Low-Complexity and Distributed Energy Minimization in Multi-hop Wireless Networks

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1 Low-Complexity an Distribute Energy inimization in ulti-hop Wireless Networks Longbi Lin, Xiaojun Lin, an Ness B. Shroff Center for Wireless Systems an Applications (CWSA) School of Electrical an Computer Engineering, Purue University West Lafayette, IN 47907, USA {llin, linx, Abstract In this work, we stuy the problem of minimizing the total power consumption in a multi-hop wireless network subject to a given offere loa. It is well-known that the total power consumption of multi-hop wireless networks can be substantially reuce by jointly optimizing power control, link scheuling, an routing. However, the known optimal crosslayer solution to this problem is centralize, an with high computational complexity. In this paper, we evelop a lowcomplexity an istribute algorithm that is provably powerefficient. In particular, uner the noe exclusive interference moel, we can show that the total power consumption of our algorithm is at most twice as large as the power consumption of the optimal (but centralize an complex) algorithm. Our algorithm is not only the first such istribute solution with provable performance boun, but its power-efficiency ratio is also tighter than that of another sub-optimal centralize algorithm in the literature. Inex Terms Energy Aware Routing, Duality, athematical Programming/Optimization, Cross-Layer Optimization, Simulations I. INTRODUCTION There has been significant recent interest in eveloping control protocols for multi-hop wireless networks. any applications can benefit from the eployment of these networks. For instance, sensors can form multi-hop wireless networks for such applications as habitat monitoring [], an the management of sewer overflow events []. Vehicles can form multihop wireless networks to exchange safety messages an traffic information [3]. Wireless LAN evices can form multi-hop mesh networks to provie wireless broaban access [4]. A key issue in eveloping control protocols for multihop wireless networks is to reuce the energy or power consumption. This is obviously an important issue for batterypowere networks since the power consumption often limits the lifetime of the network. Even for networks with access to power sources, the transmission power of the communication links may still nee to be properly controlle, e.g., ue to health or regulatory concerns. In this work, we are intereste in the problem of minimizing the total power consumption of a multi-hop wireless network, subject to a given offere loa. It is well-known that the total power consumption of multi-hop wireless networks can be substantially reuce by jointly optimizing power control, link scheuling, an routing. However, known optimal solutions require centralize computation an high computational complexity. In this paper, we propose a new low-complexity an istribute solution to this problem uner a wiely-use interference moel, calle the noe-exclusive interference moel. Using this moel, the work in [5] evelope a centralize solution that yiele a 3-approximation ratio (the resultant power consumption is within a factor of 3 from the optimal power consumption). In contrast, in this paper, we will obtain a ( + ε)-approximation algorithm that is fully istribute, where ε > 0 is an arbitrarily small constant. To the best of our knowlege, our propose algorithm is the first istribute solution in the literature with a provable performance boun. Our solution approach is inspire by the recent progress in using imperfect scheuling algorithms to evelop istribute cross-layer congestion control an scheuling algorithms in multi-hop wireless networks. We first formulate the energy minimization problem into a special form that naturally leas to a istribute solution. We then map the solution to corresponing components of the cross-layer control protocols, an rigorously establish the power-efficiency of the resulting algorithm. Our work is also relate to the stuy of energy-aware routing protocols for minimizing energy consumption an extening network lifetime [6] [0]. These works assume that the system capacity is battery-limite instea of interference-limite an therefore o not consier scheuling constraints. In contrast, our work explicitly consiers scheuling, jointly with power control an routing. The intellectual contribution of this work is summarize as follows: We evelop a low-complexity an istribute joint routing, power control, an scheuling algorithm for multihop wireless networks with provable power-efficiency ratio. Further, our algorithm can guarantee a better powerefficiency level than some existing centralize algorithms. Our solution cannot be obtaine by extening the known optimal solution in the literature [], []. Instea, we evelop an optimization approach to the energy minimization problem that naturally leas to istribute solutions. We also evelop rigorous techniques to ientify the

2 convergence properties an quantify the power-efficiency of the resulte control protocol. The rest of this paper is organize as follows: in Section II, we present the system moel an formulate the energy minimization problem. In Section III, we present our algorithm an iscuss its implications. In Section IV, we iscuss our main result on the convergence properties an the power-efficiency of the ual algorithm. Numerical results are provie in Section V. Concluing remarks are presente in Section VI. II. PROBLE FORULATION We moel a wireless multi-hop network by a irecte graph G(V, E), where V is the set of vertices representing the noes, an E is the set of eges representing the communication links. We use N o (v) an N i (v) to enote the sets of outgoing an incoming links of noe v, respectively. Their union N(v) forms the set of all links incient on noe v. The system is time-slotte. We aopt the following noeexclusive interference moel that is use to characterize FH- CDA an UWB system with perfect orthogonal spreaing coes an low power-spectrum ensity. Uner this moel, a noe can only receive from or transmit to at most one noe at any time-slot m. Further, each link is power-controlle. That is, if noe-exclusiveness is satisfie, we assume that the possible ata rate of link e is a result of its power assignment, p e. We use h( ) to enote the power consumption of supporting ata rate of. It is assume that h( ) is a nonecreasing, convex function satisfying h(0) = 0. An example of h( ) is the power-rate relationship in an Aitive White Gaussian Noise (AWGN) channel. Other interference moels can be incorporate in this formulation, an we will iscuss this in Section VI. Each packet may take multiple hops to be elivere from source to estination. Let T v enote the long-term average ata rate of the flow that nees to be supporte from source noe v to estination noe. We use D to enote the set of estinations. The joint energy minimization problem is now formulate as follows: (*) min f, R lim subject to h( (m)), () m= e E R(m) satisfies noe-exclusiveness, m, f e = lim (m), e, () m= fe fe T v 0,, an v, (3) where the quantity fe can be interprete as the average amount of ata rate on link e allocate for estination. The constraints in () require that the long-term average ata rate, etermine by the power allocation, shoul be able to support the total average ata rate ( f e ) on each link. The constraints in (3) require that the total outgoing flow of a noe shoul be able to support the total incoming an locally generate flow, for all estinations. Note that we use as a shorthan notation for all noes belonging to D, the set of estinations. We will refer to the above problem as Problem ( ). III. SOLUTION ETHODOLOGY A. Approximating the Energy inimization Problem The optimal solutions evelope in [], [] attempt to solve Problem ( ). However, their solutions contain a scheuling component with high computational complexity. In orer to compute at which power an at what time each link e shoul be activate, these solutions nee to solve a complex global optimization problem in each time-slot. In this paper, in orer to obtain a low-complexity an istribute solution, we take a ifferent approach. We first approximate ( ) by another optimization problem that is easier to solve. The following Lemma [5] provies the first step in this irection. Lemma : For all time-slot m when link e is activate, the instantaneous ata rate (m) is inepenent of m in any power-optimal scheme. It is worth noting that (m) = hols when link e is activate. As a result, lim m= (m) is equal to the prouct of an the frequency that link e is activate. Therefore, by (), the objective function of Problem ( ) can now be written as f e h( ), f e e E where is the fraction of time-slots that link e is activate. Further, using the results from low-complexity scheuling [3], we have Fact : In the optimal solution to ( ), we must have f e, v V. Fact : Uner the noe-exclusive interference moel, if f e η, v V, where η > 0 is an arbitrarily small number, then a maximal scheule can be compute such that each link is activate for f e fraction of time-slots [4], [5]. We will iscuss more about the role of maximal scheuling in our solution in Section III-D. Base on these two facts, in the rest of the paper, we will replace the scheuling constraints by f e β, v V. (4)

3 Problem ( ) can then be reformulate as the following problem: (A) min f e f, R h( ), (5) e E subject to (3) an (4) ( f, R) X, where X = {( f, R) : 0, f e 0, e, }. The formulation in (A) is not only easier to solve, but it also prouces natural bouns for proving the power efficiency ratio of our solution. Inee, solving (A) with β = provies a lower boun on the minimum power of ( ), while β = η provies an upper boun. Remark: This problem appears in the formulation of [5]. However, our solution is ifferent from this point on. As we mentione earlier, their solution is a centralize one with an approximation ratio of 3, while our solution is a istribute one with a better approximation ratio. B. Hanling the Non-Convexity In Problem (A), the objective function an constraints (4) are non-convex. Problems of this type are consiere to be ifficult in general. To overcome this ifficulty, the following change of variable is performe: t e = f e, e E. The physical meaning of t e is the fraction of time-slots that link e is activate. It can also be interprete as the loa on link e. The latter interpretation seems more appropriate when we eal with t e (m) for each time-slot in the ual solution. Note that t e = 0 implies that fe = 0 for any estination an any link e: if a link is not activate (or the loa is zero), it cannot support any non-zero ata rate. The long-term average power consumption Θ from link e can therefore be efine as { 0, Θ( f te = 0, e, t e ) = t e h( f (6) e t e ), t e > 0. Using the above notation, Problem (A) can be transforme into (B) min f, t subject to Θ( f e, t e ), (7) e E t e β, v V, (8) f e f e T v 0,, an v, (9) ( f, t) Y, where Y = {( f, t) : fe 0, e, ; 0 t e, f e at e, e, for some a 0}. Remark: The conition f e at e, for some a 0, in the efinition of Y is require to guarantee that the optimal solution from Problem (B) can be transforme to the optimal solution of Problem (A). Without this conition, Problem (B) can be trivially solve by setting t = 0 an f large enough for the flow constraint. Such a solution cannot be transforme to an optimal solution of Problem (A), since the instantaneous ata rate R must be finite. To show that there is no uality gap in solving Problem (B), we nee the following Lemma: Lemma : Let θ(f, t) = { 0, t = 0, th( f t ), t > 0. (0) If h( ) is a convex function on R +, then θ(f, t) is also convex on C = a [0,+ ) {(f, t) : 0 f at, t 0}. In our online technical report [6], we have inclue a proof of Lemma. Remark: A function of the form g(f, t) = th( f t ), where t > 0 is known as the perspective of function h(f). As shown in [7, p89], the perspective operation is one of the transformations that preserve convexity. The transformation in the above Lemma inclues the t = 0 case, an therefore can be viewe as a slightly generalize version of the perspective operation. Our proof in [6] takes a ifferent approach than [7]. Note that θ is non-convex over the entire R + R +. However, the convexity of θ over C is enough for our purpose, since ( f, t) Y in Problem (B). Due to Lemma, the objective function Θ( f, t) of (B) is convex, an the entire problem is a convex program. We can then use the uality approach to solve the problem. C. Distribute Algorithm Base on Lagrange Duality To use the uality approach to solve the above problem, we first form the Lagrangian: L( f, t, µ, q) µ v = Θ( f e, t e ) + e E v V qv fe v, t e β fe T v, where µ 0 an q 0 are the Lagrange multipliers, an f e = {fe } D. For ease of notation, we efine q = 0, for all. By rearranging the orer of summation, the above equation can be transforme into the following: = e E L( f, t, µ, q) c e ( f e, t e ) β v µ v + v, (q vt v ),

4 where c e ( f e, t e ) = Θ( f e, t e ) + (µ x(e) + µ r(e) )t e (q x(e) q r(e) )f e, () an x(e) an r(e) are the transmission noe an reception noe of link e, respectively. The ual objective function is = min D( µ, q) ( f, t) Y = e E [ L( f, t, µ, q), min c e ( f e, t e ) ( f e,t e) Y e ] β v µ v + v, (q vt v ), () where Y e enotes the constraint set on link e, namely Y e = {( f e, t e ) : 0 fe H, ; 0 t e ; fe at e, for some a 0}. (3) To ensure convergence, we have ae fe H, where H is a large constant to be etermine later. See the proof of Theorem 4 (in Appenix) for etails. The minimization of the Lagrangian can now be ecompose into a minimization on each link. Note that all the information neee in minimizing c e ( f e, t e ) is local to link e. The ual optimization problem is (C) max D( µ, q). (4) µ 0, q 0 As mentione before, the convexity of Θ( f, t) can be inferre from Lemma. Base on this, the following Theorem [6] establishes the relationship between the primal problem (B) an the ual problem (C) above. Due to space constraint, we state it without proof. Theorem 3: (Strong Duality) Problem (B) is feasible an its optimal value Θ is finite. Further, there is no uality gap: Θ can be foun by solving its ual optimization problem. The next step is to solve the ual problem in a istribute fashion. We can show that D( µ, q) is convex an its subgraient is given by D µ v = D q v = t e β, f e fe T v. We can then use the following subgraient metho to solve the ual problem. Distribute Energy inimization Algorithm At each iteration m, ) At link e, the ata rate f e an the link assignment t e are etermine by: ( f e (m), t e (m)) = argmin c e ( f e, t e, m) ( f e,t e) Y e = argmin ( f e,t e) Y e [ Θ( f e, t e ) + (µ x(e) (m) + µ r(e) (m))t e (q x(e) (m) q r(e) (m))f e ]. (5) ) At noe v, the ual variables are upate by: µ v (m + ) = µ v(m) + α m t e (m) β qv(m + ) = q v(m) α m f e (m) fe (m) T v + + (6). (7) Remark: At each iteration m, the ata rate an scheule are chosen to minimize the total cost c e, given the current implicit costs ( µ, q). To maximize the ual objective function, the ual variables are upate accoring to the subgraient ascent proceure in (6) an (7), where {α m } are the stepsizes. In the above algorithm, we use the same stepsize for µ an f at each iteration. This is simply for ease of notation: the stepsizes can be ifferent for each ual variable. Since the primal cost function is not strictly convex, the ual function may not be ifferentiable. The stepsizes nee to be chosen carefully to guarantee the convergence of the ual variables. We will iscuss this issue in the Section IV. The above exercise of using Lagrange uality is stanar. Nonetheless, there are some questions that are not answere by the above algorithm alone. One of them is how the algorithm maps to ifferent protocol components. We will now stuy this problem. D. apping to Network Protocol Components To ecie the routing an scheuling, each link solves (5) by minimizing c e ( f e, t e, m). Note that there are three components in c e ( f e, t e, m): The term Θ( f e, t e ) is the power cost, i.e., the power consumption of supporting an average ata rate of f e while link e being activate for t e fraction of time. The term (µ x(e) (m) + µ r(e) (m))t e is the scheuling cost. We will show that t e (m) will either be 0 or in orer for link e to minimize c e ( f e, t e ). If t e (m) is chosen to be, then µ x(e) (m + ) µ x(e) (m) from (6): scheuling the transmission on link e gives rise to increase (or at least the same) scheuling costs on the

5 transmission/reception noes. If t e (m) is chosen to be 0, then µ x(e) (m + ) µ x(e) (m): an ile link gives the system more flexibility in scheuling other links, which leas to a potential ecrease in the scheuling costs on the transmission/reception noes. The term (q x(e) (m) q r(e) (m))f e is the utility of supporting the total ata rate on link e. The term q v can be interprete as an approximation of the scale version of the queue length at noe v for estination if constant stepsizes are use [4], [8]. As in (7), if f e (m) > 0, it helps to reuce the workloa at the transmission noe, but a to the workloa at the reception noe. The following transformation leas to a better unerstaning of the minimization of c e ( f e, t e ): f e (m) = R e(m)t e (m), (8) where R e(m) is the instantaneous ata rate allocate on link e for estination, an f e (m) is the resultant average ata rate when transmitting for t e (m) fraction of the time. Substitute the above equation into c e ( f e, t e, m), we have (all time inex roppe for ease of notation): where l e ( ) is efine as l e ( ) = h( c e ( f e, t e ) = t e l e ( ), (9) R e) + (µ x(e) + µ r(e) ) [ ] (qx(e) q r(e) ). (0) Since t e 0, to minimize (9), we shoul first minimize l e ( ) as a function of. Note that function h( ) takes as input parameter the sum of the ata rates allocate for all the estinations on this link. In other wors, from the viewpoint of power consumption, it is inifferent which estination the ata rate is allocate for, as long as the total ata rate is the same. As a result, the minimum of l e ( ) is attaine when all the ata rates are allocate to the estination with the maximum positive backlog ifference. Let ˆ = argmax(qx(e) q r(e) ), () then R ˆ > 0 if (q ˆ x(e) q ˆ r(e) ) > 0, an R e = 0 for all ˆ. With this observation, the minimization of l e ( ) is relatively simple. For example, if h(x) = e x, an max (q x(e) qr(e) ) > 0, then R ˆ e = [log(q ˆ x(e) q ˆ r(e) )] +, an Re = 0 for all other estination. Now that R e has been chosen to minimize l e ( R e ) in (0), the next step is to etermine the value of t e over the interval [0, ] to minimize c e ( f e, t e ) = t e l e ( R e ). Clearly, the optimal t e value is ˆt e = {, if min Re l e ( R e ) 0, 0, if min Re l e ( R e ) > 0. () Remark: There are two possible scenarios where ˆt e = 0 (no loa is assigne in this time-slot): All backlog ifferences (qx(e) q r(e) ) are negative, which means the utility oes not increase by transporting ata to next hop on this link for any estination. Although some backlog ifferences (qx(e) q r(e) ) are positive, as in (0), the utility of transporting ata to next hop is not large enough to outweigh the importance of saving more energy (the h( ) term) an/or the nee to gain more flexibility in scheuling (the (µ x(e) + µ r(e) ) term). As a result, optimal l e ( R e ) is non-negative, an no loa is assigne to this link. To summarize, the minimization of c e ( f e, t e ) on each link naturally translates into the following protocol components: ) Routing: Choose only the flow ˆ with maximum positive backlog ifference (cf. ()). This is the flow that shoul receive service. ) Power control: Choose R e to minimize l e ( R e ). This is the power link e shoul use. 3) Link assignment: Choose t e to minimize t e l e ( R e ) in such a way that t e takes its maximum value if the optimal l e ( R e ) is less than or equal to 0; an 0 otherwise. This etermines the amount of time link e shoul be on. 4) aximal Scheuling: As mentione before, given the fraction of up-time, or loa, on each link, a scheuling component is neee to etermine the exact time-slots link e shoul be on. In other wors, the link assignment from previous step may have scheuling conflicts, an therefore cannot be use as a transmission scheule as it is. A scheuling component can resolve such conflicts by elaying the transmission by a certain number of time-slots. Clearly, given the link assignment t(m), the elay can be ifferent for ifferent links. With the noeexclusive interference moel, by choosing β = η, it can be shown that a scheuling policy base on maximal matching can stabilize the system (i.e., the elay of transmission mentione above is boune) [4], [5], [9] []. aximal scheules such as this can be implemente in a istribute fashion. We refer reaers to [3] for more etails on the istribute implementation of maximal scheules. As we have seen thus far, the uality approach exploits the problem structure an ecomposes the primal problem into sub-problems on each link that are much simpler. In aition, some of the quantities prouce/monitore by this algorithm actually help network engineering. For instance, the Lagrange multiplier qv can also be interprete as the shaow price of the corresponing constraint. If a small change occurs in the amount of supporte traffic from noe v to noe, qv measures the sensitivity of the optimal power consumption with respect to this perturbation. In a ifferent networking setting where the network has some control over the traffic matrix T, information such as q can be use as guielines to optimize power consumption.

6 IV. PERFORANCE ANALYSIS In this section, first we are intereste in the following question: Uner what conition o the ual variables in the Distribute Energy inimization algorithm converge? Furthermore, as we have seen in Section III, there is no guarantee that ( f(m), t(m)) will converge, even if the ual variables converge. For example, t e (m) is either 0 or from the istribute ual algorithm. A natural question to ask then is the following: In what sense is the primal solution optimal? The two Theorems below answer both of the above questions from two ifferent perspectives. A. Convergence Result with Constant Stepsizes The following theorem establishes the stability an the optimality of the propose ual algorithm. Theorem 4: (Stability an Optimality) Let the stepsizes in the Distribute Energy inimization algorithm be constant, i.e., α m = h, for all m. Let Φ be the set of ( µ, q) that maximizes D( µ, q), an (( µ, q), Φ) = min ( µ, q ) Φ ( µ, q) ( µ, q ). Given any ε > 0, there exists some h 0 > 0 such that, for any h h 0 an any initial implicit costs ( µ 0, q 0 ), there exists a time 0 such that for all m > 0, (( µ(m), q(m)), Φ) < ε, (3) an lim sup m m m Θ( f e (τ), t e (τ)) < Θ + ε. (4) τ= e E The proof of Theorem 4 is inclue in the Appenix. The above Theorem shows that, when stepsizes are small, the ual variables eventually converge to within a small neighborhoo of the optimal ual solution. Further, the power consumption from the istribute algorithm is asymptotically optimal: although the primal variables ( f(m), t(m)) may not converge, by using ( f(m), t(m)) for each time-slot m, the long-term average of the resultant power consumption is arbitrarily close to the optimal power consumption. B. Optimality of Primal Variables As shown in Theorem 4, with constant stepsizes, the ual variables eventually converge to within a small neighborhoo of the optimal ual solution. When appropriate iminishing stepsizes are use in the ual algorithm, the following Theorem shows that the entire sequence of the ual variables converges to one point. In this case, we also obtain a ifferent interpretation of the primal optimality. Theorem 5: (Primal Optimality with Diminishing Stepsizes) (a) Let the stepsizes {α m } in the Distribute Energy inimization algorithm satisfy the following conitions: αm < +, α m = +. (5) m= m= Then for any nonnegative starting point ( µ 0, q 0 ), the ynamics of the Distribute Energy inimization algorithm converges to the optimal value of Problem (B): lim m ( µ, q ), (6) lim m Θ, (7) where ( µ, q ) is a maximum point of D( µ, q). (b) Let the stepsizes {α m } be chosen as α m = κ m + ρ, (8) where κ an ρ are some positive scalars (note that the above efinition of {α m } satisfies the conitions (5)). If the longterm average of the vector ( f(m), t(m)) converges, i.e., lim f(m) = f, m= lim t(m) = t, (9) m= then the time-average version of ( f(m), t(m)) is the optimal solution to Problem (B): e E Θ( f e, t e) = Θ. The proof of Theorem 5 is inclue in our online technical report [6]. Remark: In the Distribute Energy inimization algorithm, t e (m), for example, can only be 0 or. From this perspective, our solution carries the same flavor as some of the relate work [], [4], namely, in the optimal solution, the resource, be it power or fraction of up-time, is use to the maximum extent, if the link is activate for the current time-slot. However, one woul expect the optimal solution of t e to Problem (B) to be a number anywhere from 0 to β for most of the links in a typical setting (cf. (8)). Theorem 5 reconciles the ifference between these two viewpoints. Although ( f(m), t(m)) is not a continuous mapping from the unerlying converging implicit costs (ual variables ( µ, q)), as long as the long-term average of primal variables ( f(m), t(m)) converges, the limit is an optimal solution to the primal problem. C. Power-Efficiency Ratio Setting β = or β = η in (4) gives necessary or sufficient conitions for scheulability, in terms of stabilizing the overall queueing system [4], [3]. From the necessary conition of β =, it is evient that the maximum loss in throughput uner the noe exclusive interference moel is η. This is the throughput loss ratio in approximating ( ) using Problem (B) with β = η. To erive the approximation ratio of our algorithm, the throughput loss nees to be translate into power loss. We apply the same first orer approximation of the rate-power function as in [5]. ore specifically, in an AWGN channel, the total power consumption can be approximate as B e e t e, where B e is a flow-relate constant inepenent of t e. Let ( f, t) be the optimal solution to Problem (B) with β =. It is evient that ( f, t +ε ), where ε is a small positive constant, is a feasible solution to Problem (B) with β = η. This feasible solution results in a power consumption that is at most (+ε)

7 TABLE I THE TWO FLOWS SUPPORTED BY THE NETWORK source estination ata rate paths flow 7 50 kbps -7, --7 flow kbps 3--6, Total Power (mw) Distribute Algorithm Offline Computation Time (secon) Power consumption from istribute algorithm an offline computa- Fig.. tion Fig.. Network topology that of the optimal value with β =. Since the optimal value of Problem (B) with β = is a lower boun on the minimum power from ( ), we conclue that the power-efficiency ratio of our algorithm is upper-boune by ( + ε). V. NUERICAL RESULTS In this illustrative example, we consier a 7-noe network, whose topology is epicte in Figure. The power-rate function is of the following form: [ = W log + σ ] ep e, N 0 W where W =.0 Hz is the available banwith, σ e = is the channel gain of link e, N 0 = mw/hz is the noise spectral ensity, p e is the transmission power, an is the resultant instantaneous ata rate of link e. This network supports two flows, as shown in Table I. The noe-exclusive interference moel is consiere, an β = ( ) in Problem (A). The length of each timeslot is secon. The results reporte in this section are the average over a moving time winow of length 0 secons. To show that the joint energy minimization algorithm can aapt to variations in the input parameters, we apply the following changes in the system setting. At time t = 4000s, the channel gain σ (,7) of the irect link between noe an noe 7 is ecrease from to At time t = 8000s, the ata rate of flow (from noe 3 to noe 6) is reuce from 500 kbps to 50 kbps. For each setting, offline computation is carrie out to fin the optimal value of Problem (A), which is given by the ashe line in Figure. The power consumption from the propose algorithm is shown as the soli line in the same figure. This simulation result shows that the Distribute Energy inimization algorithm is capable of computing the optimal solution to Problem (A) in a istribute manner, an Average Data Rate (kbps) Fig Link (,7): flow Link (,): flow Link (3,): flow Link (3,4): flow Time (secon) Average ata rates for ifferent flows on four links automatically tracking the optimal operating point once the system parameters change. Figure 3 shows the average ata rates f e for ifferent flows on four links. We now take a closer look at the routing of the flows. In the initial state, flow concentrates on the minimum energy path, namely, link (, 7). At t = 4000s, the channel gain σ (,7) reuces by 75%, an part of flow is shifte to path 7. Since the scheuling capacity of noe is saturate, a larger percentage of flow is then route through path At t = 8000s, the traffic that the network has to support between noe 3 an noe 6 reuces (flow is reuce to 50 kbps). As a consequence, part of the scheuling capacity of noe is free, an more of flow takes path 7 to reuce the overall power consumption. The above example shows that the interaction between routing, scheuling, an power control is relatively complex even in a wireless network of small size. The correct way to eal with such interaction is ifficult to summarize into general approaches such as minimum energy approach or loa balancing approach. In our Distribute Energy inimization algorithm, as shown in Figure an Figure 3, low-complexity

8 an istribute operation on each link accomplishes this joint optimization even in networks with non-stationarity. VI. CONCLUSION In this paper, we propose a joint power control, link scheuling, an routing algorithm to minimize the power consumption in multi-hop wireless networks. The known crosslayer solution to this problem is centralize, an with high computational complexity. In contrast, our algorithm is istribute, an with low computational complexity. We establish the power efficiency ratio of our istribute solution, an show that it is provably tighter than the power efficiency ratio of a centralize solution in the literature. We map our solution to corresponing components of the cross-layer control protocols, an iscuss the implication of our result on network protocol esign. The stability an optimality of our solution is verifie via simulations. In this work, we focus on the well-stuie noe exclusive interference moel. In a more general interference moel, we can efine Γ e to be the set of links that interfere with link e. (The noe-exclusive moel can be viewe as a special case with Γ e = N(x(e)) N(r(e)).) Let ω e enote the maximum number of links that can be scheule simultaneously in Γ e, an ωmax = max e ω e. It can be shown that a istribute maximal scheule can stabilize the system while the throughput is reuce by at most a factor of /ωmax. In this case, the left han sie of the scheuling constraint (8) in our problem formulation nee to be moifie accoringly [9] []. Nonetheless, the general methoology presente in this paper can be carrie through to this type of interference moel. APPENDIX Proof of Theorem 4: The proof technique here is similar to []. We first show the bouneness of the subgraient of the ual objective function. In (3), we choose H V max v T v, where V is the number of noes in the network. Let Ȳ enote the Cartesian prouct of all Y e in (3). It is clear that Ȳ Y, where Y is efine in Problem (B). Note that the introuction of H oes not change the solution. The optimal solution ( f, t ) to Problem (B) is in Ȳ for the following reason: Θ is a non-ecreasing function of fe, an therefore any ( f, t) with fe > H for some an e is clearly suboptimal even if all the traffic is route through link e. So the solution generate from the Distribute Energy inimization algorithm coincies with the optimal solution to Problem (B). The benefit from enforcing such an upper boun on fe is that the subgraient of the ual objective function is then boune. Given the subgraient of the ual objective function is boune, it can be shown [4] that, given any ε > 0, there exists some h > 0 such that, for any h h an any initial implicit costs ( µ 0, q 0 ), there exists a time 0 such that (3) hols for all m > 0. Therefore, as time inex m increases, ( µ(m), q(m)) converges to within a small neighborhoo of the set of the maximizer of D( µ, q) if the constant stepsize is chosen to be small enough. This implies that the sequence {( µ(m), q(m))} m is boune. To show (4), we consier the following Lyapunov function: V ( µ(m), q(m)) = [q v(m)] + µ v(m). v, The subgraient of D( µ, q) at time-slot m can be written as µ v (m) = t e (m) β, qv(m) = f e (m) v fe (m) T v. Using the above notation, the one-step rift of the Lyapunov function can be calculate as follows: V ( µ(m + ), q(m + )) V ( µ(m), q(m)) = { [ (µ v (m) + h µ v (m)) +] } µ v(m) v + { [(q v(m) + h qv(m) ) ] + [ q v (m) ] } v, (from (6) an (7)) { } [µ v (m) + h µ v (m)] µ v(m) v + { [q v (m) + h qv(m) ] [ q v (m) ] } h v v, µ v (m) µ v (m) + h v, q v(m) q v(m) + h W, where W is a constant large enough to guarantee the last inequality. It is possible to choose such a constant since the subgraient of D( µ, q) is boune. Aing h e E Θ( f e (m), t e (m)) to both sies of the above formula, we have V ( µ(m + ), q(m + )) V ( µ(m), q(m)) +h e E Θ( f e (m), t e (m)) h [ Θ( f e (m), t e (m)) + e E v + v, qv(m) q v(m) + h W = hd( µ(m), q(m)) + h W (from () an (5)) hd + h W = hθ + h W, µ v (m) µ v (m) where D is the maximum ual value, which is also equal to the minimum power Θ from the primal problem.

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