rate allocation and network lifetime problems for wireless sensor networks

IEEE/ACM TRANSACTIONS ON NETWORKING, VOL. 16, NO. 2, APRIL 2008 321

Rate Allocation and Network Lifetime Problems forWireless Sensor Networks

Y. Thomas Hou, Senior Member, IEEE, Yi Shi, Member, IEEE, and Hanif D. Sherali

Abstract—An important performance consideration for wirelesssensor networks is the amount of information collected by all thenodes in the network over the course of network lifetime. Sincethe objective of maximizing the sum of rates of all the nodes inthe network can lead to a severe bias in rate allocation among thenodes, we advocate the use of lexicographical max-min (LMM) rateallocation. To calculate the LMM rate allocation vector, we developa polynomial-time algorithm by exploiting the parametric analysis(PA) technique from linear program (LP), which we call serial LPwith Parametric Analysis (SLP-PA). We show that the SLP-PA canbe also employed to address the LMM node lifetime problem muchmore efficiently than a state-of-the-art algorithm proposed in theliterature. More important, we show that there exists an elegantduality relationship between the LMM rate allocation problem andthe LMM node lifetime problem. Therefore, it is sufficient to solveonly one of the two problems. Important insights can be obtainedby inferring duality results for the other problem.

Index Terms—Energy constraint, flow routing, lexicographicmax-min, linear programming, network capacity, node lifetime,parametric analysis, rate allocation, sensor networks, theory.

I. INTRODUCTION

WIRELESS sensor networks consist of battery-powerednodes that are endowed with a multitude of sensing

modalities including multi-media (e.g., video, audio) and scalardata (e.g., temperature, pressure, light, magnetometer, infrared).Although there have been significant improvements in pro-cessor design and computing, advances in battery technologystill lag behind, making energy resource considerations the fun-damental challenge in wireless sensor networks. Consequently,there have been active research efforts on performance limitsof wireless sensor networks. These performance limits include,among others, network capacity (see e.g., [13]) and networklifetime (see e.g., [7], [8]). Network capacity typically refers tothe maximum amount of bit volume that can be successfullydelivered to the base station (“sink node”) by all the nodes inthe network, while network lifetime refers to the maximumtime limit that nodes in the network remain alive until one ormore nodes drain up their energy.

Manuscript received November 14, 2005; revised September 23, 2006 andFebruary 14, 2007; approved by IEEE/ACM TRANSACTIONS ON NETWORKING

Editor R. Srikant. The work of Y. T. Hou and Y. Shi was supported in part by theNational Science Foundation (NSF) under Grant CNS-0347390 and the Officeof Naval Research (ONR) under Grant N00014-05-1-0481. The work of H.D.Sherali was supported in part by the NSF under Grant CMMI-0552676. Thispaper was presented in part at ACM MobiHoc 2004, Tokyo, Japan.

Y. T. Hou and Y. Shi are with the Bradley Department of Electrical and Com-puter Engineering, Virginia Polytechnic Institute and State University, Blacks-burg, VA 24061 USA (e-mail: [email protected]; [email protected]).

H. D. Sherali is with the Grado Department of Industrial and Systems En-gineering, Virginia Polytechnic Institute and State University, Blacksburg, VA24061 USA (e-mail: [email protected]).

Digital Object Identifier 10.1109/TNET.2007.900407

In this paper, we consider an overarching problem that en-compasses both performance metrics. In particular, we studythe network capacity problem under a given network lifetimerequirement. Specifically, for a wireless sensor network whereeach node is provisioned with an initial energy, if all nodes arerequired to live up to a certain lifetime criterion, what is themaximum amount of bit volume that can be generated by the en-tire network? At first glance, it appears desirable to maximizethe sum of rates from all the nodes in the network, subject tothe condition that each node can meet the network lifetime re-quirement. Mathematically, this problem can be formulated asa linear programming (LP) problem (see Section II-B) withinwhich the objective function is defined as the sum of rates overall the nodes in the network and the constraints are: 1) flow bal-ance is preserved at each node, and 2) the energy constraint ateach node is met for the given network lifetime requirement.However, the solution to this problem shows (see Section VI)that although the network capacity (i.e., the sum of bit rates overall nodes) is maximized, there exists a severe bias in rate alloca-tion among the nodes. In particular, those nodes that consumethe least amount of power on their data path toward the basestation are allocated with much more bit rates than other nodesin the network. Consequently, the data collection behavior forthe entire network only favors certain nodes that have this prop-erty, while other nodes will be unfavorably penalized with muchsmaller bit rates.

The fairness issue associated with the network capacity max-imization objective calls for a careful consideration in rate allo-cation among the nodes. In this paper, we investigate the rate al-location problem in an energy-constrained sensor network for agiven network lifetime requirement. Our objective is to achievea certain measure of optimality in the rate allocation that takesinto account both fairness and bit rate maximization. We advo-cate to use the so-called Lexicographic Max-Min (LMM) crite-rion [14], which maximizes the bit rates for all the nodes for thegiven energy constraint and network lifetime requirement. Atfirst level, the smallest rate among all the nodes is maximized.Then, we continue to maximize the second level of smallest rateand so forth. The LMM rate allocation criterion is appealingsince it addresses both fairness and efficiency (i.e., bit rate max-imization) in an energy-constrained network.

A naive approach to the LMM rate allocation problemwould be to apply a max-min-like iterative procedure. Underthis approach, successive LPs are employed to calculate themaximum rate at each level based on the available energy forthe remaining nodes, until all nodes use up their energy. We callthis approach “serial LP with energy reservation” (SLP-ER).We show that, although SLP appears intuitive, unfortunatelyit usually gives an incorrect solution. To understand how this

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322 IEEE/ACM TRANSACTIONS ON NETWORKING, VOL. 16, NO. 2, APRIL 2008

could happen, we must understand a fundamental differencebetween the LMM rate allocation problem described hereand the classical max-min rate allocation in [3]. Under theLMM rate allocation problem, the rate allocation problem isimplicitly coupled with a flow routing problem, while under theclassical max-min rate allocation, there is no routing probleminvolved since the routes for all flows are given. As it turnsout, for the LMM rate allocation problem, any iterative rateallocation approach that requires energy reservation at eachiteration is incorrect. This is because, unlike max-min, whichaddresses only the rate allocation problem with fixed routesand yields a unique solution at each iteration, for the LMMrate allocation problem, there usually exist non-unique flowrouting solutions corresponding to the same rate allocation ateach level. Consequently, each of these flow routing solutionswill yield different available energy levels on the remainingnodes for future iterations and so forth, leading to a differentrate allocation vector, which usually does not coincide with theoptimal LMM rate allocation vector.

In this paper, we develop an efficient polynomial-time algo-rithm to solve the LMM rate allocation problem. We exploit theso-called parametric analysis (PA) technique [2] at each ratelevel to determine the minimum set of nodes that must depletetheir energy. We call this approach serial LP with PA (SLP-PA).In most cases when the problem is non-degenerate, the SLP-PAalgorithm is extremely efficient and only requires timecomplexity to determine whether or not a node is in the min-imum node set for each rate level. Even for the rare case whenthe problem is degenerate, the SLP-PA algorithm is still muchmore efficient than the state-of-the-art slack variable (SV)-basedapproach proposed in [6], due to fewer number of LPs involvedat each rate level.

We also extend the PA technique for the LMM rate allocationproblem to address the so-called maximum node lifetime curveproblem in [6], which we call LMM node lifetime problem. Weshow that the SLP-PA approach is much more efficient than theslack variable (SV)-based approach (SLP-SV) described in [6].More importantly, we show that there exists a simple and elegantduality relationship between the LMM rate allocation problemand the LMM node lifetime problem. As a result, it is sufficientto solve only one of these two problems. Important insights canbe obtained by inferring duality results for the other problem.

The remainder of this paper is organized as follows. InSection II, we describe the network and energy model, and for-mulate the LMM rate allocation problem. Section III presentsour SLP-PA algorithm to the LMM rate allocation problem. InSection IV, we introduce the LMM node lifetime problem andapply the SLP-PA algorithm to solve it. Section V shows an in-teresting duality relationship between the LMM rate allocationproblem and the LMM node lifetime problem. In Section VI,we present numerical results. Section VII reviews related workand Section VIII concludes this paper.

II. SYSTEM MODELING AND PROBLEM FORMULATION

We consider a two-tier architecture for wireless sensor net-works. Figs. 1(a) and (b) show the physical and hierarchicalnetwork topology for such a network, respectively. There arethree types of nodes in the network, namely, micro-sensor

Fig. 1. Reference architecture for two-tier wireless sensor networks. (a) Phys-ical topology; (b) a hierarchical view.

nodes (MSNs), aggregation and forwarding nodes (AFNs),and a base station (BS). The MSNs can be application-specificsensor nodes (e.g., temperature sensor nodes (TSNs), pressuresensor nodes (PSNs), and video sensor nodes (VSNs)) and theyconstitute the lower tier of the network. They are deployed ingroups (or clusters) at strategic locations for surveillance andmonitoring applications. The MSNs are small and low-cost.The objective of an MSN is very simple: Once triggered by anevent, it starts to capture sensing date and sends it directly tothe local AFN.1

For each cluster of MSNs, there is one AFN, which is dif-ferent from an MSN in terms of physical properties and func-tions. The primary functions of an AFN are: 1) data aggrega-tion (or “fusion”) for data flows from the local cluster of MSNs,and 2) forwarding (or relaying) the aggregated information tothe next hop AFN (toward the base station). For data fusion, anAFN analyzes the content of each data stream it receives andexploits the correlation among the data streams. An AFN alsoserves as a relay node for other AFNs to carry traffic toward thebase station. Although an AFN is expected to be provisionedwith much more energy than an MSN, it also consumes energyat a substantially higher rate (due to wireless communicationover large distances). Consequently, an AFN has a limited life-time. Upon depletion of energy at an AFN, we expect that the

1Due to the small distance between an MSN and its AFN, multi-hop routingamong the MSNs may not be necessary.

HOU et al.: RATE ALLOCATION AND NETWORK LIFETIME PROBLEMS FOR WIRELESS SENSOR NETWORKS 323

coverage for the particular area under surveillance is lost, de-spite the fact that some of the MSNs within the cluster may stillhave remaining energy.2

The third component in the two-tier architecture is the basestation. The base station is, essentially, the sink node for datastreams from all the AFNs in the network. In this investigation,we assume that there is sufficient energy resource available atthe base station and thus there is no energy constraint at the basestation. In summary, the main functions of the lower tier MSNsare data acquisition and compression while the upper-tier AFNsare used for data fusion and relaying information to the basestation.

A. Power Consumption Model

Our focus in this paper is on the communication energy con-sumption among the upper tier AFNs. For each AFN , we as-sume that the aggregated bit rate collected locally (after datafusion) is , . These collected local bit streamsmust be routed toward the base station. Our objective is to max-imize the values according to the LMM criterion (see Defini-tion 1) under a given network lifetime requirement.

For an AFN, energy consumption due to wireless communi-cation (i.e., transmitting and receiving) has been considered thedominant factor in power consumption [1]. The power dissipa-tion at a radio transmitter can be modeled as [9]

(1)

where is the power dissipated at AFN when it is transmit-ting to node , is the rate from AFN to node , is thepower consumption cost of radio link and is given by

(2)

where and are two constant terms, is the distance be-tween these two nodes, and is the path loss index, with

[16]. Typical values for these parameters areand (for ) [9].3 Since

the power level of an AFN’s transmitter can be used to controlthe distance coverage of an AFN (see, e.g., [15], [17], [20]),different network flow routing topologies can be formed by ad-justing the power level of each AFN’s transmitter.

The power dissipation at a receiver can be modeled as [9]

(3)

where (in b/s) is the rate of the received data streamat AFN . A typical value for the parameter is 50 nJ/b [9].

The above transmission and reception energy model assumesa contention-free MAC protocol, where interference fromsimultaneous transmission can be effectively minimized oravoided. For such a network, a contention-free MAC protocolis fairly easy to design (see, e.g., [18]) and its discussion isbeyond the scope of this paper.

2We assume that each MSN can only forward information to its local AFNfor processing (e.g., video fusion).

3In this paper, we use� � � in all of our numerical results.

B. The LMM Rate Allocation Problem

Before we formulate the LMM rate allocation problem, letus revisit the maximum capacity problem (with “bias” in rateallocation) that was described in Section I. For a network with

AFNs, suppose that the rate of AFN is , and that the initialenergy at this node is . For a given networklifetime requirement (i.e., each AFN must remain alive for atleast time duration ), the maximum information capacity thatthe network can collect can be found by the following linearprogram (LP):

(4)

(5)

where and are data rates transmitted from AFN to AFNand from AFN to the base station , respectively. The set

of constraints in (4) are the flow balance equations: they statethat, the total bit rate transmitted by AFN is equal to the totalbit rate received by AFN from other AFNs, plus the bit rategenerated locally at AFN . The set of constraints in (5)are the energy constraints: they state that, for a given networklifetime requirement , the energy required in communications(i.e., transmitting and receiving all these data) cannot exceed theinitial energy provisioning level.

Note that , , , and are variables and that is aconstant (the given network lifetime requirement). MaxCap isa standard LP formulation that can be solved by a polynomial-time algorithm [2]. Unfortunately, as we shall see in the numer-ical results (Section VI), the solution to this MaxCap problemlends itself into an extreme favor for those AFNs whose datapaths consume the least amount of power toward the base sta-tion. Consequently, although the network capacity is maximizedover the network lifetime , the corresponding bit rate alloca-tion among the AFNs (i.e., the values) only favors those AFNsthat have this property, while other AFNs are unfavorably allo-cated with much smaller (even close to 0) bit rates. As a result ofthis unfairness, the effectiveness of the network in performinginformation collection or surveillance could be severely com-promised.

To address this fairness issue, we advocate the so-called lexi-cographic max-min (LMM) rate allocation strategy [14] in thispaper, which has some similarity to the max-min rate allocationin data networks [3].4 Under LMM rate allocation, we start withthe objective of maximizing the bit rate for all the nodes untilone or more nodes reach their energy-constrained capacities forthe given network lifetime requirement. Given that the first levelof the smallest rate allocated among the nodes is maximized, wecontinue to maximize the second level of rate for the remainingnodes that still have available energy, and so forth. More for-mally, denote as the sorted version (i.e.,

4However, there is significant difference between max-min and LMM, whichwe will discuss shortly.


) of the rate vector ,with corresponding to the rate of node . We then have thefollowing definition for LMM-optimal rate allocation.

Definition 1: (LMM-Optimal Rate Allocation): For agiven network lifetime requirement , a sorted rate vector

yields an LMM-optimal rate alloca-tion if and only if for any other sorted rate allocation vector

with , there exists a ,, such that for and .

Based on the LMM-optimal definition, we can calculate thefirst level optimal rate easily through the following LP.

Although the first level bottleneck rate is easy to ob-tain, calculating the subsequent bottleneck rates are quitechallenging. As discussed in Section I, a naive approach thatapplies an iterative LP procedure to calculate the desired rateallocations is incorrect. This is because there is a fundamentaldifference in the nature of the LMM rate allocation problem de-scribed here and the classical max-min rate allocation problemin [3]. The LMM rate allocation problem implicitly couplesa flow routing problem (i.e., a determination of the and

for the entire network), while the classical max-min rateallocation explicitly assumes that the routes for all the flows aregiven a priori and fixed. Moreover, for the LMM rate allocationproblem, starting from the first iteration, there usually existnon-unique flow routing solutions corresponding to the samemaximum rate level. Consequently, each of these flow routingsolutions, once chosen, will yield different remaining energylevels on the nodes for future iterations and so forth, leadingto a different rate vector, which usually does not coincide withthe LMM-optimal rate vector. Therefore, any iterative rateallocation algorithm that requires energy reservation among thenodes during each iteration is unlikely to give a correct LMMrate allocation (see Section VI for numerical examples).

III. A SERIAL LP ALGORITHM BASED

ON PARAMETRIC ANALYSIS

In this section, we present an efficient (polynomial-time) al-gorithm to solve the LMM rate allocation problem correctlywithout requiring any energy reservation during each iteration.Table I lists the notation used in this paper.

We first introduce the following notation. Suppose that therate vector is LMM-optimal, with

. Note that the values of these rates maynot be all distinct. To highlight those distinct rate levels, weremove any repetitive elements in this vector and rewrite it as

such that , where ,, and . Now for each , , denote

the corresponding set of nodes that use up their energy at thisrate. Clearly, we have , where denotesthe set of all nodes.

TABLE INOTATION

The key to the LMM rate allocation problem is to findthe correct values and the corresponding sets

, respectively. This can be done iteratively. Thatis, we first determine rate level and the corresponding set ,then determine rate level and the corresponding set , andso on. In Section III-A, we will show how to determine eachrate level and in Section III-B, we will show how to determinethe corresponding node set.

A. Rate Level Determination

Denote and . For , suppose thatwe already determined and the corresponding


sets . The rate level can be found by the fol-lowing optimization problem.

(6)

(7)

(8)

(9)

Note that for , the constraints (7) and (9) do not exist. For, constraints (7) and (9) are for those nodes that have

already reached their LMM rate allocation during the previousiterations. In particular, the set of constraints in (7) say

that the sum of incoming and local data rates are equal to theoutgoing data rates for each node with its LMM-optimal rate

, . The set of constraints in (9) say that for thosenodes that have already reached their LMM-optimal rates, thetotal energy consumed for communications has reached theirinitial energy provisioning. On the other hand, the constraints in(6) and (8) are for the remaining nodes that have not yet reachedtheir LMM-optimal rates. Specifically, the set of constraints in(6) state that, for those nodes that have not yet reached theirenergy constraint levels, the sum of incoming and local datarates are equal to the outgoing data rates. Note that the objectivefunction is to maximize the additional rate for those nodes.Furthermore, for those nodes, the set of constrains in (8) statethat the total energy consumed for communications should beupper bounded by the initial energy provisioning.

To facilitate our later discussion on duality results inSection V, we further re-formulate above LP. In particular, wemultiply both sides of (6) and (7) by (which is a constantrepresenting a given network lifetime requirement) and denote

, , . Intuitively, andrepresent the bit volume that is transferred from node to

and from node to , respectively, during lifetime . Weobtain the following problem formulation.

(10)

The above LP formulation can be rewritten in the form, s.t. and , the dual problem for which

is , s.t. with being unrestricted in sign [2].Both can be solved by standard LP techniques (e.g., [2]).

Although a solution to the LMM-Rate problem gives the op-timal solution for at iteration , it remains to determine theminimum set of nodes corresponding to this , which is the keydifficulty in the LMM rate allocation problem. In the followingsection, we exploit the parametric analysis technique [2] to de-termine the minimum node set at each rate.

B. Minimum Node Set Determination

Now we show how to determine set for rate level . De-note the set of nodes for which the constraints (8) arebinding at the -th iteration in LMM-Rate, i.e., include all thenodes that achieve equality in (8) at iteration . Although it iscertain that at least one of the nodes in belong to (the min-imum node set for rate ), some nodes in may still be ableto further increase their rates under alternative flow routing so-lutions. In other words, if , then we must have ;otherwise, we must determine the minimum node setthat achieves the LMM-optimal rate allocation.

We find that the so-called parametric analysis (PA) technique[2] is most suitable to address this problem. The main idea ofPA is to investigate how an infinitesimal perturbation on somecomponents of the LMM-Rate problem can affect the objectivefunction. In particular, considering a small increase on the right-hand-side (RHS) of (10), i.e., changing to , where

, node belongs to the minimum node set if and only if. That is, node belongs to the minimum

node set if and only if a small increase in node ’s rate (interms of total volume generated at node ) leads to a decrease inthe objective function.

To compare with 0, we apply an importantduality results from LP theory. If and are the respectiveoptimal solution to the primal and dual problems, then based onthe parametric duality property [2], we have

(11)

Recall that these can be easily obtained at the same timewhen we solve the primal LP problem. Note that by the natureof the problem, we have for an optimal dual solution.Therefore, if we find that , then we can determine im-mediately that node must belong to the minimum node set .On the other hand, if we find that , it is not clear whether

is strictly negative or 0 and further analysis isthus needed.

For each node with , we must perform a complete PAto see whether a perturbation (i.e., tiny increase) on the RHS of(10) will result in any change in the objective function. If thereis no change, then we can determine that node does not belongto the minimum node set ; otherwise, node belongs to .Assume that the optimal solution is , where anddenote the set of basic and non-basic variables; and denotethe columns corresponding to the basic and non-basic variables.

and denote the objective function coefficient vectors forthe basic and non-basic variables; and denotes the objective


value. Then we have the corresponding canonical equations asfollows:

If is replaced by , where the column vector hasa single 1 element corresponding to node in the set of con-straints (10) while all the other elements are 0, then the onlychange due to this perturbation is that will be replaced by

. Consequently, the objective value for the currentbasis becomes . As long as is non-negative, the current basis remains optimal. Denote ,

, and let be an upper bound for such that thecurrent basis remains optimal. We have

(12)

If , the optimal objective value varies according tofor . Since and

, we have . Thus, the objective valuewill not change for , and consequently, the rate fornode can be increased beyond the current value. That is,node does not belong to the minimum node set .

For most problems in practice, the above procedure is suffi-cient to determine whether or not node belongs to the minimumnode set for all . But in the rare event where , theproblem is degenerate. To develop a polynomial-time algorithm,denote as the set of all nodes with and as the setof all nodes with and . Then we solve the fol-lowing LP to maximize the slack variables (SV) for nodes in .

If the optimal objective function is 0, then we conclude that nonode in can have a positive . That is, these nodes shouldall belong to and we have . On the other hand,if the optimal objective function is positive, then some nodes

must have positive values and these nodes therefore donot belong to the minimum node set . Consequently, we canremove these nodes from . If , we move on to solveanother MSV. This procedure will terminate when the optimalobjective function value is 0 or .

The following lemma ensures that MSV determines the min-imum node set correctly. Its proof is given in the Appendix.

Lemma 1: (The Minimum Node Set is Unique.): The min-imum node set for each rate level under the LMM-optimal rateallocation is unique.

In a nutshell, the complete PA procedure to determinewhether a node belongs to the minimum node set canbe summarized as follows.

Algorithm 1: (Minimum Node Set Determination with PA)

1) Initialize sets and .2) For each node ,

a) If , then .b) Otherwise (i.e., ), compute ,

, and according to (12).If , then .

3) If , then and stop;else set up the MSV problem and solve it.

4) If the optimal objective value in MSV is 0, thenand stop;

else remove all nodes with from the set andgo to Step 3.

C. Optimal Flow Routing for LMM Rate Allocation

After we solve the LMM rate allocation problem iterativelyusing the procedure in Sections III-A and III-B, the corre-sponding optimal flow routing can be obtained by dividing thetotal bit volume on each link ( or ) by , i.e.,

(13)

where is the given network lifetime requirement. Althoughthe LMM-optimal rate allocation is unique, it is important tonote that the corresponding flow routing solution is not unique.This is because upon the completion of the LMM rate alloca-tion problem (i.e., upon finding ), there usuallyexist non-unique bit volume solutions ( and values) cor-responding to the same LMM-optimal rate allocation. This re-sult is summarized in the following lemma.

Lemma 2: The optimal flow routing solution correspondingto the LMM rate allocation may not be unique.

We use the following example to illustrate the non-unique-ness of the optimal flow routing solution for an LMM rateallocation.

Example 1: Consider an 8-node network with the followingtopology (see Fig. 2). The base station is located at the origin(0,0). There are two groups of nodes, and , in the network,with each group consisting of four nodes. Group nodes con-sists of at (100, 0), at (0, 100), at ( 100,0), and at (0, 100), respectively (all in meters); Group

nodes consists of at (100, 100), at ( 100,100), at ( 100, 100), and at (100, 100), re-spectively. Assume that all nodes have the same initial energy. For a network lifetime requirement of , we can calculate

(via SLP-PA) that the final LMM-optimal rate allocation for all8 nodes are identical (perfect fairness), i.e., .We denote for .


Fig. 2. A simple example showing that the optimal flow routing to the LMMrate allocation is not unique. The range of � is � � � � ��.

Upon the completion of the SLP-PA algorithm, we also obtainan optimal flow routing solution corresponding to this LMM-op-timal rate . This optimal flow routing solution has the followingflows: ,

, and. We now show that the optimal flow routing solu-

tion is non-unique. Since the network has symmetrical property,it can be easily verified that for any , ,the LMM-optimal rate allocation can be achieved if the flowrouting solution satisfies the following two conditions: (i) eachnode in (i.e., AFNs 2, 4, 6, and 8) sends a flow of and aflow of to its two neighboring nodes as shownin Fig. 2, and a remaining flow of directly to the basestation; and (ii) each node in (i.e., AFNs 1, 3, 5, and 7) sendsa total amount of to the base station, which includes

and from its neighboring nodes, plus fromitself. Clearly, there are infinitely many flow routing solutionsthat meet these two conditions, each of which can be shown toyield the LMM-optimal rate allocation with the given networklifetime requirement .

D. Complexity Analysis

We now analyze the complexity of the SLP-PA algorithm.First we consider the complexity of finding each node’s rateand the total bit volume transmitted along each link. At eachstage, we solve an LP problem, both its primal and dual have acomplexity of [2], where is the number of con-straints or variables in the problem, whichever is larger, andis the number of binary bits required to store the data. Since thenumber of variables is and is larger than the number ofconstraints (which is ), the complexity of solving the LPis . After solving an LP at each stage, we need to deter-mine whether or not a node that just reached its energy bindingconstraint belongs to the minimum node set for this stage. Notethat and can be readily obtained when we solvethe primal LP problem. To determine whether a node, say , be-longs to the minimum node set, we examine . If , thennode belongs to the minimum node set and the complexity is

. On the other hand, if , we need to further examine

whether or not. Based on (12), the computation for is. So at each stage, the complexity in PA for each node is. The total complexity of PA at each stage for the node set

is thus or . Thus, the complexityat each stage is . As there are atmost stages, the overall complexity is .

We now analyze the complexity for the degenerate case. Uponthe completion of Step 2 in Algorithm 1, we denote .Since we need to solve at most LPs, the complexityis or . Hence, thecomplexity at each stage is

. Since there are at most stages, the overall com-plexity is .

The complexity in finding the optimal flow routing is boundedby the number of radio links in the network, which is .Hence, the overall complexity isfor the non-degenerate case andfor the degenerate case. Under either case, the computationalcomplexity is polynomial.5

E. Discussion

So far, we consider the case that each AFN generates data at aconstant rate. In practice, an AFN node may not always transmitdata and may work in on/off mode to conserve energy. In thiscase, it is necessary to construct optimal flow routing solutionfor variable bit rate source (where on/off mode is a special case).In [11], we have developed techniques to construct optimal flowrouting solution for variable bit rate, as long as its average rate isknown. Such average rate corresponds to the constant rate in thispaper. As a result, the case of on/off mode (with known averagerate) can also be handled using techniques in [11].

IV. EXTENSION TO LMM NODE LIFETIME PROBLEM

In this section, we show that our SLP-PA algorithm canbe used to solve the so-called maximum node lifetime curveproblem in [6], which we define as the LMM node lifetimeproblem. We also show that the SLP-PA algorithm is a muchmore efficient approach than the one proposed in [6], which iscurrently the state-of-the-art to address this problem.

A. The LMM-Optimal Node Lifetime Problem

The LMM node lifetime problem considers the following sce-nario. For a network with AFNs, with a given local bit rate

(fixed) and initial energy for AFN , , howcan we maximize the network lifetime for all AFNs in the net-work? In other words, the LMM node lifetime problem not onlyconsiders how to maximize the network lifetime until the firstAFN runs out of energy, but also the time for all the AFNs inthe network.

More formally, denote the lifetime for each AFN as ,. Note that ’s are fixed here, while ’s are

the optimization variables, which are different from the LMMrate allocation problem that we studied in the last section. De-note as the sorted sequence of the values in

5Note that our analysis here gives a very loose upper bound for time com-plexity. In practice, the running time for LP implementation is much faster.


nondecreasing order. Then LMM-optimal node lifetime can bedefined as follows.

Definition 2: (LMM-Optimal Node Lifetime): A sorted nodelifetime vector with isLMM-optimal if and only if for any other sorted node lifetimevector with , there exists a

, , such that for and .

B. Solution

It should be clear that, under the LMM-optimal node lifetimeobjective, we must maximize the time until a set of nodes use uptheir energy (which is also called a drop point in [6]) while min-imizing the number of nodes that drain up their energy at eachdrop point. We now show that the SLP-PA algorithm developedfor the LMM rate allocation problem can be directly applied tosolve the LMM node lifetime problem.

Suppose that with isLMM-optimal. To keep track of distinct node lifetimes (or droppoints) in this vector, we remove all repetitive elements in thevector and rewrite it as such that

, where , , and . Corre-sponding to these drop points, denote as the setsof nodes that drain up their energy at drop points ,respectively. Then , where

denotes the set of all AFNs in the network. The problem isto find the LMM-optimal values of and the cor-responding sets .

Similar to the LMM rate allocation problem, the LMM nodelifetime problem can be formulated as an iterative optimizationproblem as follows. Denote , , and .Starting from , we solve the following LP iteratively.

(14)

Comparing the above LMM-Lifetime problem to the LMM-Rate problem that we studied in Section III-A, we find that theyare exactly of the same form. The only differences are that underthe LMM-Lifetime problem, the local bit rates are constantsand the node lifetimes are variables (subject to optimization),while under the LMM-Rate problem, the are variables (sub-ject to optimization) and the node lifetimes are all identical ,

. Since the mathematical formulation for the twoproblems are identical, we can apply the SLP-PA algorithm tosolve the LMM node lifetime problem as well.

The only issue that we need to be concerned about is the op-timal flow routing solution corresponding to the LMM-optimallifetime vector. The optimal flow routing solution here is notas simple as that for the LMM rate allocation problem, whichmerely involves a simple division (see (13)). We refer readers tothe Appendix for an algorithm to obtain an optimal flowrouting solution for the LMM-optimal lifetime vector. Similarto Lemma 2, the optimal flow routing solution corresponding tothe LMM node lifetime problem may not be unique.

C. Complexity Comparison

In [6], Brown et al. studied the LMM node lifetime problemunder the so-called “maximum node lifetime curve” problem.They also developed the first procedure to solve this problemcorrectly. A key step in their procedure is the use of multipleindependent LP calculations to determine the minimum node setat each drop point, which we call serial LP with slack variableanalysis (SLP-SV). Although this approach solves the LMMnode lifetime problem correctly, its computational complexity(potentially exponential) remains an issue to be resolved.

On the other hand, the SLP-PA algorithm developed in thispaper is polynomial and is computationally more efficient thanthe SLP-SV approach. To understand the difference between thetwo, we take a closer look on the computational complexity ofthe SLP-SV approach in [6]. First, SLP-SV needs to keep trackof each sub-flow along its route from the source node towardthe base station. Such a flow-based (or more precisely, sub-flowbased) approach could make the size of the LP coefficient matrixexponential, which leads to an exponential-time algorithm [2].6

Second, even if a link-based LP formulation such as ours isadopted in [6], the computational efficiency of the SV-based ap-proach is still worse than the SLP-PA algorithm. This is becauseat each stage, the SV-based approach must solve several addi-tional LPs (up to ) to determine , which is in contrastto the simpler PA under the SLP-PA algorithm . Evenfor the degenerate case, the number of additional LPs under theSLP-PA algorithm is at most ,7 which is still no morethan .

Finally, we discuss a hybrid link-flow approach mentionedin [6]. In this approach, link-based formulations are used forsub-flows. This leads to a much fewer number of variables thanthose for the flow-based approach. But this approach still re-quires sub-flow accounting and results in an order of magnitudemore constraints than the link-based approach in SLP-PA. Al-though this approach solves the LMM node lifetime problemin polynomial-time (e.g., by using interior point methods [2]),the overall complexity is still orders of magnitude higher thanthat under the SLP-PA algorithm. Furthermore, the burden ofsolving additional LPs to determine whether a node belongs tothe minimum node set still remains.

V. DUALITY THEOREM

In this section, we present an elegant and powerful resultshowing that there is a duality relationship between the LMMrate allocation problem and the LMM node lifetime problem. As

6Incidentally, the revised simplex method proposed in [6] is not as efficientas the polynomial-time algorithm described in [2] and is itself exponential.

7Recall that � denotes � upon the completion of Step 2 in Algorithm 1.


TABLE IIDUALITY RELATIONSHIP BETWEEN LMM RATE ALLOCATION PROBLEM �

AND LMM NODE LIFETIME PROBLEM �

a result, it is only necessary to solve only one of the two prob-lems and the results for the other can be obtained via simplealgebraic calculations.

To start with, we denote the LMM rate allocation problemwhere we have AFNs in the network and all nodes have acommon given lifetime requirement (constant). Denotethe LMM-optimal rate allocation for node under ,

. Similarly, we denote the LMM node lifetimeproblem where all nodes have the same local bit rate (constant).Denote the LMM node lifetime for node under ,

. Then the following theorem shows how the solutionto one problem can be used to obtain the solution to the other.

Theorem 1: (Duality): For a given node lifetime requirementfor all nodes under problem and a given local bit rate for

all nodes under problem , we have the following relationshipbetween the solutions to the LMM rate allocation problemand the LMM node lifetime problem .

(i) Suppose that we have solved problem and obtainedthe LMM-optimal rate allocation for each node

. Then under , the LMM node lifetimefor node is

(15)

(ii) Suppose that we have solved problem and obtainedthe LMM-optimal node lifetime for each node

. Then under , the LMM rate allocationfor node is

(16)

Table II shows the duality relationship between solutions toproblems and .

Proof: We prove (i) and (ii) in Theorem 1 separately.(i) We organize our proof into two parts. First, we show that

’s are feasible node lifetimes in terms of flow balance andenergy constraints on each node . Then weshow that it is indeed the LMM-optimal node lifetime.

Feasibility: Since we have obtained the solution to problem, we have one feasible flow routing solution for sending bit

streams , , to the base station. Under problem, the bit volumes ( and values) must meet the fol-

lowing equalities under the LMM-optimal rate allocation:

Now replacing by , we see that the same bit volumesolution under yields a feasible bit volume solution to thenode lifetime problem under . Consequently, we can use Al-gorithm 2 to obtain the flow routing solution to problem

under the bit volume solution to problem and this verifiesthat , , is a feasible solution to problem .

Optimality: To prove that ’s obtained via (15) are indeedLMM-optimal for problem , we sort , ,under problem in non-decreasing order and denoteit as . We also introduce a node index

for . For example,means that actually corresponds to the rate of AFN 7, i.e.,

.Since is proportional to through the relationship

, listing , , according towill yield a sorted (in non-decreasing

order) lifetime list, denoted as . We now provethat is indeed LMM-optimal for problem .

Our proof is based on contradiction. Suppose thatis not LMM-optimal for problem . As-

sume that the LMM-optimal lifetime vector to problemis (sorted in non-decreasing order) with thecorresponding node index being . Then, byDefinition 2, there exists a such that forand .

We now claim that if , , is a feasible so-lution to problem , then obtained via ,

, is also a feasible solution to problem . The proofto this claim follows identically as above. Using this result,we can obtain a corresponding feasible solutionwith and the node index for problem . Hence,we have for but

. That is, is notLMM-optimal and this leads to a contradiction.

(ii) The proof for this part follows the same token as the aboveproof for (i) and is thus omitted here.

This duality relationship offers important insights on systemperformance issues, in addition to providing solutions to theLMM rate allocation and the LMM node lifetime problems. Forexample, in Section I, we pointed out the potential bias (fairness)issue associated with the network capacity maximization objec-tive (i.e., sum of rates from all nodes). It is interesting to see thatthere is a dual fairness issue under the node lifetime problem. Inparticular, the objective of maximizing the sum of node lifetimesamong all nodes also leads to a bias (or fairness) problem be-cause this objective would only favor those nodes that consumeenergy at a small rate. As a result, certain nodes will have muchlarger lifetimes while some other nodes will be penalized withmuch smaller lifetimes, although the sum of node lifetimes ismaximized.

VI. NUMERICAL INVESTIGATION

In this section, we use numerical results to illustrate ourSLP-PA algorithm to the LMM rate allocation problem andcompare it with other approaches. We also use numericalresults to illustrate the duality between the LMM rate allocationproblem and the LMM node lifetime problem.

We consider two network topologies, one with 10 AFNs andthe other with 20 AFNs. Under both topologies, the base station

is located at the origin while the locations for the 10 or 20AFNs are randomly generated over a 1000 m 1000 m squarearea (see Figs. 3(a) and (b) and Tables III and IV, respectively).


Fig. 3. Network topologies used in the numerical investigation. (a) A 10-AFNnetwork; (b) a 20-AFN network.

TABLE IIINODE COORDINATES FOR A 10-AFN NETWORK

TABLE IVNODE COORDINATES FOR A 20-AFN NETWORK

TABLE VRATE ALLOCATION UNDER THE THREE APPROACHES FOR THE

10-AFN NETWORK

A. SLP-PA Algorithm to the LMM Rate Allocation Problem

We will compare SLP-PA with the naive approach (seeSection II-B) that uses a serial LP “blindly” to solve the LMMrate allocation problem and performs energy reservation duringeach iteration. We call this naive approach Serial LP withEnergy Reservation (SLP-ER). As discussed in Section II-B,the SLP-ER approach will not give the correct final solution tothe LMM rate allocation problem.

We will also compare our SLP-PA algorithm to the Max-imum-Capacity (MaxCap) approach (see Section II-B). As dis-cussed in the beginning of Section II-B, the rate allocation underthe MaxCap approach can be extremely biased and in favor ofonly those AFNs that consume the least power along their datapaths toward the base station.

10-AFN Network: We assume that the initial energy at eachAFN is 50 KJ and that under the LMM rate allocation problem,the network lifetime requirement is 100 days. The power con-sumption is for transmission and reception defined in (1) and(3), respectively.

Table V shows the rate allocation for the AFNs undereach approach, which is also plotted in Fig. 4. The “sortednode index” corresponds to the sorted rates among the AFNsin non-decreasing order. Clearly, among the three rate al-location approaches, only the rate allocation under SLP-PAmeets the LMM-optimal rate allocation definition (see Def-inition 1). Specifically, comparing SLP-PA with SLP-ER,we have , ,

, and ; comparingSLP-PA with MaxCap, we have .

We also observe, as expected, a severe bias in therate allocation under the MaxCap approach. In partic-ular, alone accounts for over 48% of the sum oftotal rates among all the AFNs. Comparing the three ap-proaches, we have and

. In other words, the rateallocation vector under the SLP-PA algorithm has the smallestrate difference between the smallest rate and the largest rate

, i.e., , among the three approaches. In addition,although for the first level rate allocation,the minimum node set for is smaller than the minimumnode set for , i.e., .This confirms that the naive SLP-ER approach cannot offer thecorrect solution to the LMM rate allocation problem.


Fig. 4. Rate allocation under the SLP-PA, SLP-ER, and MaxCap approachesfor a 10-AFN network and a 20-AFN network. (a) A 10-AFN network; (b) a20-AFN network.

20-AFN Network: For the 20-AFN network (Table IV, weassume that the initial energy at each AFN is 50 KJ and thatthe network lifetime requirement under the LMM rate alloca-tion problem is 100 days. Table VI shows the sorted rate allo-cation under the three approaches, which are also displayed inFig. 4(b). It can be easily verified that all the observations forthe 10-AFN network also hold here.

B. Duality Results

We now use numerical results to verify the duality relation-ship between the LMM rate allocation problem and theLMM node lifetime problem (see Section V). Again,we use the 10-AFN and 20-AFN network configurations inFigs. 3(a) and (b), respectively. The coordinates for each AFNunder the 10-AFN network and 20-AFN network are listed inTables III and IV, respectively. For both networks, we assumethat the initial energy at each AFN is 50 KJ and that the networklifetime requirement under the LMM rate allocation problem is

. Under , we assume the local bit rate for allAFNs is Kb/s.

To verify the duality relationship (Theorem 1), we performthe following calculations. First, we solve the LMM rate al-

TABLE VIRATE ALLOCATION UNDER THE THREE APPROACHES FOR THE

20-AFN NETWORK

TABLE VIINUMERICAL RESULTS VERIFYING THE DUALITY RELATIONSHIP � � � � � � �

BETWEEN THE LMM RATE ALLOCATION PROBLEM �� AND THE LMMNODE LIFETIME PROBLEM �� FOR THE 10-AFN NETWORK

location problem and the LMM node lifetime problemindependently with the above initial conditions using the

SLP-PA algorithm. Consequently, we obtain the LMM-optimalrate allocation ( for each AFN ) under and the LMM-op-timal node lifetime ( for each AFN ) under . Then we com-pute and separately for each AFN and examine ifthey are equal to each other.

The results for the LMM-optimal rate allocation ( ,) and the LMM-optimal node lifetime ( ,) for the 10-AFN network are shown in Table VII.

We find that and are exactly equal for all AFNs,precisely as we would expect under Theorem 1. Similarly, theresults for the 20-AFN network are shown in Table VIII.

VII. RELATED WORK

Due to energy constraints in wireless sensor networks, therehas been active research on exploring the performance limitsof such networks. These performance limits include, amongothers, network capacity and network lifetime. Network ca-pacity typically refers to the maximum amount of bit volumethat can be successfully delivered to the base station (“sinknode”) by all the nodes in the network, where network lifetime


TABLE VIIINUMERICAL RESULTS VERIFYING THE DUALITY RELATIONSHIP � � � � � � �

BETWEEN THE LMM RATE ALLOCATION PROBLEM �� AND THE LMMNODE LIFETIME PROBLEM �� FOR THE 20-AFN NETWORK

refers to the maximum time that the nodes in the networkremain alive before one or more nodes deplete their energy.

The network capacity problem and network lifetime problemhave so far been studied disjointly in the literature. For example,in [13], the problem of how to maximize network capacity viarouting was studied. While, in many other efforts (see, e.g., [4],[5], [8], [12], [21],), the focus was on how to maximize the timeuntil the first node drains up its energy.

In this paper, we study the important overarching problemthat considers both network capacity and network lifetime.Under the LMM rate allocation problem, we studied how tomaximize rate allocations for all the nodes in the network undera given network lifetime requirement. Under the LMM nodelifetime problem, we studied how to maximize the lifetimefor all nodes when the local bit rate for each node is given apriori. The LMM rate allocation criterion effectively mitigatesthe unfairness issue when the objective is to maximize the totalbit volume generated by the network. Although the LMM rateallocation is somewhat similar to the classical max-min strategy[3], there is a fundamental difference between the two. In par-ticular, the LMM rate allocation problem implicitly embeds (orcouples) a flow routing problem within rate allocation, whileunder the classical max-min rate allocation, there is no routingproblem involved since the routes for all flows are given. Dueto this coupling of flow routing and rate allocation, a solutionapproach (i.e., SLP-PA) to the LMM rate allocation problem ismuch more challenging than that for the classical max-min.

In [19], Srinivasan et al. applied game theory and Nash equi-librium among the nodes to forward packets such that the totalthroughput (capacity) can achieve an optimal operating pointsubject to a common lifetime requirement on all nodes. How-ever, the fairness issue in information collection was not consid-ered. The most relevant work to the LMM node lifetime problemwas by Brown et al. [6], which has been discussed in detail inSection IV-A.

VIII. CONCLUSION

In this paper, we investigated the important problem of rateallocation for wireless sensor networks under a given networklifetime requirement. Since the objective of maximizing the sumof rates of all nodes can lead to a severe bias in rate alloca-tion among the nodes, we advocate the use of lexicographicalmax-min (LMM) rate allocation for all nodes in the network. Tocalculate the LMM-optimal rate vector, we developed a polyno-mial-time algorithm by exploiting the parametric analysis (PA)technique from linear programming (LP), which we called se-rial LP with Parametric Analysis (SLP-PA). Furthermore, weshowed that the SLP-PA algorithm can also be employed toaddress the maximum node lifetime curve problem and thatthe SLP-PA algorithm is much more efficient than an state-of-the-art algorithm. More important, we discovered a simple andelegant duality relationship between the LMM rate allocationproblem and the LMM node lifetime problem, which enables usto develop solutions and insights on both problems by solvingone of the two problems. Our results in this paper offer someimportant understanding on network capacity and network life-time problems for energy-constrained wireless sensor networks.

APPENDIX APROOF OF LEMMA 1

By the definition of LMM-optimal rate vector (see Defini-tion 1), the optimal rates ( values) are unique and the cor-responding numbers of nodes in each minimum node sets (values) are also unique. To show that the group of physical nodesin each is also unique, we employ the parametric simplex ap-proach to determine the minimum node set as follows.

In essence, the parametric simplex approach solely relies onthe PA technique without resorting to the MSV approach evenwhen the problem is degenerate. That is, when the problem isdegenerate, i.e., for some node , we have and

, then the basis can change while the optimal objectivevalue remains unchanged. We can analyze and under thenew basis to determine whether or not node belongs to theminimum node set . If we still have and , thebasis can change again with the same optimal objective value.To prevent cycling back to a previous basis, we can use an anti-cycling rule [2]. Thus, this procedure is guaranteed to terminatewithin a finite number of steps and we can determine whetheror not node indeed belongs to the minimum node set .

Note that in the above parametric simplex approach, the setof nodes corresponding to is uniquely determined since theanalysis is conducted independently for each node.8 Therefore,upon the completion of all stages, the group of AFNs in eachminimum node set is unique.

APPENDIX BOPTIMAL FLOW ROUTING SOLUTION FOR LMM-OPTIMAL

NODE LIFETIME

It is straightforward to develop an example similar to the onegiven in Section III-C that shows the non-uniqueness of the flowrouting schedule.9 Given that the optimal flow routing solution

8This can be done in parallel if so desired.9Incidentally, this result corrects an error in [16] (Lemma 3.2), which incor-

rectly stated that such a flow routing solution is unique.


is non-unique, there are potentially many flow routing solutionsthat can achieve the LMM-optimal lifetime vector. In this sec-tion, we present a simple polynomial-time algorithm that pro-vides an LMM-optimal flow routing solution.

The main task in this algorithm is to define flows from the bitvolumes ( and values), which are obtained upon the com-pletion of the last iteration in the LMM-Rate problem with ourSLP-PA algorithm. Note that the bit volumes obtained here rep-resent the total amount of bit volume being transported betweenthe nodes during , where is the time that thelast group of nodes drain up their energy. The main result here isthat if we let the total amount of outgoing flow at a node be dis-tributed proportionally to the bit volumes on each outgoing linkforall the remainingalivenodesateachstage, thenwecanachievethe drop points as well as the corresponding min-imum node sets . The algorithm is formally de-scribed as follows and its correctness proof follows that in [10].

Algorithm 2: (An Optimal Flow Routing Solution)

Upon the completion of the SLP-PA algorithm for theLMM-optimal lifetime vector, we have the drop points (instrictly increasing order) , the correspondingminimum node sets , and the total amount of bitvolume on each radio link (i.e., and ). The followingalgorithm gives an LMM-optimal flow routing solution for thetime interval , where and .

1) Denote , with . Initialize allflows to zero, i.e., , for , ,

.2) If , then stop, else choose a node from such

that:10

• node does not receive data from any other node, or• all nodes from which node receives data are not in .

3) The flow routing at node during is thendefined as

where the values, if not zero, have all been definedbefore calculating the flow routing for node .

4) Let and go to Step 2.

As shown in this algorithm, for each time interval ,, we initialize as the set of remaining alive

nodes at this stage, which is represented by .For these nodes, we compute flow routing by starting with the“boundary” nodes and then move to the “interior” nodes. Moreprecisely, we will calculate the flow routing for a node if andonly if we have calculated the flow routing for each node that

10It can be shown that an LMM-optimal solution is cycle free in terms offlow routing. Consequently, the node � under consideration must exist when� �� .

has traffic going into node . The outgoing flow from node iscalculated by distributing the aggregated flow proportionally ac-cording to the overall bit volume along its outgoing radio links.As an example, suppose that during , node 2 receives anaggregated flow with rate 2 Kb/s and generates 0.4 Kb/s locally.Assume that Kb, Kb, and Kbover . Then the outgoing flow at node 2 is routed as fol-lows: Kb/s, Kb/s, and Kb/s.

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[2] M. S. Bazaraa, J. J. Jarvis, and H. D. Sherali, Linear Programming andNetwork Flows, 2nd ed. New York: Wiley, 1990, ch. 4, 6, and 8.

[3] D. Bertsekas and R. Gallager, Data Networks. Englewood Cliffs, NJ:Prentice Hall, 1992, ch. 6.

[4] M. Bhardwaj and A. P. Chandrakasan, “Bounding the lifetime of sensornetworks via optimal role assignments,” in Proc. IEEE INFOCOM,New York, Jun. 23–27, 2002, pp. 1587–1596.

[5] D. Blough and S. Paolo, “Investigating upper bounds on network life-time extension for cell-based energy conservation techniques in sta-tionary ad hoc networks,” in Proc. ACM MobiCom, Atlanta, GA, Sep.23–28, 2002, pp. 183–192.

[6] T. X. Brown, H. N. Gabow, and Q. Zhang, “Maximum flow-life curvefor a wireless ad hoc network,” in Proc. ACM MobiHoc, Long Beach,CA, Oct. 4–5, 2001, pp. 128–136.

[7] J.-H. Chang and L. Tassiulas, “Routing for maximum system lifetimein wireless ad hoc networks,” in Proc. 37th Annu. Allerton Conf. Com-munications, Control, and Computing, Monticello, IL, Sep. 1999, vol.1, pp. 22–31.

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Y. Thomas Hou (S’91–M’98–SM’04) received theB.E. degree from the City College of New York in1991, the M.S. degree from Columbia University in1993, and the Ph.D. degree from Polytechnic Univer-sity, Brooklyn, New York, in 1998, all in electricalengineering.

Since Fall 2002, he has been with the BradleyDepartment of Electrical and Computer Engineering,Virginia Tech, Blacksburg, VA, where he is nowan Associate Professor. His current research in-terests are radio resource (spectrum) management

and networking for software-defined radio wireless networks, optimizationand algorithm design for wireless ad hoc and sensor networks, and videocommunications over dynamic ad hoc networks. From 1997 to 2002, he wasa Researcher at Fujitsu Laboratories of America, Sunnyvale, CA, where heworked on scalable architectures, protocols, and implementations for differ-entiated services Internet, service overlay networking, video streaming, andnetwork bandwidth allocation policies and distributed flow control algorithms.He holds two U.S patents and has three more pending.

Prof. Hou is a recipient of an Office of Naval Research (ONR) YoungInvestigator Award (2003) and a National Science Foundation (NSF) CAREERAward (2004). He is active in professional services and is currently servingas an Editor of IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, IEEETRANSACTIONS ON VEHICULAR TECHNOLOGY, ACM/Springer Wireless Net-works (WINET), and Elsevier Ad Hoc Networks Journal. He is Co-Chair ofTechnical Program Committee (TPC) of the Second International Confer-ence on Cognitive Radio Oriented Wireless Networks and Communications(CROWNCOM 2007), Orlando, FL, August 1–3, 2007. He was the Chair ofthe First IEEE Workshop on Networking Technologies for Software DefinedRadio Networks, September 25, 2006, Reston, VA. He will serve as Co-Chairof TPC of IEEE INFOCOM 2009, to be held in Rio de Janeiro, Brazil. He hasbeen a member of ACM since 1995.

Yi Shi (S’02–M’08) received the B.S. degree fromthe University of Science and Technology of China,Hefei, China, in 1998, the M.S. degree from theInstitute of Software, Chinese Academy of Science,Beijing, China, in 2001, a second M.S. degreefrom Virginia Tech, Blacksburg, VA, in 2003, all incomputer science, and the Ph.D. degree in computerengineering from Virginia Tech in 2007.

He is currently a Senior Research Associate in theDepartment of Electrical and Computer Engineeringat Virginia Tech. His current research focuses on al-

gorithms and optimization for cognitive radio wireless networks, MIMO andcooperative communication networks, sensor networks, and ad hoc networks.His work has appeared in some highly selective international conferences (ACMMobiCom, ACM MobiHoc, and IEEE INFOCOM) and IEEE journals.

While an undergraduate, Mr. Shi was a recipient of the Meritorious Award inthe International Mathematical Contest in Modeling in 1997 and 1998, respec-tively. He was a recipient of the Chinese Government Award for OutstandingStudents Abroad in 2006. He is active in professional services. He was a TPCmember of IEEEWorkshop on Networking Technologies for Software DefinedRadio (SDR) Networks (held in conjunction with IEEE SECON 2006), Reston,VA, Sept. 25, 2006, and ChinaCom, Hangzhou, China, April 2527, 2008. He isa TPC member of ACM International Workshop on Foundations of Wireless AdHoc and Sensor Networking and Computing (co-located with ACM MobiHoc2008), Hong Kong, China, May 26–30, 2008; IEEE ICCCN, St. Thomas, U.S.Virgin Islands, Aug. 4–7, 2008; IEEE PIMRC, Cannes, France, Sep. 15–18,2008; IEEEMASS, Atlanta, GA, Oct. 6–9, 2008; and IEEE ICC, Dresden, Ger-many, June 14–18, 2009.

Hanif D. Sherali is the W. Thomas Rice EndowedChaired Professor of Engineering in the Industrialand Systems Engineering Department at VirginiaPolytechnic Institute and State University, Blacks-burg, VA.

His area of research interest is in discrete and con-tinuous optimization, with applications to location,transportation, and engineering design problems.He has published about 200 papers in OperationsResearch journals, has co-authored four books inthis area, and serves on the editorial board of eight

journals.Dr. Sherali is a member of the U.S. National Academy of Engineering.

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