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322 IEEE/ACM TRANSACTIONS ON NETWORKING, VOL. 16, NO. 2, APRIL 2008could 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 much Fig. 1. Reference architecture for two-tier wireless sensor networks. (a) Phys-more efﬁcient than the state-of-the-art slack variable (SV)-based ical topology; (b) a hierarchical view.approach proposed in [6], due to fewer number of LPs involvedat each rate level. We also extend the PA technique for the LMM rate allocation nodes (MSNs), aggregation and forwarding nodes (AFNs),problem to address the so-called maximum node lifetime curve and a base station (BS). The MSNs can be application-speciﬁcproblem in [6], which we call LMM node lifetime problem. We sensor nodes (e.g., temperature sensor nodes (TSNs), pressureshow that the SLP-PA approach is much more efﬁcient than the sensor nodes (PSNs), and video sensor nodes (VSNs)) and theyslack variable (SV)-based approach (SLP-SV) described in [6]. constitute the lower tier of the network. They are deployed inMore importantly, we show that there exists a simple and elegant groups (or clusters) at strategic locations for surveillance andduality relationship between the LMM rate allocation problem monitoring applications. The MSNs are small and low-cost.and the LMM node lifetime problem. As a result, it is sufﬁcient The objective of an MSN is very simple: Once triggered by anto solve only one of these two problems. Important insights can event, it starts to capture sensing date and sends it directly tobe obtained by inferring duality results for the other problem. the local AFN.1 The remainder of this paper is organized as follows. In For each cluster of MSNs, there is one AFN, which is dif-Section II, we describe the network and energy model, and for- ferent from an MSN in terms of physical properties and func-mulate the LMM rate allocation problem. Section III presents tions. The primary functions of an AFN are: 1) data aggrega-our SLP-PA algorithm to the LMM rate allocation problem. In tion (or “fusion”) for data ﬂows from the local cluster of MSNs,Section IV, we introduce the LMM node lifetime problem and and 2) forwarding (or relaying) the aggregated information toapply the SLP-PA algorithm to solve it. Section V shows an in- the next hop AFN (toward the base station). For data fusion, anteresting duality relationship between the LMM rate allocation AFN analyzes the content of each data stream it receives andproblem and the LMM node lifetime problem. In Section VI, exploits the correlation among the data streams. An AFN alsowe present numerical results. Section VII reviews related work serves as a relay node for other AFNs to carry trafﬁc toward theand Section VIII concludes this paper. base station. Although an AFN is expected to be provisioned with much more energy than an MSN, it also consumes energy II. SYSTEM MODELING AND PROBLEM FORMULATION at a substantially higher rate (due to wireless communication We consider a two-tier architecture for wireless sensor net- over large distances). Consequently, an AFN has a limited life-works. Figs. 1(a) and (b) show the physical and hierarchical time. Upon depletion of energy at an AFN, we expect that thenetwork topology for such a network, respectively. There are 1Due to the small distance between an MSN and its AFN, multi-hop routingthree types of nodes in the network, namely, micro-sensor among the MSNs may not be necessary.
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HOU et al.: RATE ALLOCATION AND NETWORK LIFETIME PROBLEMS FOR WIRELESS SENSOR NETWORKS 323coverage for the particular area under surveillance is lost, de- B. The LMM Rate Allocation Problemspite the fact that some of the MSNs within the cluster may still Before we formulate the LMM rate allocation problem, lethave remaining energy.2 us revisit the maximum capacity problem (with “bias” in rate The third component in the two-tier architecture is the base allocation) that was described in Section I. For a network withstation. The base station is, essentially, the sink node for data AFNs, suppose that the rate of AFN is , and that the initialstreams from all the AFNs in the network. In this investigation, energy at this node is . For a given networkwe assume that there is sufﬁcient energy resource available at lifetime requirement (i.e., each AFN must remain alive for atthe base station and thus there is no energy constraint at the base least time duration ), the maximum information capacity thatstation. In summary, the main functions of the lower tier MSNs the network can collect can be found by the following linearare data acquisition and compression while the upper-tier AFNs program (LP):are used for data fusion and relaying information to the basestation.A. Power Consumption Model (4) 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 data (5)fusion) 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- where and are data rates transmitted from AFN to AFNtion 1) under a given network lifetime requirement. and from AFN to the base station , respectively. The set For an AFN, energy consumption due to wireless communi- of constraints in (4) are the ﬂow balance equations: they statecation (i.e., transmitting and receiving) has been considered the that, the total bit rate transmitted by AFN is equal to the totaldominant factor in power consumption [1]. The power dissipa- bit rate received by AFN from other AFNs, plus the bit ratetion at a radio transmitter can be modeled as [9] generated locally at AFN . The set of constraints in (5) are the energy constraints: they state that, for a given network (1) lifetime requirement , the energy required in communicationswhere is the power dissipated at AFN when it is transmit- (i.e., transmitting and receiving all these data) cannot exceed theting to node , is the rate from AFN to node , is the initial energy provisioning level.power consumption cost of radio link and is given by Note that , , , and are variables and that is a constant (the given network lifetime requirement). MaxCap is (2) a standard LP formulation that can be solved by a polynomial- time algorithm [2]. Unfortunately, as we shall see in the numer-where and are two constant terms, is the distance be- ical results (Section VI), the solution to this MaxCap problemtween these two nodes, and is the path loss index, with lends itself into an extreme favor for those AFNs whose data [16]. Typical values for these parameters are paths consume the least amount of power toward the base sta- and (for ) [9].3 Since tion. Consequently, although the network capacity is maximizedthe power level of an AFN’s transmitter can be used to control over the network lifetime , the corresponding bit rate alloca-the distance coverage of an AFN (see, e.g., [15], [17], [20]), tion among the AFNs (i.e., the values) only favors those AFNsdifferent network ﬂow routing topologies can be formed by ad- that have this property, while other AFNs are unfavorably allo-justing the power level of each AFN’s transmitter. cated with much smaller (even close to 0) bit rates. As a result of The power dissipation at a receiver can be modeled as [9] this unfairness, the effectiveness of the network in performing information collection or surveillance could be severely com- (3) promised. To address this fairness issue, we advocate the so-called lexi- cographic max-min (LMM) rate allocation strategy [14] in thiswhere (in b/s) is the rate of the received data stream paper, which has some similarity to the max-min rate allocationat AFN . A typical value for the parameter is 50 nJ/b [9]. in data networks [3].4 Under LMM rate allocation, we start with The above transmission and reception energy model assumes the objective of maximizing the bit rate for all the nodes untila contention-free MAC protocol, where interference from one or more nodes reach their energy-constrained capacities forsimultaneous transmission can be effectively minimized or the given network lifetime requirement. Given that the ﬁrst levelavoided. For such a network, a contention-free MAC protocol of the smallest rate allocated among the nodes is maximized, weis fairly easy to design (see, e.g., [18]) and its discussion is continue to maximize the second level of rate for the remainingbeyond the scope of this paper. nodes that still have available energy, and so forth. More for- 2We assume that each MSN can only forward information to its local AFN mally, denote as the sorted version (i.e.,for processing (e.g., video fusion). 4However, there is signiﬁcant difference between max-min and LMM, which 3In this paper, we use m = 4 in all of our numerical results. we will discuss shortly.
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324 IEEE/ACM TRANSACTIONS ON NETWORKING, VOL. 16, NO. 2, APRIL 2008 ) of the rate vector , TABLE Iwith corresponding to the rate of node . We then have the NOTATIONfollowing 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. The key to the LMM rate allocation problem is to ﬁndTable I lists the notation used in this paper. the correct values and the corresponding sets We ﬁrst introduce the following notation. Suppose that the , respectively. This can be done iteratively. Thatrate vector is LMM-optimal, with is, we ﬁrst determine rate level and the corresponding set , . Note that the values of these rates may then determine rate level and the corresponding set , andnot be all distinct. To highlight those distinct rate levels, we so on. In Section III-A, we will show how to determine eachremove any repetitive elements in this vector and rewrite it as rate level and in Section III-B, we will show how to determine such that , where , the corresponding node set. , and . Now for each , , denote the corresponding set of nodes that use up their energy at this A. Rate Level Determinationrate. Clearly, we have , where denotes Denote and . For , suppose thatthe set of all nodes. we already determined and the corresponding
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HOU et al.: RATE ALLOCATION AND NETWORK LIFETIME PROBLEMS FOR WIRELESS SENSOR NETWORKS 325sets . The rate level can be found by the fol- The above LP formulation can be rewritten in the formlowing optimization problem. , 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- (6) timal solution for at iteration , it remains to determine the minimum set of nodes corresponding to this , which is the key (7) difﬁculty in the LMM rate allocation problem. In the following section, we exploit the parametric analysis technique [2] to de- termine the minimum node set at each rate. (8) 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) are (9) binding at the -th iteration in LMM-Rate, i.e., include all the nodes that achieve equality in (8) at iteration . Although it is Note that for , the constraints (7) and (9) do not exist. For certain that at least one of the nodes in belong to (the min- , constraints (7) and (9) are for those nodes that have imum node set for rate ), some nodes in may still be ablealready reached their LMM rate allocation during the previous to further increase their rates under alternative ﬂow routing so- iterations. In particular, the set of constraints in (7) say lutions. In other words, if , then we must have ;that the sum of incoming and local data rates are equal to the otherwise, we must determine the minimum node setoutgoing data rates for each node with its LMM-optimal rate that achieves the LMM-optimal rate allocation. , . The set of constraints in (9) say that for those We ﬁnd that the so-called parametric analysis (PA) techniquenodes that have already reached their LMM-optimal rates, the [2] is most suitable to address this problem. The main idea oftotal energy consumed for communications has reached their PA is to investigate how an inﬁnitesimal perturbation on someinitial energy provisioning. On the other hand, the constraints in components of the LMM-Rate problem can affect the objective(6) and (8) are for the remaining nodes that have not yet reached function. In particular, considering a small increase on the right-their LMM-optimal rates. Speciﬁcally, the set of constraints in hand-side (RHS) of (10), i.e., changing to , where(6) state that, for those nodes that have not yet reached their , node belongs to the minimum node set if and only ifenergy constraint levels, the sum of incoming and local data . That is, node belongs to the minimumrates are equal to the outgoing data rates. Note that the objective node set if and only if a small increase in node ’s rate (infunction is to maximize the additional rate for those nodes. terms of total volume generated at node ) leads to a decrease inFurthermore, for those nodes, the set of constrains in (8) state the objective function.that the total energy consumed for communications should be To compare with 0, we apply an importantupper bounded by the initial energy provisioning. duality results from LP theory. If and are the respective To facilitate our later discussion on duality results in optimal solution to the primal and dual problems, then based onSection V, we further re-formulate above LP. In particular, we the parametric duality property [2], we havemultiply both sides of (6) and (7) by (which is a constantrepresenting a given network lifetime requirement) and denote (11) , , . Intuitively, and represent the bit volume that is transferred from node to Recall that these can be easily obtained at the same time and from node to , respectively, during lifetime . We when we solve the primal LP problem. Note that by the natureobtain the following problem formulation. of 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 is thus needed. (10) For each node with , we must perform a complete PA to see whether a perturbation (i.e., tiny increase) on the RHS of (10) will result in any change in the objective function. If there is no change, then we can determine that node does not belong to the minimum node set ; otherwise, node belongs to . Assume that the optimal solution is , where and denote the set of basic and non-basic variables; and denote the columns corresponding to the basic and non-basic variables. and denote the objective function coefﬁcient vectors for the basic and non-basic variables; and denotes the objective
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326 IEEE/ACM TRANSACTIONS ON NETWORKING, VOL. 16, NO. 2, APRIL 2008value. Then we have the corresponding canonical equations as The following lemma ensures that MSV determines the min-follows: 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 rate allocation is unique.If is replaced by , where the column vector has In a nutshell, the complete PA procedure to determinea single 1 element corresponding to node in the set of con- whether a node belongs to the minimum node set canstraints (10) while all the other elements are 0, then the only be summarized as follows.change due to this perturbation is that will be replaced by . Consequently, the objective value for the current Algorithm 1: (Minimum Node Set Determination with PA)basis becomes . As long as is non-negative, the current basis remains optimal. Denote , 1) Initialize sets and . , and let be an upper bound for such that the 2) For each node ,current basis remains optimal. We have a) If , then . b) Otherwise (i.e., ), compute , , and according to (12). (12) If , then . 3) If , then and stop;If , the optimal objective value varies according to else set up the MSV problem and solve it. for . Since and 4) If the optimal objective value in MSV is 0, then , we have . Thus, the objective value and stop;will not change for , and consequently, the rate for else remove all nodes with from the set andnode can be increased beyond the current value. That is, go to Step 3.node does not belong to the minimum node set . For most problems in practice, the above procedure is sufﬁ- C. Optimal Flow Routing for LMM Rate Allocationcient to determine whether or not node belongs to the minimum After we solve the LMM rate allocation problem iterativelynode set for all . But in the rare event where , the using the procedure in Sections III-A and III-B, the corre-problem is degenerate. To develop a polynomial-time algorithm, sponding optimal ﬂow routing can be obtained by dividing thedenote as the set of all nodes with and as the set total bit volume on each link ( or ) by , i.e.,of all nodes with and . Then we solve the fol-lowing LP to maximize the slack variables (SV) for nodes in . (13) where is the given network lifetime requirement. Although the LMM-optimal rate allocation is unique, it is important to note 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 usually exist 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 corresponding to 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 rate allocation. Example 1: Consider an 8-node network with the following topology (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-If the optimal objective function is 0, then we conclude that no sists of at (100, 0), at (0, 100), at ( 100,node in can have a positive . That is, these nodes should 0), and at (0, 100), respectively (all in meters); Groupall belong to and we have . On the other hand, nodes consists of at (100, 100), at ( 100,if the optimal objective function is positive, then some nodes 100), at ( 100, 100), and at (100, 100), re- must have positive values and these nodes therefore do spectively. Assume that all nodes have the same initial energynot belong to the minimum node set . Consequently, we can . For a network lifetime requirement of , we can calculateremove these nodes from . If , we move on to solve (via SLP-PA) that the ﬁnal LMM-optimal rate allocation for allanother MSV. This procedure will terminate when the optimal 8 nodes are identical (perfect fairness), i.e., .objective function value is 0 or . We denote for .
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HOU et al.: RATE ALLOCATION AND NETWORK LIFETIME PROBLEMS FOR WIRELESS SENSOR NETWORKS 327 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 complexity at each stage is . As there are at most stages, the overall complexity is . We now analyze the complexity for the degenerate case. Upon the completion of Step 2 in Algorithm 1, we denote . Since we need to solve at most LPs, the complexity is or . Hence, the complexity at each stage is . Since there are at most stages, the overall com- plexity is . The complexity in ﬁnding the optimal ﬂow routing is bounded by the number of radio links in the network, which is .Fig. 2. A simple example showing that the optimal ﬂow routing to the LMM Hence, the overall complexity is rate allocation is not unique. The range of x is 0 x 39g=62. for the non-degenerate case and for the degenerate case. Under either case, the computational complexity is polynomial.5 Upon the completion of the SLP-PA algorithm, we also obtainan optimal ﬂow routing solution corresponding to this LMM-op- E. Discussiontimal rate . This optimal ﬂow routing solution has the followingﬂows: , So far, we consider the case that each AFN generates data at a , and constant rate. In practice, an AFN node may not always transmit . We now show that the optimal ﬂow routing solu- data and may work in on/off mode to conserve energy. In thistion is non-unique. Since the network has symmetrical property, case, it is necessary to construct optimal ﬂow routing solutionit can be easily veriﬁed that for any , , for variable bit rate source (where on/off mode is a special case).the LMM-optimal rate allocation can be achieved if the ﬂow In [11], we have developed techniques to construct optimal ﬂowrouting solution satisﬁes the following two conditions: (i) each routing solution for variable bit rate, as long as its average rate isnode in (i.e., AFNs 2, 4, 6, and 8) sends a ﬂow of and a known. Such average rate corresponds to the constant rate in thisﬂow of to its two neighboring nodes as shown paper. As a result, the case of on/off mode (with known averagein Fig. 2, and a remaining ﬂow of directly to the base rate) can also be handled using techniques in [11].station; and (ii) each node in (i.e., AFNs 1, 3, 5, and 7) sendsa total amount of to the base station, which includes IV. EXTENSION TO LMM NODE LIFETIME PROBLEM and from its neighboring nodes, plus fromitself. Clearly, there are inﬁnitely many ﬂow routing solutions In this section, we show that our SLP-PA algorithm canthat meet these two conditions, each of which can be shown to be used to solve the so-called maximum node lifetime curveyield the LMM-optimal rate allocation with the given network problem in [6], which we deﬁne as the LMM node lifetimelifetime requirement . problem. We also show that the SLP-PA algorithm is a much more efﬁcient approach than the one proposed in [6], which isD. Complexity Analysis currently the state-of-the-art to address this problem. We now analyze the complexity of the SLP-PA algorithm. A. The LMM-Optimal Node Lifetime ProblemFirst we consider the complexity of ﬁnding each node’s rateand the total bit volume transmitted along each link. At each The LMM node lifetime problem considers the following sce-stage, we solve an LP problem, both its primal and dual have a nario. For a network with AFNs, with a given local bit ratecomplexity of [2], where is the number of con- (ﬁxed) and initial energy for AFN , , howstraints or variables in the problem, whichever is larger, and can we maximize the network lifetime for all AFNs in the net-is the number of binary bits required to store the data. Since the work? In other words, the LMM node lifetime problem not onlynumber of variables is and is larger than the number of considers how to maximize the network lifetime until the ﬁrstconstraints (which is ), the complexity of solving the LP AFN runs out of energy, but also the time for all the AFNs inis . After solving an LP at each stage, we need to deter- the network.mine whether or not a node that just reached its energy binding More formally, denote the lifetime for each AFN as ,constraint belongs to the minimum node set for this stage. Note . Note that ’s are ﬁxed here, while ’s arethat and can be readily obtained when we solve the optimization variables, which are different from the LMMthe primal LP problem. To determine whether a node, say , be- rate allocation problem that we studied in the last section. De-longs to the minimum node set, we examine . If , then note as the sorted sequence of the values innode belongs to the minimum node set and the complexity is 5Note that our analysis here gives a very loose upper bound for time com- . On the other hand, if , we need to further examine plexity. In practice, the running time for LP implementation is much faster.
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328 IEEE/ACM TRANSACTIONS ON NETWORKING, VOL. 16, NO. 2, APRIL 2008nondecreasing order. Then LMM-optimal node lifetime can be The only issue that we need to be concerned about is the op-deﬁned as follows. timal ﬂow routing solution corresponding to the LMM-optimal Deﬁnition 2: (LMM-Optimal Node Lifetime): A sorted node lifetime vector. The optimal ﬂow routing solution here is notlifetime vector with is as simple as that for the LMM rate allocation problem, whichLMM-optimal if and only if for any other sorted node lifetime merely involves a simple division (see (13)). We refer readers tovector with , there exists a the Appendix for an algorithm to obtain an optimal ﬂow , , such that for and . routing solution for the LMM-optimal lifetime vector. Similar to Lemma 2, the optimal ﬂow routing solution corresponding toB. Solution the LMM node lifetime problem may not be unique. It should be clear that, under the LMM-optimal node lifetime C. Complexity Comparisonobjective, we must maximize the time until a set of nodes use up In [6], Brown et al. studied the LMM node lifetime problemtheir energy (which is also called a drop point in [6]) while min- under the so-called “maximum node lifetime curve” problem.imizing the number of nodes that drain up their energy at each They also developed the ﬁrst procedure to solve this problemdrop point. We now show that the SLP-PA algorithm developed correctly. A key step in their procedure is the use of multiplefor the LMM rate allocation problem can be directly applied to independent LP calculations to determine the minimum node setsolve the LMM node lifetime problem. at each drop point, which we call serial LP with slack variable Suppose that with is analysis (SLP-SV). Although this approach solves the LMMLMM-optimal. To keep track of distinct node lifetimes (or drop node lifetime problem correctly, its computational complexitypoints) in this vector, we remove all repetitive elements in the (potentially exponential) remains an issue to be resolved.vector and rewrite it as such that On the other hand, the SLP-PA algorithm developed in this , where , , and . Corre- paper is polynomial and is computationally more efﬁcient thansponding to these drop points, denote as the sets the SLP-SV approach. To understand the difference between theof nodes that drain up their energy at drop points , two, we take a closer look on the computational complexity ofrespectively. Then , where the SLP-SV approach in [6]. First, SLP-SV needs to keep track denotes the set of all AFNs in the network. The problem is of each sub-ﬂow along its route from the source node towardto ﬁnd the LMM-optimal values of and the cor- the base station. Such a ﬂow-based (or more precisely, sub-ﬂowresponding sets . based) approach could make the size of the LP coefﬁcient matrix Similar to the LMM rate allocation problem, the LMM node exponential, which leads to an exponential-time algorithm [2].6lifetime problem can be formulated as an iterative optimization Second, even if a link-based LP formulation such as ours isproblem as follows. Denote , , and . adopted in [6], the computational efﬁciency of the SV-based ap-Starting from , we solve the following LP iteratively. proach is still worse than the SLP-PA algorithm. This is because at each stage, the SV-based approach must solve several addi- tional LPs (up to ) to determine , which is in contrast to the simpler PA under the SLP-PA algorithm . Even for the degenerate case, the number of additional LPs under the (14) SLP-PA algorithm is at most ,7 which is still no more than . Finally, we discuss a hybrid link-ﬂow approach mentioned in [6]. In this approach, link-based formulations are used for sub-ﬂows. This leads to a much fewer number of variables than those for the ﬂow-based approach. But this approach still re- quires sub-ﬂow accounting and results in an order of magnitude more constraints than the link-based approach in SLP-PA. Al- though this approach solves the LMM node lifetime problem in polynomial-time (e.g., by using interior point methods [2]), the overall complexity is still orders of magnitude higher than that under the SLP-PA algorithm. Furthermore, the burden of Comparing the above LMM-Lifetime problem to the LMM- solving additional LPs to determine whether a node belongs toRate problem that we studied in Section III-A, we ﬁnd that they the minimum node set still remains.are exactly of the same form. The only differences are that under V. DUALITY THEOREMthe LMM-Lifetime problem, the local bit rates are constantsand the node lifetimes are variables (subject to optimization), In this section, we present an elegant and powerful resultwhile under the LMM-Rate problem, the are variables (sub- showing that there is a duality relationship between the LMMject to optimization) and the node lifetimes are all identical , rate allocation problem and the LMM node lifetime problem. As . Since the mathematical formulation for the two 6Incidentally, the revised simplex method proposed in [6] is not as efﬁcientproblems are identical, we can apply the SLP-PA algorithm to as the polynomial-time algorithm described in [2] and is itself exponential.solve the LMM node lifetime problem as well. 7Recall that U denotes U upon the completion of Step 2 in Algorithm 1.
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HOU et al.: RATE ALLOCATION AND NETWORK LIFETIME PROBLEMS FOR WIRELESS SENSOR NETWORKS 329 TABLE II under the bit volume solution to problem and this veriﬁes DUALITY RELATIONSHIP BETWEEN LMM RATE ALLOCATION PROBLEM P that , , is a feasible solution to problem . AND LMM NODE LIFETIME PROBLEM P Optimality: To prove that ’s obtained via (15) are indeed LMM-optimal for problem , we sort , , under problem in non-decreasing order and denote it as . We also introduce a node index for . For example, means that actually corresponds to the rate of AFN 7, i.e.,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 simple Since is proportional to through the relationshipalgebraic calculations. , listing , , according to To start with, we denote the LMM rate allocation problem will yield a sorted (in non-decreasingwhere we have AFNs in the network and all nodes have a order) lifetime list, denoted as . We now provecommon given lifetime requirement (constant). Denote that is indeed LMM-optimal for problem .the LMM-optimal rate allocation for node under , Our proof is based on contradiction. Suppose that . Similarly, we denote the LMM node lifetime is not LMM-optimal for problem . As-problem where all nodes have the same local bit rate (constant). sume that the LMM-optimal lifetime vector to problemDenote the LMM node lifetime for node under , is (sorted in non-decreasing order) with the . Then the following theorem shows how the solution corresponding node index being . Then, byto one problem can be used to obtain the solution to the other. Deﬁnition 2, there exists a such that for Theorem 1: (Duality): For a given node lifetime requirement and . for all nodes under problem and a given local bit rate for We now claim that if , , is a feasible so-all nodes under problem , we have the following relationship lution to problem , then obtained via ,between the solutions to the LMM rate allocation problem , is also a feasible solution to problem . The proofand the LMM node lifetime problem . to this claim follows identically as above. Using this result, (i) Suppose that we have solved problem and obtained we can obtain a corresponding feasible solution the LMM-optimal rate allocation for each node with and the node index for problem . Hence, . Then under , the LMM node lifetime we have for but for node is . That is, is not (15) LMM-optimal and this leads to a contradiction. (ii) The proof for this part follows the same token as the above (ii) Suppose that we have solved problem and obtained proof for (i) and is thus omitted here. the LMM-optimal node lifetime for each node This duality relationship offers important insights on system . Then under , the LMM rate allocation performance issues, in addition to providing solutions to the for node is LMM rate allocation and the LMM node lifetime problems. For (16) example, in Section I, we pointed out the potential bias (fairness) issue associated with the network capacity maximization objec- Table II shows the duality relationship between solutions to tive (i.e., sum of rates from all nodes). It is interesting to see thatproblems and . there is a dual fairness issue under the node lifetime problem. In Proof: We prove (i) and (ii) in Theorem 1 separately. particular, the objective of maximizing the sum of node lifetimes (i) We organize our proof into two parts. First, we show that among all nodes also leads to a bias (or fairness) problem be- ’s are feasible node lifetimes in terms of ﬂow balance and cause this objective would only favor those nodes that consumeenergy constraints on each node . Then we energy at a small rate. As a result, certain nodes will have muchshow that it is indeed the LMM-optimal node lifetime. larger lifetimes while some other nodes will be penalized with Feasibility: Since we have obtained the solution to problem much smaller lifetimes, although the sum of node lifetimes is , we have one feasible ﬂow routing solution for sending bit maximized.streams , , to the base station. Under problem , the bit volumes ( and values) must meet the fol- VI. NUMERICAL INVESTIGATIONlowing equalities under the LMM-optimal rate allocation: In this section, we use numerical results to illustrate our SLP-PA algorithm to the LMM rate allocation problem and compare it with other approaches. We also use numerical results to illustrate the duality between the LMM rate allocation problem and the LMM node lifetime problem. We consider two network topologies, one with 10 AFNs andNow replacing by , we see that the same bit volume the other with 20 AFNs. Under both topologies, the base stationsolution under yields a feasible bit volume solution to the is located at the origin while the locations for the 10 or 20node lifetime problem under . Consequently, we can use Al- AFNs are randomly generated over a 1000 m 1000 m squaregorithm 2 to obtain the ﬂow routing solution to problem area (see Figs. 3(a) and (b) and Tables III and IV, respectively).
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330 IEEE/ACM TRANSACTIONS ON NETWORKING, VOL. 16, NO. 2, APRIL 2008 TABLE V RATE 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 (see Section II-B) that uses a serial LP “blindly” to solve the LMM rate allocation problem and performs energy reservation during each iteration. We call this naive approach Serial LP with Energy Reservation (SLP-ER). As discussed in Section II-B, the SLP-ER approach will not give the correct ﬁnal solution to the 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 under the MaxCap approach can be extremely biased and in favor of only those AFNs that consume the least power along their data paths toward the base station. 10-AFN Network: We assume that the initial energy at each AFN 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 underFig. 3. Network topologies used in the numerical investigation. (a) A 10-AFN each approach, which is also plotted in Fig. 4. The “sortednetwork; (b) a 20-AFN network. node index” corresponds to the sorted rates among the AFNs in non-decreasing order. Clearly, among the three rate al- TABLE III location approaches, only the rate allocation under SLP-PA NODE COORDINATES FOR A 10-AFN NETWORK meets the LMM-optimal rate allocation deﬁnition (see Def- inition 1). Speciﬁcally, comparing SLP-PA with SLP-ER, we have , , , and ; comparing SLP-PA with MaxCap, we have . We also observe, as expected, a severe bias in the rate allocation under the MaxCap approach. In partic- ular, alone accounts for over 48% of the sum of TABLE IV NODE COORDINATES FOR A 20-AFN NETWORK total rates among all the AFNs. Comparing the three ap- proaches, we have and . In other words, the rate allocation vector under the SLP-PA algorithm has the smallest rate 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 minimum node set for , i.e., . This conﬁrms that the naive SLP-ER approach cannot offer the correct solution to the LMM rate allocation problem.
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HOU et al.: RATE ALLOCATION AND NETWORK LIFETIME PROBLEMS FOR WIRELESS SENSOR NETWORKS 331 TABLE VI RATE ALLOCATION UNDER THE THREE APPROACHES FOR THE 20-AFN NETWORK TABLE VII = NUMERICAL RESULTS VERIFYING THE DUALITY RELATIONSHIP T 1 g R1t ( ) BETWEEN THE LMM RATE ALLOCATION PROBLEM P AND THE LMM ( ) NODE LIFETIME PROBLEM P FOR THE 10-AFN NETWORKFig. 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, we location problem and the LMM node lifetime problemassume that the initial energy at each AFN is 50 KJ and that independently with the above initial conditions using thethe network lifetime requirement under the LMM rate alloca- SLP-PA algorithm. Consequently, we obtain the LMM-optimaltion problem is 100 days. Table VI shows the sorted rate allo- rate allocation ( for each AFN ) under and the LMM-op-cation under the three approaches, which are also displayed in timal node lifetime ( for each AFN ) under . Then we com-Fig. 4(b). It can be easily veriﬁed that all the observations for pute and separately for each AFN and examine ifthe 10-AFN network also hold here. they are equal to each other. The results for the LMM-optimal rate allocation ( ,B. Duality Results ) and the LMM-optimal node lifetime ( , We now use numerical results to verify the duality relation- ) for the 10-AFN network are shown in Table VII.ship between the LMM rate allocation problem and the We ﬁnd that and are exactly equal for all AFNs,LMM node lifetime problem (see Section V). Again, precisely as we would expect under Theorem 1. Similarly, thewe use the 10-AFN and 20-AFN network conﬁgurations in results for the 20-AFN network are shown in Table VIII.Figs. 3(a) and (b), respectively. The coordinates for each AFNunder the 10-AFN network and 20-AFN network are listed in VII. RELATED WORKTables III and IV, respectively. For both networks, we assume Due to energy constraints in wireless sensor networks, therethat the initial energy at each AFN is 50 KJ and that the network has been active research on exploring the performance limitslifetime requirement under the LMM rate allocation problem is of such networks. These performance limits include, among . Under , we assume the local bit rate for all others, network capacity and network lifetime. Network ca-AFNs is Kb/s. pacity typically refers to the maximum amount of bit volume To verify the duality relationship (Theorem 1), we perform that can be successfully delivered to the base station (“sinkthe following calculations. First, we solve the LMM rate al- node”) by all the nodes in the network, where network lifetime
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332 IEEE/ACM TRANSACTIONS ON NETWORKING, VOL. 16, NO. 2, APRIL 2008 TABLE VIII VIII. CONCLUSIONNUMERICAL RESULTS VERIFYING THE DUALITY RELATIONSHIP T 1 g = r1t ( ) BETWEEN THE LMM RATE ALLOCATION PROBLEM P AND THE LMM In this paper, we investigated the important problem of rate ( ) NODE LIFETIME PROBLEM P FOR THE 20-AFN NETWORK allocation for wireless sensor networks under a given network lifetime requirement. Since the objective of maximizing the sum of rates of all nodes can lead to a severe bias in rate alloca- tion among the nodes, we advocate the use of lexicographical max-min (LMM) rate allocation for all nodes in the network. To calculate 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, we showed that the SLP-PA algorithm can also be employed to address the maximum node lifetime curve problem and that the SLP-PA algorithm is much more efﬁcient than an state-of- the-art algorithm. More important, we discovered a simple and elegant duality relationship between the LMM rate allocation problem and the LMM node lifetime problem, which enables us to develop solutions and insights on both problems by solving one of the two problems. Our results in this paper offer some important understanding on network capacity and network life- time problems for energy-constrained wireless sensor networks. APPENDIX Arefers to the maximum time that the nodes in the network PROOF OF LEMMA 1remain alive before one or more nodes deplete their energy. By the deﬁnition of LMM-optimal rate vector (see Deﬁni- The network capacity problem and network lifetime problem tion 1), the optimal rates ( values) are unique and the cor-have so far been studied disjointly in the literature. For example, responding numbers of nodes in each minimum node sets (in [13], the problem of how to maximize network capacity via values) are also unique. To show that the group of physical nodesrouting was studied. While, in many other efforts (see, e.g., [4], in each is also unique, we employ the parametric simplex ap-[5], [8], [12], [21],), the focus was on how to maximize the time proach to determine the minimum node set as follows.until the ﬁrst node drains up its energy. In essence, the parametric simplex approach solely relies on In this paper, we study the important overarching problem the PA technique without resorting to the MSV approach eventhat considers both network capacity and network lifetime. when the problem is degenerate. That is, when the problem isUnder the LMM rate allocation problem, we studied how to degenerate, i.e., for some node , we have andmaximize rate allocations for all the nodes in the network under , then the basis can change while the optimal objectivea given network lifetime requirement. Under the LMM node value remains unchanged. We can analyze and under thelifetime problem, we studied how to maximize the lifetime new basis to determine whether or not node belongs to thefor all nodes when the local bit rate for each node is given a minimum node set . If we still have and , thepriori. The LMM rate allocation criterion effectively mitigates basis can change again with the same optimal objective value.the unfairness issue when the objective is to maximize the total To prevent cycling back to a previous basis, we can use an anti-bit volume generated by the network. Although the LMM rate cycling rule [2]. Thus, this procedure is guaranteed to terminateallocation is somewhat similar to the classical max-min strategy within a ﬁnite number of steps and we can determine whether[3], there is a fundamental difference between the two. In par- or not node indeed belongs to the minimum node set .ticular, the LMM rate allocation problem implicitly embeds (or Note that in the above parametric simplex approach, the setcouples) a ﬂow routing problem within rate allocation, while of nodes corresponding to is uniquely determined since theunder the classical max-min rate allocation, there is no routing analysis is conducted independently for each node.8 Therefore,problem involved since the routes for all ﬂows are given. Due upon the completion of all stages, the group of AFNs in eachto this coupling of ﬂow routing and rate allocation, a solution minimum node set is unique.approach (i.e., SLP-PA) to the LMM rate allocation problem is APPENDIX Bmuch more challenging than that for the classical max-min. OPTIMAL FLOW ROUTING SOLUTION FOR LMM-OPTIMAL In [19], Srinivasan et al. applied game theory and Nash equi- NODE LIFETIMElibrium among the nodes to forward packets such that the totalthroughput (capacity) can achieve an optimal operating point It is straightforward to develop an example similar to the onesubject to a common lifetime requirement on all nodes. How- given in Section III-C that shows the non-uniqueness of the ﬂowever, the fairness issue in information collection was not consid- routing schedule.9 Given that the optimal ﬂow routing solutionered. The most relevant work to the LMM node lifetime problem 8This can be done in parallel if so desired.was by Brown et al. [6], which has been discussed in detail in 9Incidentally, this result corrects an error in [16] (Lemma 3.2), which incor-Section IV-A. rectly stated that such a ﬂow routing solution is unique.
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HOU et al.: RATE ALLOCATION AND NETWORK LIFETIME PROBLEMS FOR WIRELESS SENSOR NETWORKS 333is non-unique, there are potentially many ﬂow routing solutions has trafﬁc going into node . The outgoing ﬂow from node isthat can achieve the LMM-optimal lifetime vector. In this sec- calculated by distributing the aggregated ﬂow proportionally ac-tion, we present a simple polynomial-time algorithm that pro- cording to the overall bit volume along its outgoing radio links.vides an LMM-optimal ﬂow routing solution. As an example, suppose that during , node 2 receives an The main task in this algorithm is to deﬁne ﬂows from the bit aggregated ﬂow with rate 2 Kb/s and generates 0.4 Kb/s locally.volumes ( and values), which are obtained upon the com- Assume that Kb, Kb, and Kbpletion of the last iteration in the LMM-Rate problem with our over . Then the outgoing ﬂow at node 2 is routed as fol-SLP-PA algorithm. Note that the bit volumes obtained here rep- lows: Kb/s, Kb/s, and Kb/s.resent the total amount of bit volume being transported betweenthe nodes during , where is the time that the REFERENCESlast group of nodes drain up their energy. The main result here is [1] I. F. Akyildiz, W. Su, Y. Sankarasubramaniam, and E. Cayirci, “Wire-that if we let the total amount of outgoing ﬂow at a node be dis- less sensor networks: A survey,” Comput. Netw. (Elsevier), vol. 38, no.tributed proportionally to the bit volumes on each outgoing link 4, pp. 393–422, 2002. [2] M. S. Bazaraa, J. J. Jarvis, and H. D. Sherali, Linear Programming andfor all the remaining alive nodes at each stage, then we can achieve Network Flows, 2nd ed. New York: Wiley, 1990, ch. 4, 6, and 8.the drop points as well as the corresponding min- [3] D. Bertsekas and R. Gallager, Data Networks. Englewood Cliffs, NJ: Prentice Hall, 1992, ch. 6.imum node sets . The algorithm is formally de- [4] M. Bhardwaj and A. P. 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Allerton Conf. Com-volume on each radio link (i.e., and ). The following munications, Control, and Computing, Monticello, IL, Sep. 1999, vol.algorithm gives an LMM-optimal ﬂow routing solution for the 1, pp. 22–31. [8] J.-H. Chang and L. Tassiulas, “Energy conserving routing in wirelesstime interval , where and . ad hoc networks,” in Proc. IEEE INFOCOM, Tel Aviv, Israel, Mar. 1) Denote , with . Initialize all 26–30, 2000, pp. 22–31. [9] W. Heinzelman, “Application-speciﬁc protocol architectures for wire- ﬂows to zero, i.e., , for , , less networks,” Ph.D. dissertation, Massachusetts Inst. Technol., Cam- . bridge, MA, Jun. 2000. [10] Y. T. Hou, Y. Shi, and H. D. Sherali, “On node lifetime problem for 2) If , then stop, else choose a node from such energy-constrained wireless sensor networks,” ACM/Springer Mobile that:10 Netw. Applicat., vol. 10, no. 6, pp. 865–878, Dec. 2005. • node does not receive data from any other node, or [11] Y. T. Hou, Y. Shi, J. H. Reed, and K. Sohraby, “Flow routing for vari- able bit rate source nodes in energy-constrained wireless sensor net- • all nodes from which node receives data are not in . works,” in Proc. IEEE Int. Conf. Communications, Seoul, Korea, May 3) The ﬂow routing at node during is then 16–20, 2005, pp. 3057–3062. deﬁned as [12] K. Kalpakis, K. Dasgupta, and P. Namjoshi, “Maximum lifetime data gathering and aggregation in wireless sensor networks,” in Proc. IEEE Int. Conf. Networking (ICN’02), Atlanta, GA, Aug. 26–29, 2002, pp. 685–696. [13] K. Kar, M. Kodialam, T. V. Lakshman, and L. Tassiulas, “Routing for network capacity maximization in energy-constrained ad hoc net- works,” in Proc. IEEE INFOCOM, San Francisco, CA, Mar. 30–Apr. 3 2003, pp. 673–681. [14] H. Luss and D. R. Smith, “Resource allocation among competing ac- tivities: A lexicographic minimax approach,” Oper.s Res. Lett., vol. 5, no. 5, pp. 227–231, Nov. 1986. [15] R. Ramanathan and R. Rosales-Hain, “Topology control of multihop wireless networks using transmit power adjustment,” in Proc. IEEE where the values, if not zero, have all been deﬁned INFOCOM, Tel Aviv, Israel, Mar. 26–30, 2000, pp. 404–413. before calculating the ﬂow routing for node . [16] T. S. Rappaport, Wireless Communications: Principles and Practice. Englewood Cliffs, NJ: Prentice Hall, 1996. 4) Let and go to Step 2. [17] V. Rodoplu and T. H. Meng, “Minimum energy mobile wireless net- works,” IEEE J. Sel. Areas Commun., vol. 17, no. 8, pp. 1333–1344, Aug. 1999. As shown in this algorithm, for each time interval , [18] K. Sohrabi, J. Gao, V. Ailawadhi, and G. Pottie, “Protocols for self- , we initialize as the set of remaining alive organizing of a wireless sensor network,” IEEE Pers. Commun. Mag.,nodes at this stage, which is represented by . vol. 7, pp. 16–27, Oct. 2000. [19] V. Srinivasan, P. Nuggehalli, C. F. Chiasserini, and R. Rao, “Coop-For these nodes, we compute ﬂow routing by starting with the eration in wireless ad hoc networks,” in Proc. IEEE INFOCOM, San“boundary” nodes and then move to the “interior” nodes. More Francisco, CA, Mar. 30–Apr. 3 2003, pp. 808–817.precisely, we will calculate the ﬂow routing for a node if and [20] R. Wattenhofer, L. Li, P. Bahl, and Y.-M. Wang, “Distributed topology control for power efﬁcient operation in multihop wireless ad hoc net-only if we have calculated the ﬂow routing for each node that works,” in Proc. IEEE INFOCOM, Anchorage, AK, Apr. 22–26, 2001, pp. 1388–1397. 10It can be shown that an LMM-optimal solution is cycle free in terms of [21] G. Zussman and A. Segall, “Energy efﬁcient routing in ad hoc dis-ﬂow routing. Consequently, the node i under consideration must exist when aster recovery networks,” in Proc. IEEE INFOCOM, San Francisco,U 6= ;. CA, Mar. 30–Apr. 3 2003, pp. 405–421.
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334 IEEE/ACM TRANSACTIONS ON NETWORKING, VOL. 16, NO. 2, APRIL 2008 Y. Thomas Hou (S’91–M’98–SM’04) received the Hanif D. Sherali is the W. Thomas Rice Endowed B.E. degree from the City College of New York in Chaired Professor of Engineering in the Industrial 1991, the M.S. degree from Columbia University in and Systems Engineering Department at Virginia 1993, and the Ph.D. degree from Polytechnic Univer- Polytechnic Institute and State University, Blacks- sity, Brooklyn, New York, in 1998, all in electrical burg, VA. engineering. His area of research interest is in discrete and con- Since Fall 2002, he has been with the Bradley tinuous optimization, with applications to location, Department of Electrical and Computer Engineering, transportation, and engineering design problems. Virginia Tech, Blacksburg, VA, where he is now He has published about 200 papers in Operations an Associate Professor. His current research in- Research journals, has co-authored four books in terests are radio resource (spectrum) management this area, and serves on the editorial board of eightand networking for software-deﬁned radio wireless networks, optimization journals.and algorithm design for wireless ad hoc and sensor networks, and video Dr. Sherali is a member of the U.S. National Academy of Engineering.communications 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 from the University of Science and Technology of China, Hefei, China, in 1998, the M.S. degree from the Institute of Software, Chinese Academy of Science, Beijing, China, in 2001, a second M.S. degree from Virginia Tech, Blacksburg, VA, in 2003, all in computer science, and the Ph.D. degree in computer engineering from Virginia Tech in 2007. He is currently a Senior Research Associate in the Department of Electrical and Computer Engineering at 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.
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