven links and we assume eight routes in the network -

r1 = {l1}, r2 = {l2}, r3 = {l3, l1}, r4a = {l4a, l1}, r4b = {l4b, l2}, r5 = {l5, l2}, r6a =

{l6, l4a, l1}, r6b = {l6, l4b, l2} - with sources 4 and 6 having two routes each while the remaining

nodes have a single route to the sink.

We assume a maximum link rate of 250 kbps as in the IEEE 802.15.4 standard [18] and require

that each node have a minimum source rate of 1 kbps. All sensornodes are assumed to have equal

initial energy of 1/6 J each, i.e., the total energy,Etotal = 1 J in the network. The parameters for

the node power dissipation model of (2)Es, Erx, andEtx are chosen to be 100 nJ/bit, 158 nJ/bit,

and 150 nJ/bit based on the power dissipation measurements in the IEEE 802.15.4-compliant

CC2420 [26] RF transceiver. Each link is assumed to be scheduledfor 1/3 fraction of time by

an underlying medium access protocol and the weightswn = 1/N, n = 1, . . . , N are equal.

In order to illustrate the trade-off between utility and lifetime in this network, we solve

the convex optimization problem in (21) (using MOSEK [27], an optimization toolbox) for

different values of the lifetime-control factorγ. Fig. 3 shows the optimal trade-off curve (with

multipath routing, Total energy = 1 J) between network utility and network lifetime and the

corresponding source rates are shown in Fig. 4. By decreasingγ, the network lifetime can be

traded for greater network utility through increased source rates until some links become saturated

(γ ≈ 1). Alternatively, by increasingγ network utility can be traded for greater network lifetime

by reducing the source rates until they cannot be reduced below the minimum requirement of

1 kbps (γ ≈ 4 × 104). The network lifetime in the these results is only of the order of seconds

since the initial energy of each node is assumed to be less than a joule. In practice, with typical

battery capacities in hundreds of kilojoules, the maximum operational network lifetime is of the

order of several days. Further, by duty-cycling the nodes the network can be operated for much

longer periods of time.

There are three regimes of interest in the utility-lifetimetrade-off curve - network lifetime

close to minimum where source rates are almost constant since some links are saturated and the

corresponding link-prices are non-zero, network lifetimeclose to maximum where source rates

approach the minimum required rate and thusx 6> xmin, and finally for a wide range of network


lifetimes between these extremes where link-prices are zero andx > xmin. Using (24),γ v2 = 1

in the third regime, which approximately corresponds to network lifetime between 8 s and 176

s andγ between 60 and3.1 × 104 in Fig. 3 (with multipath routing, Total energy = 1 J). Thus

the lifetime control factor allows aprecisecontrol of network lifetime for a significant portion

of the optimal utility-lifetime trade-off curve.

The optimal source rates are a function of the optimal dual prices (see (19) and (20)), which

are non-zero only for bottleneck resources due to the complementary slackness conditions (23).

In all three regimes mentioned above, nodes 1 and 2 which havethe highest power dissipation

among all nodes are bottleneck resources and have non-zero node-prices. Due to symmetry in

the network, optimal prices of nodes 1 and 2 are equal and so are the optimal link-prices ofl1

andl2. Therefore, as shown in Fig. 4, nodes 3, 4, 5, and 6 have equal source rates since they use

equally priced bottleneck resources to route data to the sink. Nodes 1 and 2 have a higher rate

than the other nodes because of the difference inEs andErx values that appear in the route-price

throughE (see (20)).

The optimal utility-lifetime trade-off curve of Fig. 3 gives an upper bound on the achievable

application performance for a desired network lifetime in apractical network and could be a

useful design tool. The above formulation can also be used tocompare the performance of

different routing policies (for example minimum hop-countrouting or nearest-neighbor routing)

since each routing policy could result in a different set of routes between each source and the

sink and thus differentH, A, andE matrices in our formulation.

Consider then the network of Fig. 2 with linkl4b removed and thus consisting only of single-

path routing from all nodes (nodes 4 and 6 in particular) to the sink. The optimal utility-lifetime

trade-off curve (with single-path routing) for the above parameters is shown in Fig. 3 and shows

the improved utility-lifetime performance of multipath routing compared to single-path routing

for this simple network. The corresponding source rates forthe network with single-path routing

are shown in Fig. 5. As explained earlier, only bottleneck resources have non-zero prices which

in this network are nodes 1 and 2. However, due to asymmetry inthe amount of traffic routed

through these nodes, node 1 has a higher node lifetime price than node 2 and this is reflected

in the difference in the source rates.

We now discuss the effect of changes in the system parameterson the source rates and

the network lifetime. For a fixed set of weights{wn}, the optimal trade-off curves between


source rates and network lifetime Fig. 4 and Fig. 5 do not varywith the exact nature of node

utility functions (log(x), −1/x etc.) and lifetime-penalty function (v2, v3 etc.) as long as these

functions satisfy conditions (1) and (2) of Theorem 3.1. Thevalue of the lifetime control factor

γ corresponding to a particular network lifetime (and the corresponding source rates) could

however be different for different functions.

Fig. 3 (with multipath routing, Total energy = 0.5 J) illustrates the utility-lifetime trade-

off curve with only half the total initial energy in the network. Comparing these two curves

(with multipath routing), the network lifetime is halved inthe first and second regimes which

correspond to nearly constant source rates. However, in thethird regime withγ v2 = 1, decreasing

the initial energy does not affect the network lifetime and only results in a decrease in the network

utility.

Variations in the link capacities only affect the utility-lifetime trade-off curve in the first regime

corresponding to small values ofγ when links are saturated. An increase in link capacities of

bottleneck links results in a lower minimum network lifetime and a correspondingly higher

maximum network utility. Finally, varying the parameters of the power dissipation model (Es,

Etx, andErx) changes the optimal node lifetime prices and the optimal source rates (see (12)

and (10)).

B. Algorithm 1: Numerical Results

We consider the network of Fig. 2 without linkl4b to illustrate the performance of Algorithm

1. For γ ≈ 2.5 × 103 and step sizeβ = 0.75, the primal variable updates - source rates and

network lifetime - are plotted in Fig. 6 and Fig. 7, respectively. The corresponding dual node-

price updates are shown in Fig. 8. All the dual link-price updates converge to zero within a few

iterations and are not shown due to space constraints. The optimal primal and dual values are

plotted as dashed lines to illustrate convergence of the iterates to the optimal values.

Note that only nodes 1 and 2 have non-zero resource prices andthus represent bottleneck

resources, while all other nodes and all links have zero prices, as discussed in the previous

subsection. Therefore, nodes 3, 4, and 6 have the same sourcerate as can be seen from Fig.

6, since they all route their traffic through node 1 and thus have the same route-price equal to

µ1(Erx + Etx). Node 1 has a route-price equal toµ1(Es + Etx) and has a higher source rate since

Es is less thanErx in our numerical model.


C. Algorithm 2: Numerical Results

In order to illustrate the performance of Algorithm 2 of Section IV, we consider the network

of Fig. 2 with multipath routing i.e., with linkl4b included. Forγ ≈ 2.5 × 103 and step size

β = 0.75, the optimal source rates, route flows, and network lifetimeare plotted in Fig. 9, Fig.

10, and Fig. 11, respectively. The corresponding dual node-price updates are shown in Fig. 12.

As earlier, all the dual link-prices converge to zero withina few iterations.

Note that nodes 1 and 2 are the only bottleneck resources and ideally they have equal optimal

node-price due to symmetry. However, in practice, they are not exactly equal and as a result

routes4a and4b (also6a and6b) do not have equal route prices. Based on the algorithm, nodes

4 and 6 route their data only through the route with a lower price which in turn increases the

price of the node (1 or 2) carrying this flow and decreases the price of the other node. Thus,

the prices of nodes 1 and 2 oscillate and so do the flows along routes through these nodes as

shown in Fig. 10. At the packet-level, this oscillation is equivalent to nodes routing their packets

alternately between routes with nearly equal prices.

Algorithms 1 and 2 exhibit convergence within at most a few hundred iterations in most

practical cases or even faster as illustrated in the numerical results above. Therefore, the energy

expended in message passing during the transient phase of these algorithms corresponds to only

a few hundred packets communicated by each node. Typical battery capacity in sensors allows

several orders of magnitude more packets to be transmitted during the lifetime of the node and

thus the energy expended in the execution of the above distributed algorithms can be negligible.

VI. D ISCUSSION ANDCONCLUSION

The energy constrained nature of wireless sensor networks results in an inherent trade-off

between application performance that depends on the gathered data and the operational lifetime of

the network. In this paper, we present a framework that allows a systematic design of distributed

protocols for the transport and network layers that allow the network to be operated on the

optimal trade-off curve between application performance and network lifetime. We apply our

framework to networks with proportionally fair rate allocation and study practical design issues

by characterizing the optimal utility-lifetime trade-offcurve and through numerical simulations

of the proposed distributed algorithms.


Although the analysis in Sections III and IV considers a single sink in the network, the

analysis and the algorithms can be readily extended to the case of multiple sinks. Note that the

above algorithms exploit thecentralizednature of the sink in computing the network lifetime

by passing the weighted node lifetime prices to the sink. In the case of single-path routing,

a network with multiple sinks trivially partitions into subnetworks each with a single sink. In

the case of multipath routing, since each node can have routes to multiple sinks, the weighted

node-prices could be communicated to every sink so that the network inverse-lifetime update is

computed by all sinks in the network. The associated communication burden on the nodes can

be significantly reduced if the sinks are connected by a backbone network which can be used to

exchange the weighted node lifetime prices between sinks. Alternatively, by forming a spanning

tree within the network with a particular sink as its root, the weighted node-prices only need to

be communicated to this sink which computes the network lifetime updates.

The analysis and the proposed algorithms in this paper can also be extended to another

generalization of the system model involving multiple commodities within the network. An

example of such a network could involve nodes with multi-modal sensing capabilities (e.g.,

temperature, pressure, vibration) with different utilityfunctions for different modes. Any such

node in this network can be modeled as being equivalent to multiple sources each with its

own utility function and having the same set of potential routes for communicating with the

sinks. Finally, even though the developments in this paper are in a sensor network setting, the

methodology is generally extensible to any energy constrained wireless ad hoc network.

There are several directions for future work. The cross-layer design framework can be extended

to include the MAC and physical layers of the protocol stack as well. However, the design of

distributed MAC layer protocols is a challenging task in wireless networks due to interference

resulting from the broadcast nature of such networks. Another interesting direction relates to the

design and convergence analysis of fully asynchronous versions of Algorithm 1 and Algorithm

2 of this paper.

REFERENCES

[1] M. Chiang, S. H. Low, R. A. Calderbank, and J. C. Doyle, “Layering as Optimization Decomposition: A mathematical

theory of network architectures,” Proc. IEEE, Jan 2007, to be published.

[2] S. Shenker, “Fundamental design issues for the future internet,”IEEE J. Select. Areas Commun., vol. 13, no. 7, pp.

1176–1188, Sep 1995.


[3] F. P. Kelly, A. Maulloo, and D. Tan, “Rate control for communicationnetworks: Shadow prices, proportional fairness and

stability,” J. Oper. Res. Soc., vol. 49, no. 3, pp. 237–252, Mar 1998.

[4] S. H. Low and D. E. Lapsley, “Optimization flow control, I: Basic algorithm and convergence,”IEEE/ACM Trans.

Networking, vol. 7, no. 6, pp. 861–874, Dec 1999.

[5] X. Lin and N. Shroff, “Utility maximization for communication networks with multipath routing,”IEEE Trans. Automat.

Contr., vol. 51, no. 5, pp. 766–781, May 2006.

[6] L. Chen, S. Low, M. Chiang, and J. Doyle, “Optimal cross-layer congestion control, routing, and scheduling design in ad

hoc wireless networks,” inProc. IEEE Infocom, Apr 2006.

[7] L. Xiao, M. Johansson, and S. Boyd, “Simultaneous Routing and Resource Allocation via Dual Decomposition,”IEEE

Trans. Commun., vol. 52, no. 7, pp. 1136–1144, Jul 2004.

[8] A. Eryilmaz and R. Srikant, “Joint congestion control, routing and MAC for stability and fairness in wireless networks,”

in Proc. of IZS’06, ETH Zurich, Switzerland, Feb 2006.

[9] M. Chiang, “Balancing transport and physical layers in wireless multihop networks: Jointly optimal congestion control

and power control,”IEEE J. Select. Areas Commun., vol. 23, no. 1, pp. 104–116, Jan 2005.

[10] J. H. Chang and L. Tassiulas, “Routing for maximum system lifetime inwireless ad-hoc networks,” in37th Annual Allerton

Conference on Communications, Control and Computing, Sep 1999.

[11] A. Sankar and Z. Liu, “Maximum lifetime routing in wireless ad hoc networks,” in Proc. IEEE INFOCOM, Mar 2004.

[12] R. Madan and S. Lall, “Distributed algorithms for maximum lifetime routing in wireless sensor networks,” IEEE Trans.

Wireless Commun., to be published.

[13] R. Madan, S. Cui, S. Lall, and A. Goldsmith, “Cross-layer design for lifetime maximization in interference-limited wireless

sensor networks,” inProc. IEEE INFOCOM, Mar 2005.

[14] V. Srinivasan, C. Chiasserini, P. S. Nuggehalli, and R. R. Rao, “Optimal rate allocation for energy-efficient multipath

routing in wireless ad hoc networks,”IEEE Trans. Wireless Commun., vol. 3, no. 3, pp. 891–899, May 2004.

[15] C. E. Perkins and P. Bhagwat, “Highly dynamic destination-sequenced distance-vector routing (DSDV) for mobile

computers,” inACM SIGCOMM Conference on Communications Architectures, Protocols and Applications, 1994, pp.

234–244.

[16] C. Perkins, E. Belding-Royer, and S. Das, “Ad hoc on-demanddistance vector (AODV) routing,” RFC 3561, July 2003.

[Online]. Available: http://rfc.sunsite.dk/rfc/rfc3561.html

[17] D. B. Johnson and D. A. Maltz, “Dynamic source routing in ad hoc wireless networks,” inMobile Computing, Imielinski

and Korth, Eds. Kluwer Academic Publishers, 1996, vol. 353.

[18] Wireless Medium Access Control (MAC) and Physical Layer (PHY) Specifications for Low-rate Wireless Personal Area

Networks, IEEE Computer Society Std. 802.15.4, 2003.

[19] S. Boyd and L. Vandenberghe,Convex Optimization. Cambridge University Press, 2004. [Online]. Available:

http://www.stanford.edu/∼boyd/cvxbook

[20] H. Nama, M. Chiang, and N. Mandayam, “Optimal utility-lifetime trade-off in self-regulating wireless sensor networks:

A distributed approach,” inProc. 40th Annual Conference on Information Sciences and Systems (CISS), Mar 2006.

[21] D. P. Bertsekas and J. N. Tsitsiklis,Parallel and Distributed Computation: Numerical Methods. Prentice-Hall, 1989.

[22] S. Athuraliya, D. Lapsley, and S. H. Low, “An enhanced RandomEarly Marking algorithm for Internet flow control,” in

Proc. IEEE Infocom, Mar 2000.


[23] H. Nama, M. Chiang, and N. Mandayam, “Utility-lifetime trade-off in wireless sensor networks: Characterization and

distributed algorithms,” WINLAB, Rutgers University, NJ, Tech. Rep.,Aug. 2006.

[24] D. P. Bertsekas,Nonlinear Programming, 2nd ed. Athena Scientific, 1999.

[25] N. Z. Shor,Minimization methods for non-differentiable functions. Springer-Verlag, 1985.

[26] CC2420, True single-chip 2.4 GHz IEEE 802.15.4/ZigBee RF transceiver with MAC support, Chipcon Inc. [Online].

Available: http://www.chipcon.com/index.cfm?katid=2&subkatid=12&dok id=115

[27] MOSEK. [Online]. Available: http://www.mosek.com


Fig. 1. Vertical and Horizontal Decomposition

−150 −100 −50 0 50 100 150

−250

−200

−150

−100

−50

0

Source1 Source2

Source3 Source4 Source5

Source6

Sink

l1 l2

l3 l4a l4b

l5

l6

Fig. 2. Sensor network with multipath routing


0 50 100 150 2006.5

7

7.5

8

8.5

9

9.5

10

10.5

Network Lifetime (in sec)

Net

wor

k U

tility

Multipath routing, Total energy = 1 JSingle−path routingMultipath routing, Total energy = 0.5 J

γ ≈ 4 x 104

γ ≈ 1

γ ≈ 2.5 x 103

γ ≈ 2 x 104

Fig. 3. Network Utility vs. Network Lifetime

0 50 100 150 2000

0.005

0.01

0.015

0.02

0.025

0.03


Sou

rce

Rat

es (

in M

bps)

Source 1Source 2Source 3Source 4Source 5Source 6

γ ≈ 1

γ ≈ 4 x 104

γ ≈ 2.5 x 103

Fig. 4. Source rates vs. Network Lifetime with multipath routing


0 50 100 1500

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

0.045


Sou

rce

Rat

es (

in M

bps)


γ ≈ 1

γ ≈ 2 x 104

γ ≈ 2.5 x 103

Fig. 5. Source rates vs. Network lifetime with single-path routing

0 50 100 150 200 2501

2

3

4

5

6

7

8

9

10x 10

−3

Number of Iterations

Sou

rce

Rat

es (

in M

bps)


Dotted lines represent optimal solutions obtained using centralized computation

Fig. 6. Distributed Algorithm 1 (single-path routing): Source rates


0 50 100 150 200 2505

10

15

20

25

30

35

40

45

50

55


Net

wor

k lif

etim

e (in

s)

Dotted line represents optimal solution obtained using centralized computation

Fig. 7. Distributed Algorithm1 (single-path routing): Network lifetime

0 50 100 150 200 2500

0.05

0.1

0.15

0.2

0.25

0.3

0.35


Dua

l var

iabl

es −

µ



Fig. 8. Distributed Algorithm 1 (single-path routing): Dual variableµ


0 50 100 150 200 2501

1.5

2

2.5

3

3.5

4

4.5

5

5.5

6x 10

−3


Sou

rce

Rat

es (

in M

bps)



Fig. 9. Distributed Algorithm 2 (multipath routing): Source rates

0 50 100 150 200 2500

1

2

3

4

5

6x 10

−3


Rou

te F

low

s (in

Mbp

s)

Route 1Route 2Route 3Route 4aRoute 4bRoute 5Route 6aRoute 6b


Fig. 10. Distributed Algorithm 2 (multipath routing): Route flows


0 50 100 150 2005

10

15

20

25

30

35

40

45

50

55


Net

wor

k lif

etim

e (in

s)

Dotted line represents optimal solution obtained using centralized computation

Fig. 11. Distributed Algorithm 2 (multipath routing): Network lifetime

0 50 100 150 200 2500

0.05

0.1

0.15

0.2

0.25

0.3

0.35


Dua

l var

iabl

es −

µ



Fig. 12. Distributed Algorithm 2 (multipath routing): Dual variableµ

hithesh nama et al. 1 optimal utility-lifetime trade-off in ...chiangm/utilitylife.pdfhithesh nama...

Documents