Joint Congestion Control, Routing and Media Access Control Optimization via Dual Decomposition for Ad Hoc Wireless Networks

Transcription

1 Joint Congestion Contro, Routing and Media Access Contro Optimization via Dua Decomposition for Ad Hoc Wireess Networks Francesco Lo Presti Dipartimento di Informatica Università de L Aquia opresti@di.univaq.it ABSTRACT In this paper we present a mode for the joint congestion contro, routing and MAC ink access for ad hoc wireess networks. We formuate the probem as a utiity maximization probem with routing and ink access constraints. For the soution we expoit the separabe structure of the probem via dua decomposition and the sub-gradient agorithm. The resuting agorithm directy transates into a distributed cross-ayer scheme for joint congestion contro, routing and ink scheduing of the wireess inks which revoves around ink ayer pricing. The convex probem formuation and the use of the sub-gradient agorithm ensures that the soution converges within an interva of the optima vaue. We iustrate the agorithm behavior through exampes. Categories and Subject Descriptors C.. [Computer-Communications Networks]: Network Architecture and Design Wireess Communications Genera Terms Performance, Design Keywords Congestion Contro, Ad Hoc Wireess Networks, Cross-Layer Design, Convex Optimization, Dua Decomposition.. INTRODUCTION In the ast few years, the fast growing demand for wireess services has greaty stimuated the research in the area of wireess networking. Today the issues under study range from congestion contro, node mobiity and routing, to MAC agorithms and power contro just to name a few. From the research perspective, wireess networking poses new chaenges with respect to traditiona wired networks: on one Permission to make digita or hard copies of a or part of this work for persona or cassroom use is granted without fee provided that copies are not made or distributed for profit or commercia advantage and that copies bear this notice and the fu citation on the first page. To copy otherwise, to repubish, to post on servers or to redistribute to ists, requires prior specific permission and/or a fee. MSWiM, October,, Montrea, Quebec, Canada. Copyright ACM //...$.. hand, there are distinctive issues which are not present in the wired counterpart, e.g., power contro; on the other hand, there are more traditiona topics, e.g., congestion contro, which need to be revisited in ight of the inherent characteristics of the wireess domain, as the reative scarcity of the radio resources, shared medium contention and the varying channe condition. Cross-ayer design has recenty received increased attention as an efficient soution to address these chaenges []. The basic idea is to do away with the strict ayer separation in the protoco stack design: rather than designing/optimizing the different ayers in isoation, cross-ayer design takes advantage of (possiby strict) coordination among the different ayers. In this paper we address the probem of congestion contro over a muti-hop ad-hoc wireess network as a cross-ayer design probem, whereby we formuate the congestion contro probem as a joint optimization with the routing and the ink scheduing probems at the ower network ayers. We consider a muti-hop ad hoc network composed of fixed wireess nodes. The network serves a set of traffic sources which send traffic to a set of destinations. Sources can adjust their transmission rate to changes in network conditions. Network nodes cooperate to route traffic aong one or more paths to destination. Since nodes communications take pace over a shared medium, it is interference imited. A suitabe MAC agorithm is used to schedue inks for transmission. Buiding on the utiity based optimization framework for congestion contro [,, 6, ], we formuate the optima congestion contro probem for the wireess network described above as an utiity based optimization probem. Each traffic source is associated with a utiity function which is concave and increasing with the transmission rate and subject to resource constraints; the objective is to maximize the sum of the sources utiities. The probem constraints comprise the routing and the MAC/physica channe constraints. As in [] we use a fow mode for traffic routing, aowing mutipe paths from source to destination. The constraints takes the form a fow conservation at network nodes []. The MAC/physica constraints, instead, account for the inks transmitting over a shared medium. We express these constraints in term of a ink contention graph which captures interference among inks []. Based on the proposed mode, the congestion contro probem takes the form of a convex optimization probem with

2 inear constraints. For the soution we resort to standard convexity theory resuts. In particuar, to take advantage of the probem structure, we sove the associated dua probem. The Lagrangian of the dua probem is separabe and naturay decomposes into different subprobems: a congestion contro subprobem for each fow, a routing subprobem and a ink access contro subprobem. Each subprobem has a we known structure: the congestion contro probem takes the form of a utiity-based user rate optimization []; the routing probem resuts into a minimum cost path routing probem; finay, the ink access probem takes the form of a scheduing probem. Since the dua function is nondifferentiabe, soution of dua cannot be soved via usua descent methods []. We resort to the sub-gradient method which guarantees convergence within an interva of the optima dua soution irrespectivey of the initia soution[]. Because of the subprobems structure, the computation can be carried out in a distributed way by network nodes. For our mode strong duaity hods; as a consequence, the soution of the dua provides an optima soution of the origina congestion contro probem. We must observe, though, that because the objective is not stricty concave, the soution of the probem might not be unique (it is sti possibe to consider a modified objective function which woud make the objective stricty convex and the soution unique). The agorithm that soves the dua probem can be decomposed into three distributed agorithms for congestion contro, routing and ink scheduing, respectivey. The key observation is that the agorithms directy transate into a cross-ayer scheme whereby the transport, the network and the ink ayer cooperate to drive the network to the optima operating point, i.e. the point which maximizes the probem objective function. The three ayers operations are coordinated by ink pricing as foows. At the MAC eve each ink has an associated price which evoves over time according to the aw of suppy and demand: when demand for transmission over a ink by the routing ayer exceeds the ink assigned capacity, the ink price increases, otherwise it decreases. At each iteration, the MAC ayer schedues inks for transmission (thus assigning capacity to each ink) according to ink prices and scheduing constraints as to maximize the price of the schedued inks. At the routing eve, the routing agorithm updates the routes using a minimum cost path agorithm with ink price as cost. Traffic can be spitted over minimum cost paths. Finay, at the transport eve, each source adjusts its rate as to maximize its net utiity. A the required computations can be carried out at the different ayers in a distributed way where nodes ony require oca information (but knowedge of the topoogy up to two-hop away is required to sove the ink scheduing probem though). We iustrate the agorithm behavior through exampes which iustrates the interactions between ink pricing, ink scheduing and routing. In this paper we do not consider power constraints, mobiity and/or fow dynamics. Power constraints can be easiy accounted for within this framework. Foowing the approach in [], we can express power constraints by introducing additiona inear constraints. The probem is again soved by dua decomposition. In this case, though, we have two pricing mechanism: the inks prices as above; and additiona prices which is associated to node power consumption. Both prices must be taken into account when adjusting fow rate via the congestion contro probem and fow routes via routing. Detais can be find in []. We have aso studied the agorithm behavior under time varying condition, fow dynamics and nodes mobiity through simuations. Our preiminary resuts are promising. A more quantitative anaysis wi be object of future work. There is a arge body of iterature on utiity based congestion contro schemes for wired networks (see [,, 6, 7, ] just to name a few). A these works assume traffic sources are associated with concave utiity functions increasing with the source transmission rate. The optimization probem ies in maximizing the aggregate utiity without vioating the the ink bandwidth constraints. The soution agorithms are based on the notion of resources prices which are set as function of the network resources congestion eve. Source nodes adjust their transmission rate as to maximize their net utiity. Depending on whether the agorithm is derived from the prima or dua formuation of the optimization probem, different congestion contro agorithms are then obtained. The resuts, first derived under the assumption of a fixed singe path for each fow, have been extended to the muti-path case. In this case, the anaysis is compicated by the non strict concavity of the objective function. The same framework has been more recenty appied to wireess networks [8,, 7, ]. The present paper is motivated by the recent work by Chen and a. [7]. In [7] the authors consider the congestion contro probem for mutihop ad hoc wireess networks and present a joint design for congestion contro and MAC contro under the assumption of fixed routing. The joint optimization is thus imited to ayers two and four. Our work extends the resuts in [7] by incuding the network ayer into the joint design and by considering muti-path routing. We show that a the three ayers can jointy cooperate to achieve system optimum and that the very same pricing mechanism is used for determine fow route and rates. The rest of the paper is organized as foows. In Section we present the system mode. In Section we appy duaity theory to derive a distributed agorithm for joint congestion contro, routing and MAC ink scheduing. We iustrate the agorithm behavior with numerica exampes in Section. Section concudes the paper.. MODEL AND PROBLEM FORMULATION We consider a static muti-hop ad-hoc wireess network. Ony nodes that are within the transmission range R of each other can communicate directy. A nodes communicate over the same ogica channe with capacity C. Because of the shared ogica channe, a successfu transmission precudes any node in the neighborhood of either the transmitter or the receiver from engaging in another simutaneous packet transmission/reception.. Network and Fow Mode We mode the network with a directed graph G = (N, L) (for the sake of simpicity we aways assume connected) where nodes represent the wireess stations and edges (i, j) L the communication inks between nodes. We assume that a communication ink between node i and j exists if and ony if node i and j are within the transmission range of each other, i.e., if and ony if they can communicate directy. For symmetry, we assume that if (i, j) L then (, i) L as we.

3 The network topoogy can be represented by a node-ink incidence matrix A, with entries A n, n N, L as foows if originates at n A n = if terminates at n otherwise. We mode network traffic by a set of end-to-end muti-hop fows F. Each fow f F is characterized by a source node s(f) and a destination node d(f). Fow traffic is routed by the network aong one or mutipe paths from source to destination. For each fow f, we denote by x f the amount of fow (measured in bit/s) injected into the network per unit of time and by y f the amount of fow f traversing ink per unit of time. We wi denote by x = (x f ) f F and by y f = (y f ) L the fow rate vector and fow f ink rate vector, respectivey. Uness otherwise stated, we wi assume vectors are aways coumn vectors. For a vector z we wi denote by z T its transpose. We assume ossess transmission. For a node n, denote by O(n) and I(n) the set of outgoing and incoming inks, respectivey. The fow conservation aw impies that for f F, n N: y f x f if n = s(f) y f = x f if n = d(f) otherwise. () O(n) I(n) () can be compacty written as () A y f = w f, f F () where w f = (w fn ) n N, and w fn denotes the amount of fow f traffic injected/removed from the network at node n, i.e., and w fn = x f if n = s(f), w fn = x f if n = d(f) and w fn = otherwise.. Link Contention Mode Because of the shared ogica channe, transmission over different inks causes interference when the sender or the receiver of one is within the interference range of the sender or the receiver of the other. Link contention can be represented by means of a contention graph. In this ink contention graph, each vertex represents a wireess ink. An edge between two nodes denotes that transmission aong those inks contend which each other and those inks cannot be active at the same time. An accurate ink contention graph depends on protocos detais. In this paper, as in [, 7, ] we assume that transmission over two inks cause contention when either ends of one of the two inks fa within carrier sense range of the either end of the other inks. For a graph G, the ink contention graph is then the undirected graph H = (N H, L H ), where. N H = L;. ((i, j), (i, j )) L H iff (i, j ) L such that {i, j} {i, j } = and {i, j } {i, j } = As an exampe, consider the wireess network in Figure. The graph has inks (each segment counts twice, one ink for each direction). For the sake of simpicity, et us ony consider ony the inks directed from eft to right. Figure shows the corresponding ink contention graph. Links contention over the shared medium imits the achievabe inks transmission rates. We now provide two different types of constraints on the inks transmission rate. The first ones, the feasibiity constraints, provide a compete set of constraints, but are difficut to use in practice. As in [7] we wi utimatey resort to the second type of constraints, the cique feasibe constraints (9), which abeit being approximate for a genera graph, are more easy to dea with... Feasibiity Constraints To derive the constrains imposed by the MAC ayer, observe first that at, any given time, ony inks that form an independent set in the ink contention graph can transmit simutaneousy without causing interference. For the previous exampe, we distinguish seven independent sets: I = {(, ), (, 6)}, I = {(, )}, I = {(, )}, I = {, }, I = {, }, I 6 = {(, )} and I 7 = {(, 6)} (the ast two are not maxima in that they are subset of other independent sets). Thus ony inks (, ) and (, 6) can transmit simutaneousy. For transmitting, any other ink, e.g., ink (, ), requires a other ink to be sient. For a ink contention graph H, et I be the set of a independent sets of H. A transmission schedue over the network S can be defined as in infinite sequence of independent sets I, I,..., I k,..., I k I. Given a schedue S, ink L average transmission rate in schedue S is c = im t t k= S(, k) t where S(, k) = C if I k and S(, k) = otherwise. A rate vector c = (c ) L is feasibe if there exists a schedue such that ink rate is c. In other words, a ink rate vector c is feasibe, if and ony if, it is possibe to schedue groups of inks for transmission over time - with each group corresponding to an independent sets in the ink contention graph - so that the average transmission rate rate defined by () equas c. For a genera contention graph, Theorem in [] (actuay [] deas with ink frequencies which can be regarded as normaized rates) provides a characterization of feasibe rate vectors. Let r I = (r I ) L the characteristic vector of the independent set (of rates) I I, with r I = C if I and r I = otherwise. The feasibe rate region R is then defined as the convex hu of the independent sets: () R = { r r = I airi, ai, I I ai = } () R is a cosed and connected compact space. The feasibiity (MAC) constraint takes the form c R. (6).. Cique Constraints We now ook at the same probem from another viewpoint. Consider the maxima ciques in H. A maxima cique Q = (N Q, L Q) in H is a maxima compete subgraph of H. A maxima cique in the ink contention graph denotes a distinct contention region because at any time ony one ink (i, j) N Q can be in transmission. Each maxima cique thus represents a distinct channe resource of capacity C with the upstream nodes of the inks in Q contending for access to it. Hence, ony inks that are in different ciques can transmit simutaneousy (observe that inks in an independent set a beong to different ciques). Ciques deter-

4 6 We associate a utiity function U f (x f ) with each fow f F representing the user utiity when transmitting at a rate x f. We assume U f continuousy differentiabe and stricty concave. We further assume utiity is additive so that the aggregate traffic utiity is U(x) = f U f (x f ). Our goa is to determine the source rates which maximizes the overa utiity U(x) subject to the routing and MAC constraints Figure : Ad Hoc Wireess Network. Sma Network Topoogy (,) (,) (,) Figure : Sma Network Topoogy. Contention Graph. mine constraints on the inks transmission rate. Consider a ink L. needs to transmit for a fraction c of time to C sustain an average transmission rate of c. Since inks within the same cique cannot transmit simutaneousy we obtain the foowing constraints c Q Q. (7) C N Q Define the cique contention matrix F with entries { /C if NQ F Q = otherwise and rewrite (7) as (,) (,) (,6) (8) F c, (9) where denotes a Q dimensiona unit vector. A rate vector c is said to be cique feasibe if c F [], where F = {c F c }. () The notions of feasibiity and cique feasibiity differ. Indeed, it is possibe to show that (9) is a necessary but not sufficient condition for c to be feasibe (this is due to the fuid-eve type of argument to derive the constraints, see []). Hence, in genera, R F. Nevertheess, for the cass of perfect graphs, and ony for the cass of perfect graph, feasibiity and cique feasibiity coincide, i.e., (9) is aso sufficient and R = F.. Probem Formuation A graph is perfect if and ony if it has no induced subgraph that is isomorphic to an odd cyce of ength at east five without chords, or the compement of such cyce [9]. P : max U(x) = f U f (x f ) subject to: Ay f = w f f F f y f c c = I a Ir I I ai =, ai y f () where the additiona constraint f y f c simpy requires that the aggregate traffic over each ink does not exceed the ink transmission rate.. JOINT CONGESTION CONTROL, ROUT- ING AND MAC DESIGN VIA DUAL DE- COMPOSITION In this section we present our agorithm to sove the system probem based on Lagrangian duaity and show how it can be impemented in a distributed way.. Dua Probem Probem P is a convex optimization probem. The feasibe set is a convex and compact space. The objective function () is stricty concave in x f but it is not stricty concave in {x f, y f, c, a}. As a consequence, we do not have, in genera, a unique soution. Direct soution of P woud require a centra entity with compete knowedge (from the wireess network topoogy to the user utiity functions) and woud not be of practica interest in a wireess network. Here we take advantage of the probem structure and derive a distributed agorithm via dua decomposition. There are many way to formuate the dua probem of P depending on which Lagrangian mutipiers we introduce (when not a mutipiers are considered we sha speak of partia duaity). Here we form the dua probem by reaxing (introducing Lagrangian mutipiers) the constraints f y f c. This resuts in the partia Lagrangian L(x, y, c, p) = f = f U f (x f ) p T ( f y f c) (U f (x f ) p T y ) + p T c () with mutipiers p = (p ) L. The dua probem associated with P is then D : min D(p) = subject to p with partia dua function D(p) = max L(x, y, c, p) x,y,c,a Ay f = w f f F c = I a Ir I I a I =, a I y f () () Since the dua function is aways convex, D is a convex optimization probem. Moreover, since Sater s conditions for

5 constraints quaifications are satisfied [6, ] ( P has concave objective and affine constraints) strong duaity hods; the optima vaues of the dua and the prima probems are thus equa, and we can sove the atter via soution of the former. We now consider the two key ingredients for the soution of the dua, namey a method for computing the dua function D(p) and the iterative soution strategy for soving the the dua probem D.. Dua Function Evauation Since the Lagrangian () is separabe in the fow variabes x f, y f, f F, and the ink variabes c, the evauation of dua function () naturay decomposes in the foowing subprobems where D(p) = f D NET f (p) = max x f,y f D MAC(p) = max c,a D NET f (p) + D MAC(p) () { U f (x f ) p T y f Ay f = w f y f { p T c } (6) c = I a } Ir I I ai =, ai.(7) The first set of subprobems are user rate optimization probems []. They differ from the formuation in [] in that rate optimization is couped with the routing probem. The ast subprobem is the MAC ink scheduing probem. Computation of D NET f (p) requires the soution of the foowing probem NET f max U f (x f ) p T y f subject to: Ay f = w f y f. (8) NET f is an uncapacitated fow probem with concave objective. If we interpret the dua variabe p as the price per unit bandwidth at ink, then p T y f = p y f is the price to transmit fow f traffic at rate x f and to route it aong the network according to y f. The objective function U f (x f ) p T y f is thus user f net utiity. The constraints are fow f conservation aw: traffic is generated at rate x f at node s(f), and without oss traverses the network to the destination d(f) via a possibe paths. Probem NET f thus consists in determining the traffic rate x f and routing it to destination (by determining the vector y f ) as to maximize fow net utiity. The soution of NET f is quite simpe. Because there are no inks capacity constraints, for any rate x f the net utiity is maximized when a traffic is sent aong minimum cost paths (sending any fraction of traffic aong non minimum cost paths woud increase the cost). The soution of NET f is computed in two steps:. Compute the optima source rate x f (p) as the maximizer of U f (x f ) x f p(f) (where by p(f) we denote the cost of a minimum path from s(f) to d(f) for the given set of prices p). We readiy obtain x f (p) = U f (p(f)) (9) where U is the derivative of the inverse of the utiity function.. Spit the traffic among the minimum cost paths. If there is a unique minimum cost path the soution is unique as we, with a traffic routed aong this path. If there are mutipe minimum cost paths, we have infinite soutions corresponding to a possibe ways to spit traffic. In our exampes we equay divided traffic among a minimum cost paths. NET f defines both congestion contro and routing behavior. Routing is based on minimum cost path with the ink prices as costs. Mutipe paths can be determined by means of a suitabe muti-path routing protoco (see [, 9] for proposas). Congestion contro is based on (9) with source f adjusting its rate according to the path price p(f). For a distributed impementation, the traffic source must earn about the price p(f). One simpe soution to convey the price is to use packet marking, e.g., using congestion-indication bit marking scheme as Random Exponentia Marking (REM)[] or Sef-normaized Additive Marking (SAM) []. We now turn to our attention to D MAC(p). Computation of D MAC (p) is given by the soution of the probem MAC max p T c subject to: c = I a Ir I I a I =, a I. () MAC is the ink scheduing probem and takes the form of a inear program where the objective is the weighted sum of the ink transmission rates with the ink prices p as weights. For this probem, a soution is provided by a maxima independent set r I. To see this observe that the theory of inear programming ensures that a soution can be found ooking ony at the extreme points of R. Since R is a convex hu of the independent sets r I, its extreme points are the independent sets themseves. Let p I = I p denote the price associated with an independent set. It is not difficut to verify that the optimizer c (p) satisfies c (p) = arg max ri p I, () i.e., the soution of MAC is given by the independent set of maxima price. Since prices are non-negative, it is aways possibe to take an maxima independent set as soution. Soution of MAC requires knowedge of the independent set r I. This is difficut to determine in a distributed manner as required for impementation in an ad-hoc network. An approach which eads to a distributed soution consists in repacing the feasibe constraints c R with the cique feasibe constraints c F, and presented in [7], is described ater in this section. It must be noted, though, that such an approach may ead to approximate soution when the ink contention graph is not perfect.. Dua Probem Soution via Subgradient Given the convexity of dua functions [], a natura strategy to the soution of dua probems ies in the use of the descent methods. The situation is somewhat compicated by the the fact that U(x) is not stricty concave in a variabes {x f, y f, c, a}. As a consequence, D(p) is ony piecewise differentiabe, and the usua gradient methods, which requires the objective function to be differentiabe, cannot be used. There are severa methods to sove non-differentiabe convex probem. Here we resort to the subgradient method. The choice is motivated by its simpicity and the possibiity to impement it in a distributed way.

6 Given a non-differentiabe convex function V, a vector h is a subgradient of V at a point p if V (q) V (p) + h T (q p) () for a q. For p, et x f (p) and y f (p) be an optima soution of NET f, f F, and c (p) an optima soution MAC. A subgradient h of D(p) at p is then [] h(p) = f y f (p) c (p). () The subgradient method generates a sequence of dua feasibe points p (k) using subgradients h (k) = h(p (k) ). Here we use the simpest form p (k+) = = [ p (k) p (k) ] + γ (k) h (k) () γ (k) ( f y f (p (k) ) c (p (k) )) () where [.] + denote projection on the non-negative orthant and γ (k) the stepsize (possiby step-dependent). Convergence of the sub-gradient method depend on the stepsize choice. For a constant stepsize γ, which is here of interest for practica purposes, the agorithm is guaranteed to converge to within a interva of the optima vaue. More specificay, if the norm of the subgradient is bounded, i.e., h (k), as is in our case since both y f and c are bounded, then D(p (k) ) converges within γ of the optima vaue D(p ) []. It is important to observe that () can be be computed in a distributed way, independenty by each node, since it is based on oca information ony, namey, for ink, the actua oad f y f(p (k) ) and the aocated capacity c (p (k) ). We concude by observing that the agorithm has a natura interpretation in terms of aw of suppy and demand. When ink is under-utiized, i.e., when f y f < c the price p is decreased, otherwise it is increased. The pricing adjustment wi induce the sources to adjust their rates, the network to reroute traffic (both source and network decisions are obtained via the soution of NET f), and the ink ayer to schedue ink for transmission (by soving MAC).. Distributed Scheduing Agorithm To derive a distributed scheduing agorithm, we foow the approach in [7]. The first step ies in repacing the feasibe constraint with the cique feasibiity constraint F c. Since R F (with the equaity hoding ony for perfect graph), a soution to the modified probem may not be feasibe when the graph is not perfect. In such case, the optimizer is not necessariy an an independent set. For a distributed agorithm, we consider the dua probem of MAC. Observe, though, that because of the inear objective function soving the dua probem woud not provide the prima soution (which is the ink scheduing we are trying to compute). To circumvent this probem, in [7] the objective function is made stricty concave by subtracting the term δc T c where δ is a sma positive constant. The scheduing probem thus becomes MAC cique (δ) max p T c δc T c subject to: F c c + (6) Figure : Simuation Resuts: Average Rate over each Link. The thickness is proportiona to the ink average rate. As δ approaches zero, the soution approaches a soution of the origina probem. The dua probem is then where λ = (λ Q ) Q Q and L : L(λ) = max c min L(λ) = (7) subject to λ (8) { } p T c δc T c λ T (F c ). (9) The gradient agorithm then yieds, component-wise, [ λ (k+) Q = λ (k) Q + β( F Q c (λ (k) ) )] + () where β is a positive stepsize and c (λ (k) ) is the maximizer of L(λ (k) ). From (9) after some agebra we readiy obtain c (λ (k) ) = [ p Q λ(k) Q δ Convergence requires β to satisfy [] < β < δ O Q F Q ] + () () where O and Q denotes the maxima size of a cique and maximum number of ciques containing the same ink. Computation of () and () can be carried out in a distributed way. For ink = (i, j), we assume () is carried out by node i. For a cique Q Q, () can be carried out by (some or a) nodes which are transmitting over a ink beonging to the cique, i.e. by nodes n such that O(n) N Q. In order to compute () and () a nodes need to find out which ciques they beong to and a rates of the inks in these ciques. This can be accompished by having nodes distributing connectivity and ink rate information up to two-hops away []. This is sufficient for each node to compute a ciques its outgoing inks beong to e.g., using the Bierstone agorithm [], and to acquire the rates of the inks forming these ciques. A probem with this soution is that the scheduing agorithm ()-() just soves a singe instance of MAC and thus must be executed at faster time scae to ensure convergence within an iteration of the dua agorithm. This poses a ower bound on the iteration duration.

7 Source Rate (Mbit/s)..... Dua Function D(p) Figure : Simuation Resuts: Source Rate and Dua Function with stepsize γ =.. k = nt k = nt + k = nt + k = nt + k = nt + k = nt + Figure : Simuation Resuts for Figure : Link Scheduing. The optima schedue is periodic with period 6.. NUMERICAL EXAMPLES In this section we iustrate the behavior of the agorithm through numerica exampes. We impemented the agorithms presented in the previous Section with Matab [8]. In the exampes presented, we assume a utiity functions are ogarithmic U f (x f ) = og(x f ). This ensures that the resuting fow rates are proportionay fair []. Channe capacity is assumed to be C=Mbit/s. We first consider the sma wireess network of Figure. We assume there is ony one fow originating at node and terminating at node 6. For this network the optima fow rate is C. Figure shows the fow rates and the dua function evoution over time for γ =.. Here we assume one iteration for unit of time. Both rates and dua function approaches the optima vaues very fast but not monotonicay. The osciations are due to the non-differentiabiity of the objective function and the use of the subgradient method with constant step size which ensures the dua approaches the optima dua vaue p but, rather than reaching it, osciates around it, exhibiting a imit-cyce behavior. As we wi show ater, the osciations we observe in steady-state are a consequence of the scheduing process and are thus inherent to the agorithm behavior. Figure shows the average amount of traffic carried over each ink (ink thickness is proportiona to amount of transported fow). As expected, traffic is eveny divided between the two avaiabe routes. To understand how traffic is actuay spitted over time, we take a cose ook at the ink scheduing. As shown in Figure, the optima schedue S =..., I, I, I, I, I, I,... is periodic with period T = 6. Traffic is thus aternativey routed over the two avaiabe routes. At each iteration, transmission is granted to the maxima independent set which maximizes max I p I. The mechanism that makes a different independent set chosen for transmission at each iteration is easiy expained in terms of the price update, whereby inks schedued for transmission (and the independent set they form) have their price decreased, whie the other inks (and the corresponding independent sets) have the price increased. Now we turn to a arger exampe in Figure 6, which has been generated by randomy pacing nodes in a unit square. A ink exists between two nodes if their distance is beow the threshod. (chosen as the smaest vaue which ensures that the graph is connected). The graph has nodes and 8 inks (corresponding to the 6 bidirectiona inks shown in the figure). We randomy chose source-destinations pairs. Figure 8 and 9 show the fow rates x f for the different fows and the dua function evoution for different vaues of the stepsize. We observe that a rates converge at the same speed, with faster convergence, but aso bigger osciations for the bigger stepsize. Figure 7 show the average amount of traffic transported over each ink (thicker ink transport more traffic). For a set of representative fows, in Figure we show the routes used and the amount of traffic aong each ink. Source and destination are indicated by the two bigger nodes. We observe that, for each fow, traffic foows mutipe paths from source to destination. Here, though, traffic is not eveny spit among routes: the argest fraction of traffic is aways sent aong the shortest routes (measured in terms of hops). This is no surprise as the onger routes are more ikey to share inks with routes of other fows and are

8 Figure 6: Ad Hoc Wireess Network. Topoogy of a randomy generated topoogy with nodes and 6 bidirectiona inks. Figure 7: Simuation Resuts: Average Rate over each Link. The thickness is proportiona to the ink average rate. Source Rate (Mbit/s)..... Dua Function D(p) Figure 8: Simuation Resuts: Source Rate and Dua Function with stepsize γ =.. thus can sustain smaer rates. We observed this behavior throughout a our experiments.. CONCLUSIONS We have presented a mode for the joint congestion contro, routing and MAC ink access for ad hoc wireess networks. We have formuated the congestion contro probem as a convex optimization probem with routing and ink access constraints. We have soved the probem via dua decomposition and sub-gradient agorithm. More interestingy, the resuting agorithm directy transated into a distributed cross-ayer scheme for joint congestion contro, routing and ink scheduing of the wireess inks which revoves around ink ayer pricing. Much work remains to be done in order to make this scheme practica. Here we have assumed idea condition with strict synchronization among nodes and among ayer operation, no propagation deay, no computation and signaing overhead. We pan to address these issues in future work. 6. REFERENCES [] S. Athuraiya, V. H. Li, S. H. Low and Q. Yin, REM: Active Queue Management, IEEE Networks, vo., no., pp. 8-,. [] M. Ader, J. Y. Cai, J. K. Shapiro and D. Towsey, SAM: Estimation of Congestion Price Using Probabiistic Packet Marking,, in Proc. of IEEE Infocom, Apri. [] J. G. Augustson and J. Minker, An Anaysis of Some Graph Theoretica Custer Techniques, Journa of the ACM, vo. 7, no., pp. 7-86, 97. [] A. Bar-Noy, A. Mayer, B. Schieber and M. Sudan, Guaranteeing Fair Service to Persistent Dependent Tasks, SIAM J. Computing, Vo. 7, no., pp , August 998. [] D. Bertseaks, Noninear Programming. Athena Scientific, second ed., 999. [6] S. Boyd and L. Vandenberghe, Convex Optimization. Cambridge University Press,. [7] L. Chen, S. H. Low and J. C. Doye, Joint Congestion Contro and Media Access Contro Design for Ad Hov Wireess Networks, in Proc. of IEEE Infocom, March. [8] M. Chiang, To ayer or not to ayer: baancing transport and physica ayers in wireess muti-hop networks, in Proc. IEEE Infocom, March. [9] G. Cornuejos, The strong perfect graph conjecture, in Internationa Congress of Mathematicians, Bejing, China,. [] A. Nasipuri and S. Das, On-Demand Muti-path Routing for Mobie Ad Hoc Networks, in Proc. of the IEEE Conference of Computer Communications and Networks, (Boston, MA), October 999. [] Z. Fang and B. Bensaou, Fair Bandwidth Sharing Agorithms based on Game Theory Frameworks for Wireess ad-hoc Networks, in Proc. IEEE Infocom, March.

9 Source Rate (Mbit/s)..... Dua Function D(p) Figure 9: Simuation Resuts: Source Rate and Dua Function with stepsize γ =.. Figure : Simuation Resuts for Figure 6: Routes used for seected source-destination pairs. Link thickness is proportiona to the amount of transported traffic. [] F. P. Key, Charging and Rate Contro for Eastic Traffic, European Transactions on Teecommunications, vo. 8, no., pp. -7, 997. [] F. P. Key, A. K. Mauoo and D. K. H. Tan, Rate Contro for Communications Networks: Shadow Prices, Proportiona Fairness and Stabiity, Journa of Operations Research Society, vo. 9, no., pp. 7-, March 998. [] S. Kunniyur and R. Srikant, End-to-end congestion contro schemes: Utiity functions, random osses and ECN marks, IEEE/ACM Transactions on networking, vo., no., pp , October. [] F. Lo Presti, Cross Layer Optimization via Dua Decomposition for Ad Hoc Wireess Networks, CS-Technica Report, Università de Aquia, in preparation. [6] S. H. Low and D. E. Lapsey, Optima fow contro, I: Basic agorithm and convergence, IEEE/ACM Transactions on networking, vo. 7, no. 6, pp , December 999. [7] S. H. Low, Muti-path optimization fow contro, in Proc. IEEE Int. Conf. Networks,. [8] Matab version 6. [9] J. Raju and J. Garcia-Luna-Aceves, A New Approach to On-Demand-Loop-Free Muti-path Routing, in Proc. of the IEEE Conference of Computer Communications and Networks, (Boston, MA), October 999. [] S. Shakkottai, T. S. Rappaport and P. C. Karsson, Cross ayer design for wireess networks, IEEE Communications Magazine, Apri. [] V. Srinivasan, C. F. Chiasserini, P. Nuggehai, R. R. Rao, Optima Rate Aocation for Energy Efficient Muti-path Routing in Ad Hoc Wireess Networks, IEEE Transactions on Wireess Communications, vo., pp , May. [] L. Xiao, M. Johansson, S. Boyd, Simutaneous Routing and Resource Aocation via Dua Decomposition, in Proc. of IEEE Infocom, March. [] Y. Xue, B. Li and K. Nahrstedt, Price Based Resource Aocation in Wireess Ad Hoc Networks, in Proc. of Eeventh Internationa Workshop on Quaity of Service (IWQoS ), (Monterey, CA), June,. [] Y. Yi and S. Shakkottai, Hop-by-hop congestion contro over a wireess muti-hop network, in Proc. IEEE Infocom, March.