Cross-Layer Throughput Optimization With Power Control in Sensor Networks

3300 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 60, NO. 7, SEPTEMBER 2011

Cross-Layer Throughput Optimization WithPower Control in Sensor Networks

Maggie X. Cheng, Xuan Gong, Lin Cai, Senior Member, IEEE, and Xiaohua Jia, Senior Member, IEEE

Abstract—In wireless sensor networks, transmission power hasa significant impact on network throughput as wireless interfer-ence increases with transmission power, and interference nega-tively impacts the network throughput. In this paper, we try toimprove the network throughput through cross-layer optimiza-tion. We first present two algorithms to compute the transmis-sion power of each node with the objectives of minimizing thetotal transmission power and minimizing the total interference,respectively, from which we can obtain a network topology thatensures a connected path from each source to the sink; then, wecompute the maximum achievable throughput from the obtainedtopology by using joint routing and link rate control. The powercontrol algorithms can generate symmetric links or asymmetriclinks if so desired. Based on different link models, we use differentalgorithms to compute the maximum achievable throughput. Sincecomputing the maximum throughput is an NP-hard problem, weuse efficient heuristics that use a sufficient condition instead of thecomputationally expensive-to-get optimal condition to capture themutual conflict relation in a collision domain. The formal proof forthe sufficient condition is provided, and the proposed algorithmsare compared with previous work. Simulation results show thatthe proposed algorithms improve the network throughput andreduce the energy consumption, with significant improvement overprevious work on both aspects.

Index Terms—Clique, cross-layer design, interference, lin-ear programming, optimization, power control, sensor network,throughput, topology control.

I. INTRODUCTION

IN WIRELESS sensor networks, due to the broadcast na-ture of wireless transmission, the signal from one sensor

could reach many unintended receivers and interfere with thereception of these neighbors. The higher transmission powerit uses, the more neighbors with which it interferes. As the

Manuscript received January 14, 2011; revised May 6, 2011 and June 20,2011; accepted June 21, 2011. Date of publication June 27, 2011; date ofcurrent version September 19, 2011. The works of M. X. Cheng and X. Gongwere supported in part by the National Science Foundation under Grant CNS-0841388. The work of L. Cai was supported in part by grants from the NaturalSciences and Engineering Research Council of Canada. The work of X. Jiawas supported in part by the National Science Foundation of China underGrant 60970117 and in part by the Research Grants Council of Hong Kongunder Project CityU 114710. The review of this paper was coordinated byDr. G. Mao.

M. X. Cheng is with the Missouri University of Science and Technology,Rolla, MO 65409 USA (e-mail: [email protected]).

X. Gong is with Automatic Data Processing, Inc., Seattle, WA 98104 USA(e-mail: [email protected]).

L. Cai is with the University of Victoria, Victoria, BC V8W 2Y2, Canada(e-mail: [email protected]).

X. Jia is with the City University of Hong Kong, Kowloon, Hong Kong(e-mail: [email protected]).

Color versions of one or more of the figures in this paper are available onlineat http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TVT.2011.2160883

interference level increases, the network throughput decreases.To intuitively understand how transmission power works on thenetwork throughput, we can picture a multihop wireless sensornetwork with a fixed number of nodes. If two nodes can heareach other, then we build a link between them. When one link isactive, any other link that interferes with it should not be. Whenthe transmission power increases, the link density increases,and consequently, a wireless link will have many other linksinterfering with it. All these conflicting links cannot be activeat the same time; they must carefully be scheduled to transmitat a different time; otherwise, their transmissions will interferewith each other. Although the wireless link capacity remainsthe same, the spatial reuse of the wireless spectrum decreasesas the transmission power increases. As a result, the networkthroughput drops.

To increase the network throughput, we can address theproblem from different layers: At the physical layer, we canadjust the transmission power to reduce interference; at thenetwork layer, we can route data packets to the least interferedpath; and at the medium access control (MAC) layer, we canschedule transmissions to avoid simultaneous transmissionsfrom interfering links. To make sure all transmissions canbe scheduled without conflict, we also need to control thetransmission data rate to make sure a node’s channel occupationtime is proportional to its data rate. Overall, it takes a cross-layer design scheme to achieve the maximum throughput.

In this paper, a cross-layer optimization framework is pro-vided. We first try to decide the transmission power of eachnode toward optimizing throughput, and then, we use a jointrouting and link rate control scheme to achieve the maximumthroughput. The second part computes the maximum achiev-able throughput on a given topology, therefore serving as theassessment of the power control schemes.

The main contributions of this paper include the following:1) We formulated the maximum throughput power control prob-lem into two linear programs and designed efficient algorithmsto solve them. The power control algorithms can generate sym-metric or asymmetric links, as required. 2) For both symmetricand asymmetric links, we provided mathematical optimizationmodels to compute the maximum achievable throughput on agiven topology. Part of it requires to accurately capture themutual conflicting relation among wireless links, which is awell-known NP-hard problem. We proposed a polynomial-term constraint that can sufficiently capture the mutual conflictrelation among wireless links and is tighter than all knownpolynomial-term approximations.

Although the objective of this paper is to achieve maximumthroughput, we found that the power control schemes also

0018-9545/$26.00 © 2011 IEEE

CHENG et al.: CROSS-LAYER THROUGHPUT OPTIMIZATION WITH POWER CONTROL IN SENSOR NETWORKS 3301

reduce the total energy consumption of the sensor network.Through cross-layer optimization, we show that it is possibleto achieve higher throughput with longer lifetime.

The rest of this paper is organized as follows: In Section II,we briefly survey the most related work in cross-layer optimiza-tion. In Section III, we present the mathematical optimizationmodels and algorithms for power control. In Section IV, wepresent models and algorithms for joint routing and link rateallocation. In Section V, we compare the protocol model used inthis paper with the physical model. In Section VI, we compareour algorithms with previous work and show the effectivenessof power control on throughput improvement. In the Appendix,we show the theoretical foundation of the optimization modelwith formal proof.

II. RELATED WORK

Along the line of maximizing the network throughputthrough transmission power control, the most related workis [1]. In [1], two pruning algorithms were presented to as-sign transmission power to nodes to minimize the maximalinterference or total interference, respectively; then, linear pro-gramming models were used for data routing to maximize thenetwork throughput. The power control algorithms do not useoptimization methods; instead, they start with a graph Gmax

generated by using the maximum transmission power and thenprune edges to reduce power consumption. We compared ourlinear programming-based power control algorithms with thepruning algorithms in [1] and found significant performanceimprovement. Reference [1] is the most relevant work since italso jointly considered three layers that involve power control,routing, and transmission rate control.

Most of the other cross-layer design schemes only involvetwo layers, such as joint routing and link rate allocation [2]–[4],and joint power control and scheduling when the routing in-formation is given [5]. In [5], links that share a common nodeare not allowed to transmit in the same slot; for disjoint links,whether a node’s reception is interfered by others is decidedby a physical model, i.e., if the receiver’s signal-to-noise ratioexceeds the threshold, then it is considered not interfering. In[5], the interference model is a hybrid of the protocol andphysical models. The physical model is applicable only whenthe routing information is given and the traffic demand oneach link is given as input. However, in our work, the routinginformation is not given and the traffic demand on each link isunknown; therefore, a pure protocol model is used, in whichthe interfering relation is determined by network link topologyrather than the actual signal strength.

Throughput modeling and optimization in wireless networksstarted as early as 1987 [6]–[8], and at that time, it was forpacket radio networks. In recent years, it has become a hottopic again when multihop wireless networks became popular.Some researchers attempted to give asymptotic results withoutinput on traffic and network topology [9]–[11], and most otherstried to find the exact solutions [4], [12]–[18]. To find theexact throughput, part of the effort is to extend the conceptof flow networks to multihop wireless networks. To come upwith the capacity constraint, some scholars used link capacity

as the upper bound of the data rate of a single link withoutconsidering the interference from other links [13], [19]; someattempted to model interference but used global informationsuch as cliques on a conflict graph (see [16]), which is NP-hardto get in the first place; and some proposed polynomial-termconstraint but simply considered that all links within two hopsof a common link are conflicting links and required that the totaldata rate of these links be bounded by the wireless link capacity.We have demonstrated in this paper that our polynomial-termconstraint is more accurate than this simplified model and cansufficiently capture the interference relation. Our interferencemodel represents the tightest sufficient condition known so far.

III. TRANSMISSION POWER CONTROL

Given a sensor network of N nodes with adjustable trans-mission power, the objective of power control is to compute thetransmission power for all nodes such that the network through-put is maximized. Depending on whether DATA packets needto be acknowledged by the next hop, links can be symmetricor asymmetric. The algorithms presented in the following canproduce either symmetric or asymmetric links.

Since the network throughput is related to interference, andinterference is related to total transmission power, we use mini-mum total power and minimum interference as the optimizationobjectives, respectively, in the following for transmission powercontrol.

A. For Minimum Total Power

1) Linear Programming Model:Variables: Let Pi be the transmission power of node i, Rij

be the data rate on link (i, j), and Xij be the decision variable:Xij = 1 if there is a link from i to j, and Xij = 0 otherwise.

Constants: Pij is the transmission power needed for nodei to reach node j, Pmax is the maximum transmission powernode i can use, Ri is the source rate of node i, and B is thewireless link capacity. At this stage, the objective is to geta connected topology with minimum total power (connectedmeans there is a connected path from each source to the sink),and there is no concern about the data rate; therefore, Ri is usedas a constant. If i is a source node, then Ri > 0, if i is a sinknode, then Ri < 0, and if i is neither a source nor a sink, thenRi = 0.

Now, we can formulate the minimum power topology controlproblem as follows:

Minimize∑

i

Pi (1)

subject to∑

j

Xij ≥ 1 ∀ i ∈ sources (2a)

∑

j

(Rij − Rji) = Ri ∀ i (2b)

Pi ≥ XijPij ∀ (i, j) (2c)


Fig. 1. (a) Network with six nodes. (b) Network topology resulting fromusing Pmax. (c) Network topology resulting from the minimum power topologycontrol algorithm.

Xij ≥ Rij/B ∀ link(i, j) (2d)

Xij = {0, 1} ∀ link(i, j) (2e)

0 ≤ Rij ≤ B ∀ link(i, j) (2f)

0 ≤ Pi ≤ Pmax ∀ i. (2g)

In the foregoing formulation, (2a) requires that each sourcemust have at least one outgoing link; (2b) requires that the datarate satisfies flow conservation; (2c) requires that to establish alink from i to j, node i must use enough transmission powerto reach j; and (2d) requires that if the data rate from i to j isnonzero, then there must be a link from i to j.

Solving the foregoing linear program can get Xij , fromwhich we can obtain a connected topology with minimumtotal power. However, since Xij is a 0–1 integer variable, theproblem remains NP-hard. An LP-rounding-based heuristic ispresented in the next section.

The foregoing linear program is for asymmetric links, i.e.,Xij and Xji can be different, which implies i can hear j butj cannot hear i or vice versa. If the links are required to besymmetric because DATA packets need to be acknowledged bythe next hop, then the following constraint is added: Xij = Xji.In practice, Pij could be different from Pji, but to ensure that iand j can reach each other, we only need Xij = Xji and applycondition (2c) on both (i, j) and (j, i) as follows: Pi ≥ XijPij ,and Pj ≥ XjiPji.

Fig. 1 shows a simple network with six nodes. Nodes 1 and 4are two sources, and node 6 is the sink. Suppose that the nodesare evenly spaced. Without using any power control algorithm,if each node transmits at the maximum power level, then itresults in the network topology in (b). Solving the foregoing lin-ear program for symmetric links results in a network topologywith the minimum transmission power, in which P2 = P3 = 0;therefore, there are no edges coming out of nodes 2 and 3;X14 = X45 = X56 = 1 (links are symmetric), and therefore, itensures a connected path from each source to the sink.

2) LP-Rounding-Based Algorithm: We first relax the integerconstraint of Xij and solve the problem as a real-valued linearprogram. The solution includes fractional values for Xij . Wewill use a rounding-based algorithm to construct the networktopology.

We introduce two variables Cij and Mi. Cij = 1 means link(i, j) is established; otherwise, Cij = 0. Mi = 1 means nodei has a connected path to the sink; otherwise, Mi = 0. In thefollowing algorithm, the real-valued Xij from the solution ofLP is rounded to Cij , and the network topology is derivedfrom Cij . Initially, each source node i has Mi = 0. When thealgorithm terminates, all the sources have Mi = 1.

MINPOWER

1 sort Xij in nonincreasing order into a listset Cij = 0 for all pairs of i, jset Mi = 0 for all sources;

2 while∑

i∈sources Mi < |sources|3 do remove the largest Xij from the list

set Cij = 1set Pi = maxj{CijPij};[for symmetric links, add the following

set Cji = 1set Pj = maxi{CjiPji}remove Xji from the list;]

4 for all j,if Pi ≥ Pij

set Cij = 1remove Xij from the list;

5 if there is a connected path fromnode i to the sink, set Mi = 1;

6 return Cij and Pi.

Remark: If there is a tie in choosing the largest Xij

in line 3, choose the link (i, j) that leads to the small-est increase in total power: for symmetric links, (i, j) =arg min(i,j){(Pij − Pi) + (Pji − Pj)}; for asymmetric links,(i, j) = arg min(i,j){Pij − Pi}.

B. For Minimum Total Interference

Since interference has a more direct impact on networkthroughput than transmission power does, we try to use min-imum total interference as the objective of power control.Intuitively, we will have a better chance of finding the topologythat maximizes the network throughput. The simulation resultsin Section VI verified the intuition.

The interference model we adopted here is the “protocolmodel”: if node j is in the interference range of node i, then jis interfered by i. We try to minimize the total number of nodesthat are subject to interference.

1) Linear Programming Model:Variables: In addition to the variables defined in

Section III-A, we define a few new variables: Yik = 1 if node iuses power level k, k = 0, 1, 2, . . . ,K (k = 0 means the nodeis not transmitting); each node can only choose one transmis-sion power. Ii is the number of nodes interfered by node i’stransmission.

Constants: Nik is the number of nodes in node i’s inter-ference range when node i uses power level k, i.e.,

Minimize∑

i

Ii (3)

Subject to∑

j

Xij ≥ 1 ∀ i ∈ sources (4a)

∑

j

(Rij − Rji) = Ri ∀ i (4b)

Pi ≥ XijPij ∀ (i, j) (4c)


∑

k

Yik = 1 ∀ i (4d)

Ii =∑

k

NikYik ∀ i (4e)

Pi =∑

k

kYik ∀ i (4f)

Xij ≥ Rij/B ∀ link(i, j) (4g)

Xij = {0, 1} ∀ link(i, j) (4h)

Yik = {0, 1} ∀ i, k (4i)

0 ≤ Rij ≤ B ∀ link(i, j. (4j)

Equation (4d) indicates that each node can only choose onepower level. The lowest power level is 0 when the node isnot transmitting. Equation (4e) defines the number of nodesinterfered by node i’s transmission. Equation (4f) translates a0–1 variable Yik into a discrete-valued power Pi. The constantPij is also given in discrete power levels.

Similarly, this integer linear program is NP-hard to solve. Wewill describe an LP-rounding-based scheme in the following.

2) Rounding: The rounding algorithm will generate the net-work topology defined by Cij and the transmission power Pi.The rounding algorithm is largely the same as the roundingalgorithm MinPower, except that when there is a tie in choosingthe largest Xij in line 3, we will choose the link (i, j) that leadsto the smallest increase in the total interference: For symmetriclinks, if link (i, j) is chosen, then Pi = maxj{CijPij}, andPj = maxi{CjiPji}. Update Yik and Yjk, and then calculatethe total increase in interference δI = (

∑k NikYik − Ii) +

(∑

kNjkYjk − Ij). Set (i, j) = arg min(i,j) δI . For asymmetriclinks, δI =

∑k NikYik − Ii, and set (i, j) = arg min(i,j) δI .

IV. MAXIMUM ACHIEVABLE THROUGHPUT

The output from the power control algorithms is the trans-mission power of each node and the resulting topology. It isguaranteed that each source has a connected path to the sink.However, how much throughput can be achieved depends notonly on the topology but also on the upper layer protocols, suchas routing and MAC. Without presumption about what routingand MAC algorithms are used, we calculate the maximumachievable throughput on the resulting topology, which is ameasure of the effectiveness of power control algorithms.

A. Asymmetric Links for One-Way Communication

If DATA packets do not need to be acknowledged, then linksdo not need to be symmetric. A directed path from source tosink consisting of asymmetric links will suffice.

We define Ni as the group of nodes that i can reach, i.e.,Ni = {j|Cij = 1}, where Cij = 1 means that there is a di-rected link from i to j; and we define N+

i as the group ofnodes that can reach node i : N+

i = {j|i ∈ Nj}. Ni and N+i

are obtained from the results of power control and are given asinputs to the following optimization model. Let variable Ri bethe source rate of node i. If node i is neither a source nor thesink, then Ri is set to be zero. We also introduce a decision

variable fi : fi = 1 if i is expected to receive data, i.e., i is arelay node on the routing path or a sink node. The joint routingand link rate allocation problem can be formulated as follows:

maximize∑

i∈sources

Ri (5)

subject to∑

j∈Ni

Rij −∑

j∈N+i

Rji = Ri ∀ i (6a)

∑

j∈Ni

Rij + fi

∑

j∈N+i

∑

k∈Nj

Rjk ≤ B ∀ i (6b)

0 ≤ Rij ≤ B ∀ i,∀ j (6c)

fi = {0, 1} ∀ i. (6d)

Equation (6a) is for flow conservation, and (6b) is the ca-pacity constraint for wireless transmissions. Inequality (6b) is asufficient condition to capture the mutual conflict relationshipamong links. In our previous work [3], a formal proof for itssufficiency is provided.

To linearize inequality (6b) so that we can solve it as a linearprogram, we set the initial value of fi as follows and use theiterative approach to find the solution. Initially, we set fi = 1for all the nodes that have

∑j∈Ni

Cij ≥ 1, and then, we setfi = 1 if i is the sink and fi = 0 if i is a source; it takes two tothree iterations to converge.

B. Symmetric Links for Two-Way Communication

If the DATA packets must be followed by ACKs, then linksmust be symmetric, i.e., Cij = Cji. In the following, we as-sume that the links are symmetric and communication on a linkis two way; therefore, if two links are within two hops of eachother, then they interfere with each other, i.e.,

Maximize∑

i∈sources

Ri (7)

subject to∑

j∈Ni

(Rij − Rji) = Ri ∀ i (8a)

rij +∑

l∈Ni,l �=j

ril +∑

k∈Nj ,k �=i

rjk

+∑

(k,l)∈N2ij

rkl ≤ B ∀ link(i, j) (8b)

rij = Rij + Rji ∀ link(i, j) (8c)

0 ≤ Rij , rij ≤ B ∀ link(i, j). (8d)

In this linear program, (8a) is for flow conservation, and(8b) defines the capacity constraint. The capacity constraint isthe reason for not being able to further increase throughput.Equation (8b) ensures that all the links possibly in the samecollision domain have a total demand of less than B. Since


Fig. 2. Capacity constraint.

the interference model is based on an undirected graph, thetotal flow amount rij is used in (8b) to meet the capacityconstraint. rij is the sum of data rates Rij (from i to j) and Rji

(from j to i).In inequality (8b), N2ij is defined as N2ij = {(k, l)|, where

link (k, l) is a two-hop neighbor of link (i, j), and the sum ofdistance from k to (i, j) and from l to (i, j) via a differentpath is ≤ 4}. If there is no other path, then the distance iscounted as ∞.

For example, in Fig. 2(a), links (k1, l1) and (k2, l2) belongto N2ij , but (k2, l1) does not, because (k2, l1) is not a two-hopneighbor of link (i, j); in Fig. 2(b), link (k, l) does not belong toN2ij since there is only one path to reach link (i, j) from k andl; the distance from k to (i, j) is 1, and the distance from l to(i, j) is ∞. In this case, the mutual conflicting relation among(i, j), (j, k), and (k, l) is captured when we apply the constraint(8b) on link (j, k) : rjk + rij + rkl ≤ B.

Inequality (8b) is a sufficient but not necessary condition forcapturing all the conflict relation in wireless communication.The accurate condition, which is both sufficient and necessary,includes no more than necessary links in the left-hand side ofthe inequality. However, it is an NP-hard problem to identifythese links. To identify these links, we need to first constructa conflict graph [4], in which a link is represented as a node,and a pair of wireless links that are mutually conflicting witheach other are connected by an edge. Then, we need to computeall the cliques on the graph and make sure all the nodes ina clique have total data rate of no more than B. However, ittakes exponential time to list all the cliques. To the best ofour knowledge, the inequality (8b) is so far the most accuratepolynomial-term approximation. For links within two hops oflink (i, j), we only include the links that belong to N2ij in theinequality. Compared with previous work in which all the linkswithin two hops of (i, j) are included in the left-hand side ofthe inequality [1], our solution provides a tighter bound andtherefore enables a higher throughput.

Consider the topology in Fig. 3(a), the conflict graph is inFig. 3(b), in which each wireless link is represented as a node,and the links that are conflicting with each other are connectedby an edge. The optimal solution requires the total data rate onany clique be bounded by B; therefore, the following conditionsmust be satisfied:

rij + ril + rjk ≤ B (9a)

ril + rij + rlx1 + . . . rlxn≤ B (9b)

rjk + rij + rky1 + . . . rkyn≤ B. (9c)

Fig. 3. (a) Sample wireless network. (b) Conflict graph of (a). In (b), the singleedge from node ij to the big circle representing the clique means that there isan edge from ij to every vertex of the clique.

Our solution, which was derived from inequality (8b), re-quires that the following conditions be satisfied. It is the sameas the optimal solution, i.e.,

for (i, j) : rij + ril + rjk ≤ B (10a)

for (i, l) : ril + rij + rlx1 + . . . rlxn≤ B (10b)

for (j, k) : rjk + rij + rky1 + . . . rkyn≤ B. (10c)

However, the previous work that simply includes all the linksthat interfere with (i, j) requires the following condition besatisfied (see [1]), although links lx1, . . . lxn have no conflictwith links ky1, . . . kyn:

rij + ril + rjk + rlx1 + . . . rlxn+ rky1 + . . . rkyn

≤ B.

Apparently, the foregoing condition introduces a larger per-formance gap than condition (8b). Condition (8b) can suf-ficiently capture all the conflicting relation and is the mostaccurate polynomial-term condition known so far. The formalproof for its sufficiency is included in the Appendix.

V. PROTOCOL MODEL VERSUS THE PHYSICAL MODEL

The objective of power control is to compute the transmissionpower for all the nodes such that the network throughput ismaximized. In this paper, we proposed a two-step scheme thatfirst computes the transmission power of nodes, which leadsto a fixed network topology, and then computes the maximumthroughput on the given topology. Each step is formulated as aseparate mathematical optimization problem.

The maximum throughput power control problem has beenaddressed in the literature using the network utility maxi-mization approach [20], in which the problem is formulatedas a single optimization problem. The utilities are functionsof the signal-to-interference ratios that are dependent on thetransmission powers of other users. Being able to solve theproblem as a single optimization problem is indeed advanta-geous; however, the NUM approach can only work with the


Fig. 4. (a) Physical model captures interference from nodes l and m to link(i, j). (b) Protocol model accurately captures interference from links (x, u) and(y, v) to link (i, j).

physical interference model. In the protocol model, there is noutility function that directly maps the transmission power ofusers to data rates; instead, the transmission power is mappedto the transmission distance, and within the transmission range,a node can transmit at any rate between [0, B], where B is thelink capacity. Changing the transmission power will change thenetwork topology but not the data rate since there is no one-to-one mapping between transmission power and data rate.

Compared with the physical model, the protocol model hasits advantage and disadvantage. By the protocol model, onlylinks within two hops of each other are considered interferingwith each other, but in fact, transmitters farther away thantwo hops still cause interference. For example, in Fig. 4(a),by the protocol model, link (l,m) does not interfere with link(i, j), but by the physical model, nodes l and m’s transmissionsstill interfere with link (i, j), only in smaller magnitude. Onthe other hand, the protocol model also has its advantage. Inthe network shown in Fig. 4(b), by our interference modelingapproach, CDij includes {(i, j), (u, i), (v, j)}, CDui includes{(u, i), (x, u), (i, j)}, and CDvj includes {v, j), (y, v), (i, j)}.Since links (x, u) and (y, v) can transmit at the same time,therefore their interference to link (i, j) is not counted twice.This allows cooperation with a time-division multiple-accessscheme to achieve spatial reuse of wireless spectrum and leadsto higher throughput. However, by the NUM approach, nodex’s transmission and node y’s transmission are both added tothe interference on link (i, j) and thus cause lower data rate onlink (i, j).

VI. SIMULATION

We first evaluate the effect of power control algorithms onenergy savings and throughput improvement and compare ouralgorithms with those that do not use power control (referredto as “uniform” model). Then, we compare our algorithms withprevious work in [1] on energy consumption and throughput.

The network consists of 50 sensor nodes and one sink node.All the nodes are randomly deployed in a 250 × 250 region.One node is randomly chosen as the sink, and other 50 nodesare the source nodes. Each node can choose from ten differentpower levels (K = 10), and the difference in transmissionrange of adjacent power levels is 5, whereas the minimaltransmission range (at power level 1) is also 5. Although the

Fig. 5. LP-MinPower and LP-MinInterference compared with the uniformmodels at power levels 7 and 9 with symmetric links. (a) Average power level.(b) Total throughput as percentage of link capacity B.

LP-MinPower algorithm does not require discrete power levels(the model can apply to both continuous-scale and discrete-scale power assignments), in the simulation, we have useddiscrete power levels for all the algorithms. In addition, theinterference range is assumed to be twice of the correspondingtransmission range. The link capacity is assumed to be 1 unit.

In the uniform model without power control, all the nodestransmit at the same power level; therefore, the links are sym-metric. For comparison purpose, we ensure that all the links aresymmetric in our power control algorithms. Once the topologyis determined, we run the maximum throughput algorithm onthe symmetric model. Fig. 5(a) shows the average power levelof all the nodes, and Fig. 5(b) shows the throughput achieved.We compare two of our power control algorithms, i.e., theLP-MinPower and the LP-MinInterference, with uniformmodels with power levels 7 and 9. The results show thatour algorithms use less energy and achieve better through-put. The throughput is shown as a percentage of the linkcapacity B. The plots show the results for 50 test cases.On average, LP-MinPower uses average power level 4.8, andLP-MinInterference uses average power level 5.6, but theachieved throughputs are both higher than those using powerlevel 7 or 9. LP-MinPower has the lowest average power con-sumption, and LP-MinInterference has the highest throughput,and the observation is consistent for every single test case.

The second simulation is to compare the performance ofour algorithms with previous work in [1]. The network setup


Fig. 6. LP-MinPower and LP-MinInterference compared with MinMax andMinTotal in previous work [1] with asymmetric links. (a) Average power level.(b) Total throughput as percentage of link capacity B.

is the same. Results from 50 randomly deployed networks areplotted in Fig. 6. Since the algorithms in [1] produce topologieswith asymmetrical links, for comparison purpose, we alsouse asymmetric models in our algorithms. It is observed thatLP-MinPower uses the least energy, and LP-MinInterferenceachieves the highest throughput. Both LP-MinPower andLP-MinInterference achieved higher throughput than the pre-vious work. The average power level in LP-MinPower is 3.72,and in LP-MinInterference, it is 4.14; whereas the power levelson MinTotal and MinMax in the previous work [1] are 4.45and 5.97, respectively. The network throughputs from the fouralgorithms are 11.5%, 13.5%, 10.0%, and 8.1% of the linkcapacity B, respectively.

We further conducted simulations on networks of 20, 40,60, 80, and 100 nodes, and the results from 30 test cases areaveraged. We found the conclusion is the same in terms ofthe relative performance of the algorithms. Fig. 7 shows theresults on average power level and the network throughputwith asymmetrical and symmetrical links. The conclusion thatMinInterference has the highest throughput and MinPower hasthe lowest power consumption still holds.

VII. CONCLUSION

In this paper, we have addressed the question of howto achieve maximum throughput in sensor networks through

Fig. 7. For networks of different sizes. (a) and (b) Asymmetrical links.(c) and (d) Symmetrical links. (a) Average power level. (b) Average power level.(c) Throughput.


Fig. 8. For a clique of size n = 2.

cross-layer optimization. We first use transmission power con-trol to decide the link topology and then use joint routing andlink rate control to decide the maximum achievable throughputon the topology. We provided optimization models and efficientalgorithms for power control as well as for joint routing andrate control. To effectively estimate the impact of wirelessinterference on throughput, we proposed to use a sufficientcondition in the linear program and have provided vigorousmathematical proof that the condition is sufficient to capturethe interfering relation among wireless links. Although theproposed algorithms aim to optimize throughput only, they alsoreduce the energy consumption of sensor networks. For futurework, we will consider the joint optimization of throughput andenergy with specific requirement on energy or lifetime.

APPENDIX

The optimal solution to the maximum throughput problemdefined in Section III requires the total data rate of all thelinks represented by any clique be bounded by B:

∑l∈Q rl ≤

B ∀ clique Q on the conflict graph. This is a sufficient andnecessary condition. Since to list all the cliques in a graph is anNP-hard problem, hereby we use a sufficient condition in itsplace. Inequality (8b) is a sufficient condition, and it takespolynomial time to compute.

Theorem 1: If inequality (8b) is satisfied on every wirelesslink, then the following constraint is satisfied:

∑l∈Q rl ≤ B ∀

clique Q on the conflict graph.Proof: We show that for any clique found on the conflict

graph, the left-hand side of inequality (8b) includes the data rateof all the links represented in the clique.

We take an arbitrary clique of size n. When n = 2, there areonly two links concerned. Call them links i and j. Inequality(8b) requires ri + rj ≤ B when i and j are one-hop neighbors,or rk + ri + rj ≤ B when i and j are two-hop neighbors (seeFig. 8). Therefore, the sufficient and necessary condition ri +rj ≤ B (from the clique approach) is trivially satisfied.

When n ≥ 3, we distinguish two cases: In case 1, the n linksare on a network that does not have closed cycles [see Fig. 9(a)];and in case 2, the n links are on a network that has closed cycleswith zero or more open tails [see Fig. 9(b) and (c)]. We assumethat wireless links i, j, and k are on a clique. In case 1, since allthe links on a clique are within two hops of each other, andthere is no cycle, choosing the link with the check mark toapply condition (8b) can ensure ri + rj + rk ≤ B. In case 2,apparently, if i, j, and k are on a single cycle of seven ormore links (single means it does not contain any other cyclesin it), then they must be connected head to tail to have mutualconflicts and form a clique [see Fig. 9(b)]. This trivial case caneasily be solved by applying condition (8b) on the middle one.Otherwise, if the cycle has at most six links, from Fig. 9(c), itcan be shown that by applying condition (8b) to the link with

Fig. 9. For a clique of size n ≥ 3. (a) With no cycle. (b) With seven or morelinks in a single cycle. (c) With six or less links in a single cycle.

the check mark, we can ensure ri + rj + rk ≤ B. Therefore,the inequality (8b) is a sufficient condition.

Fig. 9 only shows cliques of size 3 formed by links i, j, andk. The proof still holds for cliques of size > 3. When thereare n > 3 links in a clique, the analysis for case (1) remains thesame. For case 2, it is not possible to have a clique of size n > 3when all the links in the clique are on a single cycle of seven ormore links [see Fig. 9(b)]; therefore, case 2 reduces to have atmost six links in a single cycle [see Fig. 9(c)], and the analysisfor it remains the same. �

REFERENCES

[1] D. Li, B. Wang, and X. Jia, “Topology control for throughput optimizationin wireless mesh networks,” in Proc. 4th Int. Conf. MSN, Dec. 10–12,2008, pp. 161–168.

[2] M. X. Cheng, X. Gong, and L. Cai, “Link rate allocation under bandwidthand energy constraints in sensor networks,” in Proc. IEEE GLOBECOM,2008, pp. 1–5.

[3] M. X. Cheng, X. Gong, and L. Cai, “Joint routing and link rate allocationunder bandwidth and energy constraints in sensor networks,” IEEE Trans.Wireless Commun., vol. 8, no. 7, pp. 3770–3779, Jul. 2009.

[4] K. Jain, J. Padhye, V. N. Padmanabhan, and L. Qiu, “Impact of interfer-ence on multi-hop wireless network performance,” in Proc. 9th Annu. Int.Conf. MobiCom, 2003, pp. 66–80.

[5] J. Tang, G. Xue, C. Chandler, and W. Zhang, “Link scheduling with powercontrol for throughput enhancement in multihop wireless networks,”IEEE Trans. Veh. Technol., vol. 55, no. 3, pp. 733–742, May 2006.


[6] F. Tobagi, “Modeling and performance analysis of multihop packet radionetworks,” Proc. IEEE, vol. 75, no. 1, pp. 135–155, Jan. 1987.

[7] A. Kershenbaum, R. Boorstyn, and M.-S. Chen, “An algorithm forevaluation of throughput in multihop packet radio networks with complextopologies,” IEEE J. Sel. Areas Commun., vol. SAC-5, no. 6, pp. 1003–1012, Jul. 1987.

[8] R. Boorstyn, A. Kershenbaum, B. Maglaris, and V. Sahin, “Throughputanalysis in multihop csma packet radio networks,” IEEE Trans. Commun.,vol. COM-35, no. 3, pp. 267–274, Mar. 1987.

[9] J. Li, C. Blake, D. S. De Couto, H. I. Lee, and R. Morris, “Capacity ofad hoc wireless networks,” in Proc. 7th Annu. Int. Conf. MobiCom, 2001,pp. 61–69.

[10] P. Gupta and P. Kumar, “The capacity of wireless networks,” IEEE Trans.Inf. Theory, vol. 46, no. 2, pp. 388–404, Mar. 2000.

[11] M. Gastpar and M. Vetterli, “On the capacity of wireless networks: Therelay case,” in Proc. 21st IEEE INFOCOM, 2002, vol. 3, pp. 1577–1586.

[12] X. L. Huang and B. Bensaou, “On max-min fairness and scheduling inwireless ad-hoc networks: Analytical framework and implementation,” inProc. 2nd ACM Int. Symp. MobiHoc, 2001, pp. 221–231.

[13] M. Kodialam and T. Nandagopal, “Characterizing achievable rates inmulti-hop wireless networks: The joint routing and scheduling problem,”in Proc. 9th Annu. Int. Conf. MobiCom, 2003, pp. 42–54.

[14] J. Padhye, S. Agarwal, V. N. Padmanabhan, L. Qiu, A. Rao, and B. Zill,“Estimation of link interference in static multi-hop wireless networks,” inProc. 5th ACM SIGCOMM IMC, 2005, p. 28.

[15] S. Rangwala, R. Gummadi, R. Govindan, and K. Psounis, “Interference-aware fair rate control in wireless sensor networks,” in Proc. SIGCOMM,2006, pp. 63–74.

[16] Y. Li, L. Qiu, Y. Zhang, R. Mahajan, Z. Zhong, G. Deshpande, andE. Rozner, “Effects of interference on wireless mesh networks: Patholo-gies and a preliminary solution,” in Proc. HotNets, Nov. 2007.

[17] L. Qiu, Y. Zhang, F. Wang, M. K. Han, and R. Mahajan, “A general modelof wireless interference,” in Proc. 13th Annu. ACM Int. Conf. MobiCom,2007, pp. 171–182.

[18] Y. Li, L. Qiu, Y. Zhang, R. Mahajan, and E. Rozner, “Predictable per-formance optimization for wireless networks,” in Proc. ACM SIGCOMMConf. Data Commun., 2008, pp. 413–426.

[19] M. Alicherry, R. Bhatia, and L. Li, “Joint channel assignment and routingfor throughput optimization in multiradio wireless mesh networks,” inProc. ACM MobiCom, 2005, pp. 58–72.

[20] D. Palomar and M. Chiang, “A tutorial on decomposition methods fornetwork utility maximization,” IEEE J. Sel. Areas Commun., vol. 24,no. 8, pp. 1439–1451, Aug. 2006.

Maggie X. Cheng received the Ph.D. degree incomputer science from the University of Minnesota,Minneapolis, in 2003.

She is currently an Associate Professor with theComputer Science Department, Missouri Universityof Science and Technology, Rolla. Her research in-terests include wireless networks and combinatorialoptimization.

Xuan Gong received the M.Sc. and Ph.D. degreesin computer science from the Missouri University ofScience and Technology, Rolla, in 2008 and 2011,respectively.

He is currently a Research Engineer with Auto-matic Data Processing, Inc., Seattle, WA. His areasof interests include wireless networks, mobile com-puting, and combinatorial optimization.

Lin Cai (S’00–M’06–SM’10) received the M.A.Sc.and Ph.D. degrees in electrical and computer engi-neering from the University of Waterloo, Waterloo,ON, Canada, in 2002 and 2005, respectively.

Since 2005, she has been an Assistant Professorand then an Associate Professor with the Electricaland Computer Engineering Department, Universityof Victoria, Victoria, BC, Canada. She served as As-sociate Editor for the EURASIP Journal on WirelessCommunications and Networking, the InternationalJournal of Sensor Networks, and the Journal of Com-

munications and Networks. Her research interests focus on network protocoland architecture design supporting emerging multimedia traffic over wireless,mobile, ad hoc, and sensor networks.

Dr. Cai was the recipient of the Natural Sciences and Engineering ResearchCouncil of Canada Discovery Accelerator Supplement Grant in 2010, the BestPaper Award from the 2008 IEEE International Conference on Communica-tions, and the Best Academic Paper Award from the 2011 IEEE Wireless Com-munications and Networking Conference. She served as Technical ProgramCommittee Symposium Cochair for the 2010 IEEE Global CommunicationsConference and Associate Editor for the IEEE TRANSACTIONS ON WIRE-LESS COMMUNICATIONS and the IEEE TRANSACTIONS ON VEHICULAR

TECHNOLOGY.

Xiaohua Jia (SM’01) received the D.Sc. degree ininformation science from the University of Tokyo,Tokyo, Japan, in 1991.

He is currently a Chair Professor with the De-partment of Computer Science, City University ofHong Kong, Kowloon, Hong Kong. His researchinterests include distributed systems, computer net-works, wireless sensor networks, and mobile wire-less networks.

Mr. Jia was an Editor of the IEEE TRANSAC-TIONS ON PARALLEL AND DISTRIBUTED SYS-

TEMS (2006–2009), Wireless Networks, the Journal of CombinatorialOptimization, etc. He was the General Chair of the 2008 Association for Com-puting Machinery International Symposium on Mobile Ad Hoc Networking andComputing, the Technical Program Committee (TPC) Cochair of the 2009IEEE Mobile Adhoc and Sensor Systems, the Area Chair of the 2010 IEEEInternational Conference on Computer Communications (INFOCOM’10), TPCSymposium Cochair of the 2010 IEEE Global Communications Conference,and Panel Cochair of IEEE INFOCOM’11.

Date post:	25-Sep-2016
Category:	Documents
Upload:	mx
View:	212 times
Download:	0 times

Cross-Layer Throughput Optimization With Power Control in Sensor Networks

Documents