Cross-layer design in multihop wireless networkslich1539/papers/Chen-2011-CLD...constraints at link layer for congestion control over multi-hop wireless networks. Xiao et al. [44]

Computer Networks 55 (2011) 480–496

Contents lists available at ScienceDirect

Computer Networks

journal homepage: www.elsevier .com/locate /comnet

Cross-layer design in multihop wireless networks

Lijun Chen ⇑, Steven H. Low, John C. DoyleEngineering and Applied Science Division, California Institute of Technology, Pasadena, CA 91125, USA

a r t i c l e i n f o a b s t r a c t

Article history:Available online 12 October 2010

Keywords:Theory-based approachCross-layer designDual decompositionMultihop wireless networks

1389-1286/$ - see front matter � 2010 Elsevier B.Vdoi:10.1016/j.comnet.2010.09.005

⇑ Corresponding author.E-mail address: [email protected] (L. Chen).

In this paper, we take a holistic approach to the protocol architecture design in multihopwireless networks. Our goal is to integrate various protocol layers into a rigorous frame-work, by regarding them as distributed computations over the network to solve some opti-mization problem. Different layers carry out distributed computation on different subsetsof the decision variables using local information to achieve individual optimality. Takentogether, these local algorithms (with respect to different layers) achieve a global optimal-ity. Our current theory integrates three functions—congestion control, routing and sched-uling—in transport, network and link layers into a coherent framework. These threefunctions interact through and are regulated by congestion price so as to achieve a globaloptimality, even in a time-varying environment. Within this context, this model allows usto systematically derive the layering structure of the various mechanisms of different pro-tocol layers, their interfaces, and the control information that must cross these interfaces toachieve a certain performance and robustness.

� 2010 Elsevier B.V. All rights reserved.

1. Introduction

The success of communication networks has largelybeen a result of adopting a layered architecture. With thisarchitecture, its design and implementation is divided intosimpler modules that are separately designed and imple-mented and then interconnected. A protocol stack typicallyhas five layers, application, transport (TCP), network (IP),data link (include MAC) and physical layer. Each layercontrols a subset of the decision variables, hides thecomplexity of the layer below and provides well-definedservices to the layer above. Together, they allocatenetworked resources to provide a reliable and usuallybest-effort communication service to a large pool of com-peting users.

However, the layered structure addresses only abstractand high-level aspects of the whole network protocoldesign. Various layers of the network are put togetheroften in an ad hoc manner, and might not be optimal asa whole. In order to improve the performance and achieve

. All rights reserved.

efficient resource allocation, we need to understand inter-actions across layers and carry out cross-layer design.Moreover, in wireless networks there does not exist a goodinterface between the physical and network layers. Wire-less links are unreliable and wireless nodes usually relyon random access mechanism to access the wireless chan-nel. Thus, the performance of link layer is not guaranteed,which will result in performance problems for the wholenetwork such as degraded TCP performance. So, we needcross-layer design, i.e., to exchange information betweenphysical/link layer with higher layers in order to achievebetter performance.

Motivated by the duality model of TCP congestion con-trol [16,22,18,23], one approach to understand interactionsacross layers is to view the network as an optimization sol-ver and various protocol layers as distributed algorithmssolving an optimization problem. This approach and theassociated utility maximization problem were originallyproposed as an analytical tool for reverse engineeringTCP congestion control where a network with fixed linkcapacities and prespecified routes is implicitly assumed.A natural framework for cross-layer design is then toextend the basic utility maximization problem to include

http://dx.doi.org/10.1016/j.comnet.2010.09.005

mailto:[email protected]

http://dx.doi.org/10.1016/j.comnet.2010.09.005

http://www.sciencedirect.com/science/journal/13891286

http://www.elsevier.com/locate/comnet

L. Chen et al. / Computer Networks 55 (2011) 480–496 481

decision variables of other layers, and seek a decomposi-tion such that different layers carry out distributed compu-tation on different subsets of decision variables using localinformation to achieve individual optimality, and takentogether, these local algorithms achieve the global opti-mality. This approach has come to be known as layering-as-optimization-decomposition; see [6] for an extensivesurvey.

We apply this approach to design an overall frameworkfor the protocol architecture in multihop wireless net-works, with the goal of achieving efficient resource alloca-tion through cross-layer design. We first consider thenetwork with fixed channel or single-rate devices, andformulate network resource allocation as a utility maximi-zation problem with rate constraints at the network layerand schedulability constraints at the link layer. We thenapply duality theory to decompose the system problemvertically into congestion control, routing and schedulingsubproblems that interact through congestion prices.Based on this decomposition, a distributed subgradientalgorithm for joint congestion control, routing and sched-uling is obtained, and proved to approach arbitrarily closeto the optimum of the system problem. We next extendthe dual subgradient algorithm to wireless multihop net-works with time-varying channels and adaptive multi-ratedevices. The stability of the resulting system is proved, andits performance is characterized with respect to an idealreference system. We finally apply the general algorithmto the joint congestion control and medium access controldesign over the network with single-path routing and tothe cross-layer congestion control, routing and schedulingdesign in the network without prespecified paths.

Our current theory integrates three functions—conges-tion control, routing and scheduling—in transport, networkand link layers into a coherent framework. While the inte-gration of all protocol components remain a big challenge,this framework is promising to be extended to provide amathematical theory for network architecture, and allowus to systematically derive the layering structure of thevarious mechanisms of different protocol layers, theirinterfaces, and the control information that must crossthese interfaces to achieve a certain performance androbustness. We also present a general technique and re-sults regarding the stability and optimality of the dualalgorithm in face of time-varying parameters. As the flowcontention graph that will be used to characterize feasiblerate regions of the networks is a rather general construc-tion and can be used to capture the interdependence orcontention among parallel servers of any queueing net-works, these results are applicable to any systems thatcan be modelled by a general model of queueing networkthat is served by a set of interdependent parallel serverswith time-varying service capabilities.

The remainder of this paper is organized as follows. Thenext section briefly discusses related work. Section 3 pre-sents details of the system model for the network withfixed channel or single-rate devices, and Section 4 presentsa distributed algorithm for joint congestion control, rout-ing and scheduling via dual decomposition. Section 5extends the dual algorithm to handle the network withtime-varying channel and adaptive multi-rate devices. As

specific cases of the general model and algorithm devel-oped in Sections 4–6 discusses joint congestion controland medium access control design in multihop wirelessnetworks with single-path routing, and Section 7 discussescross-layer congestion control, routing and scheduling de-sign in the network without prespecified paths. We con-clude in Section 8.

2. Related work

The utility maximization framework [16,22,18] on TCPcongestion control has been extensively applied andextended to study protocol design, especially congestioncontrol (see, e.g., [48,49]), fair channel access (see, e.g.,[25,39,19,9,35,8]), and cross-layer design (see, e.g.,[44,5,20,3,4]), in wireless networks. Xue et al. [48] and Yiet al. [49] are among the first to formulate schedulabilityconstraints at link layer for congestion control over multi-hop wireless networks. Xiao et al. [44] study joint routingand resource allocation, and are among the first to applydual decomposition to cross-layer design in wireless net-works. Chiang [5] is among the first to study joint conges-tion and power control. Lin et al. [20] and Chen et al. [3] areamong the first to study joint congestion control andscheduling. Chen et al. [3] and Wang et al. [41] are amongthe first to study cross-layer design in the network withcontention-based medium access.

The work presented in Section 6 (see also [3]) is origi-nally motivated to solve TCP unfairness problem over mul-tihop wireless networks; see, e.g., [10,37,45–47]. Themodel used in Section 7 (see also [4]) is motivated by Neelyet al. [26] that studies dynamic power control and routingfor time-varying wireless networks and by Hajeck et al.[11] and Kodialam et al. [17] that study joint routing andscheduling to determine the achievable rates in multi-hop wireless networks; similar decomposition for thenetwork with deterministic wireless channel has also beenrevealed in the journal version of [26] and in [20].

The utility maximization in time-varying wireless net-works is first studied in the context of fair scheduling. Ithas been shown that a family of primal scheduling algo-rithms maximize the sum of the utilities of the long-runaverage data rates provided to the users; see, e.g.,[39,19,35]. In contrast, the result presented in Section 5(see also [4]) is for the dual algorithms. An earlier resultfor the dual scheduling algorithm is by Eryilmaz et al. [8]that studies fair resource allocation using queue-lengthbased scheduling and congestion control. Another similarresult is by Neely et al. [27] that studies fairness and opti-mal stochastic control for heterogeneous networks. Allthese three works use stochastic Lyapunov method toestablish stability, but the technical details are somewhatdifferent. Especially, the stability and optimality result pre-sented in Section 5 is based only on general properties ofconvexity and the definition of subgradients, and can be di-rectly applied to a variety of time-varying systems that canbe solved or modelled by the dual algorithms. Anothercomparable result is by Stolyar [36] that proposes greedyprimal–dual algorithm to maximize network utility. It usesa very different technique to establish optimality.

DCA B

1

2

3

4

5

6

482 L. Chen et al. / Computer Networks 55 (2011) 480–496

Here, we focus on dual decomposition, but there aremany different ways to decompose a given problem, eachof which corresponds to a different layering scheme. Seethe survey article [6] and the references therein for variousrecent work on cross-layer design.

1

2

3

4

5

6

Fig. 1. Example of a multihop wireless network with 4 nodes and6 logical links and the corresponding flow contention graph.

3. System model

Consider a multihop wireless network with a set N ofnodes and a set L of directed logical links. We assume a sta-tic topology and each link l 2 L has a fixed finite capacity cl

bits per second when active, i.e., we implicitly assume thatthe wireless channel is fixed or some underlying mecha-nism is used to mask the channel variation so that thewireless channel appears to have a fixed rate. This assump-tion will be relaxed in Section 5. Wireless channel is inter-ference limited, where links contend with each other forexclusive access to the channel. We will use the flow con-tention graph to capture the contention relations amonglinks. The feasible rate region at link layer is then a convexhull of the corresponding rate vectors of independent setsof the flow contention graph. We will further describe rateconstraints at the network layer by linear inequalities interms of user service requirements and allocated linkcapacities. The resource allocation of the network is thenformulated as a utility maximization problem with sched-ulability and rate constraints.

3.1. Flow contention graph and schedulability constraint

The interference among wireless links is usually speci-fied by some interference model that describes physicalconstraints regarding wireless transmissions and success-ful receptions. For example, in a network with primaryinterference, links that share a common node cannot trans-mit or received simultaneously but links that do not sharenodes can do so. It models a wireless network with multi-ple channels where simultaneous communications in aneighborhood are enabled by using orthogonal CDMA orFDMA channels. In a network with secondary interference,links mutually interfere with each other whenever eitherthe sender or the receiver of one is within the interferencerange of the sender or receiver of the other. Given an inter-ference model, we can construct a flow contention graphthat captures the contention relations among the links;see, e.g., [25]. In the contention graph, each vertex repre-sents a link, and an edge between two vertices denotesthe contention between the corresponding links: two linksinterfere with each other and cannot transmit at the sametime. Fig. 1 shows an example of a simple multihop wire-less network with primary interference and the corre-sponding flow contention graph.

Given a flow contention graph, we can identify all itsindependent sets of vertices. An independent set is a setof vertices that have no edges between each other [7].The links in an independent set can transmit simulta-neously. Let E denote the set of all independent sets witheach independent set indexed by e. We represent an inde-pendent set e as a jLj-dimensional rate vector re, where thelth entry is

rel :¼

cl if l 2 e;

0 otherwise:

�

The feasible rate region P at the link layer is then definedas the convex hull of these rate vectors

P :¼ r : r ¼X

e

aere; ae P 0;X

e

ae ¼ 1

( ): ð1Þ

Thus, a link flow vector y satisfies schedulability constraintif y 2P.

The contention graph is a general construct, and cancapture the interdependence or contention among parallelservers of any queueing networks. For example, it cancharacterize the contention relations in the network wherewireless nodes are equipped with multiple radios or com-municate through multiple channels.

3.2. Rate constraint

Let fl P 0 denote the amount of capacity allocated tolink l. From the schedulability constraint, f should satisfy

f 2 P: ð2Þ

Assume that the network is shared by a set S of sources,with each source s 2 S transmitting at rate xs bits per sec-ond. In the following, we will formulate rate constraintsfor networks with different kinds of routing.

The network with single-path routingEach source s uses a path consisting of a set Ls � L of

links. The sets Ls define an jLj � jSj routing matrix

Rls ¼1 if l 2 Ls;

0 otherwise:

�

Thus, the aggregate rate over link l isP

s2SRlsxs. The rateconstraint is written as

Rx 6 f ; ð3Þ

i.e., the aggregate link rate should not exceed the linkcapacity.

The network with multipath routingEach source s can send traffic along a set Ts of given

paths. Each path r 2 Ts contains a set of Lsr � L of links,


which defines a jLj � jTsj routing matrix Hs whose (l,r) thentry is given by

Hslr ¼

1 if l 2 Lsr;

0 otherwise:

�

Denote by xrs the rate at which source s sends along path r.

Thus, the source rate xs ¼P

r2Tsxr

s. The rate constraint iswritten asXs;r2Ts

Hslrx

rs 6 fl; l 2 L: ð4Þ

The network without prespecified pathsSince no end-to-end path is given, we will use multi-

commodity flow model for routing. Let D denote the setof destination nodes of network layer flows. Let f k

i;j P 0 de-note the amount of capacity of link (i, j) allocated to theflows to destination k. Then the aggregate capacity on link(i, j) is fi;j :¼

Pk2Df k

i;j. Let xki P 0 denote the flow generated at

node i towards destination k. Then the aggregate capacityfor its incoming flows and generated flow to the destina-tion k should not exceed the summation of the capacitiesfor its outgoing flows to k

xki 6

Xj:ði;jÞ2L

f ki;j �

Xj:ðj;iÞ2L

f kj;i; i 2 N; k 2 D; i–k: ð5Þ

Eq. (5) is the rate constraint for resource allocation. Forsimplicity of presentation, we assume that there is at mostone flow between any node and destination pair[i,k] 2 S � D. Thus, xk

i ¼ xs if i is the source node of flows = [i,k], and xk

i ¼ 0 otherwise.

3.3. Problem formulation

We see from the last subsection that all three kinds ofrate constraints are expressed as linear inequalities. If werepresent the ‘‘routing” of the user (source) servicerequirement by a linear function H(x) of the source ratesx, and represent the ‘‘allocation” of the service capacityby a linear function A(f) of the link capacities f, since theservice requirement should not exceed the allocated ser-vice capacity, we have the following inequality constraint

HðxÞ 6 Aðf Þ: ð6Þ

The linear constraint (6) is a very general relation. The rateconstraints (3)–(5) are just its different concreterepresentations.

Following [16,22,18], assume each source s attains autility Us(xs) when it transmits at a rate xs. We assumeUs(�) is continuously differentiable, increasing, and strictlyconcave. Our objective is to choose source rates x and allo-cated capacities f so as to solve the following globalproblem

maxx;f

Xs

UsðxsÞ; ð7Þ

subject to HðxÞ 6 Aðf Þ; ð8Þf 2 P: ð9Þ

The system problem (7)–(9) is a convex optimization prob-lem, but it is impractical to solve it centrally in real net-

works. Distributed algorithm can be derived by solvingits Lagrange dual problem, as we will show in the nextsection.

4. Distributed algorithm via dual decomposition

4.1. Distributed algorithm

Consider the Lagrangian of the problem (7)–(9) with re-spect to the rate constraint

Lðp; x; f Þ ¼X

s

UsðxsÞ � pTðHðxÞ � Aðf ÞÞ:

Given p, the above Lagrangian has a nice decompositionstructure: it is the summation of two independent terms,of source rates and link capacities respectively. Interpret-ing p as the ‘‘congestion price” and maximizing theLagrangian over x and f for fixed p, we obtain the followingjoint congestion control and scheduling algorithm:

Congestion control: At time t, given congestion price p(t),the sources adjust flow rates x according to the congestionprice

xðtÞ ¼ xðpðtÞÞ ¼ arg maxx

Xs

UsðxsÞ � pTðtÞHðxÞ: ð10Þ

Scheduling: Over link l, send an amount of data for eachflow according to the rates f such that

f ðtÞ ¼ f ðpðtÞÞ 2 arg maxf2P

pTðtÞAðf Þ: ð11Þ

Note that there does not exist an explicit routing compo-nent in the dual decomposition. Instead, the routing isimplicitly solved in (10) if the set of paths from which asource can choose is given, and solved in (11) if no pathis prespecified for the source. We see that, by dual decom-position, the flow optimization problem decomposes intoseparate ‘‘local” optimization problems of transport, net-work and link layers respectively, and they interactthrough congestion prices.

Defining dual function D(p) = maxx,f2PL(p,x, f), by dual-ity we have (see, e.g., Chapter 5 in [2])

maxx;f

Xs

UsðxsÞ ¼minpP0

DðpÞ ¼minpP0

maxx;f2P

Lðp; x; f Þ:

The dual problem minpD(p) can be solved by using the sub-gradient method [33,2], where the Lagrangian multipliersare adjusted in the opposite direction to the subgradientof the dual function

gðpÞ ¼ Aðf ðpÞÞ � HðxðpÞÞ: ð12Þ

Congestion price update: The network (links or nodes) up-dates the congestion price, according to

pðt þ 1Þ ¼ ½pðtÞ þ ctðHðxðpðtÞÞÞ � Aðf ðpðtÞÞÞÞ�þ; ð13Þ

where ct is a positive scalar stepsize, and ‘‘+” denotes theprojection onto the set Rþ of nonnegative real numbers.The algorithm has a nice interpretation in terms of law ofsupply and demand and their regulation through pricing.Eq. (13) says that, if the demand H(x(p(t))) for servicecapacity exceeds the supply A(f(p(t))), the price p will rise,


which will in turn decrease the demand (see Eq. (10)) andincrease the supply (see Eq. (11)).

Before proceeding, we explain the notation used in thispaper. We denote a link either by a single index l or bythe directed pair (i, j) of nodes it connects. We use s or alter-natively node pair [i,k] to denote a network layer flow. Weoverload the use of the source rate x, the link capacity f andthe congestion price p throughout this paper, depending ondifferent kinds of routing involved. For example, x refers tothe source rate xs or the source rate xk

i at node i towards des-tination k, depending on the specific context. Similarly, f re-fers to both the link capacity {fi,j} and the capacity f k

i;j

n oover

link (i, j) that is allocated to destination k.

4.2. Convergence analysis

Using results on the convergence of the subgradientmethod [33,2], we show that, for constant stepsize, thealgorithm is guaranteed to converge to within a neighbor-hood of the optimal value. For diminishing stepsize, thealgorithm is guaranteed to converge to the optimal value.We would like a distributed implementation of the subgra-dient algorithm, and thus a constant stepsize ct = c is moreconvenient. Note that the dual cost usually will not mono-tonically approach the optimal value, but wander around itunder the subgradient algorithm. The usual criterion forstability and convergence is not applicable. Here we defineconvergence in a statistical sense [3]. Let �pðtÞ :¼ 1

t

Pts¼1pðsÞ

be the average price by time t.

Definition 1. Let p* denote an optimal value of the dualvariable. Algorithm (10)–(13) with constant stepsize is saidto converge statistically to p*, if for any d > 0 there exists astepsize c such that lim supt!1Dð�pðtÞÞ � Dðp�Þ 6 d.

Clearly, an optimal value p* exists. The following theo-rem guarantees the statistical convergence of the subgradi-ent method.

Theorem 2. Let p* be an optimal price. If the norm of thesubgradients is uniformly bounded, i.e., there exists G suchthat kg(p)k2 6 G for all p, then

Dðp�Þ 6 lim supt!1

Dð�pðtÞÞ 6 Dðp�Þ þ cG2

2; ð14Þ

i.e., the algorithm (10)–(13) converges statistically to p*.

Proof. The first inequality Dðp�Þ 6 lim supt!1Dð�pðtÞÞalways holds, since D(p*) is the minimum of the dual func-tion D(p). Now we prove the second inequality. By Eq. (13),we have

kpðt þ 1Þ � p�k22 ¼ k½pðtÞ � cgðpðtÞÞ�þ � p�k2

2

6 kpðtÞ � cgðpðtÞÞ � p�k22

¼ kpðtÞ � p�k22 � 2cgðpðtÞÞTðpðtÞ � p�Þ

þ c2kgðpðtÞÞk22

6 kpðtÞ � p�k22 � 2cðDðpðtÞÞ � Dðp�ÞÞ

þ c2kgðpðtÞÞk22;

where the last inequality follows from the definition of sub-gradient. Applying the inequalities recursively, we obtain

kpðt þ 1Þ � p�k22 6 kpð1Þ � p�k2

2 � 2cXt

s¼1

ðDðpðsÞÞ � Dðp�ÞÞ

þ c2Xt

s¼1

kgðpðsÞÞk22:

Since kpðt þ 1Þ � p�k22 P 0, we have

2cXt

s¼1

ðDðpðsÞÞ � Dðp�ÞÞ 6 kpð1Þ � p�k22 þ c2

Xt

s¼1

kgðpðsÞÞk22

6 kpð1Þ � p�k22 þ tc2G2:

From this inequality we obtain

1t

Xt

s¼1

DðpðsÞÞ � Dðp�Þ 6 kpð1Þ � p�k22

2tcþ cG2

2:

Since D is a convex function, by Jensen’s inequality,

Dð�pðtÞÞ � Dðp�Þ 6 kpð1Þ � p�k22

2tcþ cG2

2:

Thus, lim supt!1Dð�pðtÞÞ 6 Dðp�Þ þ cG2

2 , i.e., the algorithmconverges statistically to p*. h

The assumption of bounded norm for subgradient g(p)is reasonable, since f is finite and we always have an upperbound on x in practice. Theorem 2 implies that the conges-tion price p approaches p* statistically when the stepsize cis small enough.

Let the primal function be PðxÞ :¼P

sUsðxsÞ and achieveits optimum at x*. Define �xðtÞ :¼ 1

t

Pts¼1xðsÞ, the average

data rate up to time t. As time goes to infinity, �xðtÞ mustbe in the feasible rate region (determined by Eqs. (8) and(9)), otherwise �pðtÞ will go unbounded as time goes toinfinity, which contradicts Theorem 2.

Theorem 3. Let x* be the optimal source rates. Under thesame assumption of Theorem 2, the algorithm (10)–(13)converges statistically to within a small neighborhood of theoptimal values P(x*), i.e.,

Pðx�ÞP lim inft!1

Pð�xðtÞÞP Pðx�Þ � cG2

2: ð15Þ

Proof. The first inequality Pðx�ÞP lim inf t!1Pð�xðtÞÞ holds,since �xðtÞ is in the feasible rate region as t goes to infinity.Now we prove the second inequality. By Eq. (13), we have

kpðt þ 1Þk22 6 kpðtÞ � cgðpðtÞÞk2

2

¼ kpðtÞk22 � 2cgðpðtÞÞT pðtÞ þ c2kgðpðtÞÞk2

2

¼ kpðtÞk22 þ 2c

Xs

UsðxsðtÞÞ

� 2cX

s

UsðxsðtÞÞ � pTðtÞHðxðtÞÞ !

� 2cpTðtÞAðf ðtÞÞ þ c2kgðpðtÞÞk22

6 kpðtÞk22 þ 2c

Xs

UsðxsðtÞÞ

� 2cX

s

Us x�s� �� pTðtÞHðx�Þ

!

� 2cpTðtÞAðf ðtÞÞ þ c2kgðpðtÞÞk22

¼ kpðtÞk22 þ 2cPðxðtÞÞ � 2cPðx�Þ þ c2kgðpðtÞÞk2

2

� 2cpTðtÞðAðf ðtÞÞ � Hðx�ÞÞ6 kpðtÞk2

2 þ 2cPðxðtÞÞ � 2cPðx�Þ þ c2kgðpðtÞÞk22;


where the second inequality follows from the fact that x(t)is the maximizer in the problem (10), and the third inequal-ity follows from the fact that f(t) is the maximizer in prob-lem (11). Applying the inequalities recursively, we obtain

kpðt þ 1Þk22 6 kpð1Þk

22 þ 2c

Xt

s¼1

ðPðxðsÞÞ � Pðx�ÞÞ

þ c2Xt

s¼1

kgðpðsÞÞk22:

Since kpðt þ 1Þk22 P 0, we have

2cXt

s¼1

ðPðxðsÞÞ � Pðx�ÞÞP �kpð1Þk22 � c2

Xt

s¼1

kgðpðsÞÞk22

P �kpð1Þk22 � tc2G2:


1t

Xt

s¼1

PðxðsÞÞ � Pðx�ÞP �kpð1Þk22 � tc2G2

2tc:

Since P is a concave function, by Jensen’s inequality,

Pð�xðtÞÞ � Pðx�ÞP �kpð1Þk22 � tc2G2

2tc:

Thus, lim inf t!1Pð�xðtÞÞP Pðx�Þ � cG2

2 , i.e., the algorithm(10)–(13) converges statistically to within a small neigh-borhood of the optimal values P(x*). h

Since P(x) is continuous, Theorem 3 implies that theaverage source rate approaches the optimal x* when c issmall enough.

5. Extension to networks with time-varying channels

In the last section, we consider wireless multihop net-works with fixed channels or single-rate devices, i.e., thecapacity cl is a constant when link l is active. However, re-cent years have seen the growing popularity and demandof multi-rate wireless network devices (e.g., 802.11a cards)that can adjust transmission rate according to the time-varying channel state and improve overall network utiliza-tion. Here, we consider the networks with time-varyingchannels and adaptive multi-rate devices.


We assume that time is slotted, and the channel is fixedwithin a time slot but independently changes between dif-ferent slots.1 Let h(t) denote the channel state in time slot t.Corresponding to the channel state h, the capacity of link l iscl(h) when active and the feasible rate region at the link layeris P(h), which is defined in a similar way as in (1). We fur-ther assume that the channel state is a finite state processwith identical distribution q(h) in each time slot,2 and definethe mean feasible rate region as

1 It is straightforward to extend our results to a network where thechannel state process is modulated by a hidden Markov chain.

2 Even if the channel state is a continuous process, we only have finitechoices of modulation schemes. The corresponding capacities take discretevalues.

P :¼ �r : �r ¼X

h

qðhÞrðhÞ; rðhÞ 2 PðhÞ( )

: ð16Þ

Ideally, we would like to have a distributed algorithm thatsolves the following utility maximization problem

maxx;f

Xs

UsðxsÞ ð17Þ

subject to HðxÞ 6 Aðf Þ; ð18Þ

f 2 P: ð19Þ

However, if we solve the above problem via dual decompo-sition, we may get a link rate assignment which is infeasi-ble for the channel state at a given time slot. Instead wedirectly extend the algorithm (10)–(13) to handle thetime-varying channel.

Congestion control: At time t, given congestion price p(t),the sources adjust flow rates x according to the congestionprice

xðtÞ ¼ xðpðtÞÞ ¼ arg maxx

Xs

UsðxsÞ � pTðtÞHðxÞ: ð20Þ

Scheduling: In the beginning of period t, each node moni-tors the channel state h(t), and over link l send an amountof data for each flow according to the rates f such that

f ðtÞ ¼ f ðpðtÞÞ 2 arg maxf2PðhðtÞÞ

pTðtÞAðf Þ: ð21Þ

Congestion price update: The network (links or nodes) up-dates the congestion price, according to

pðt þ 1Þ ¼ b½pðtÞ þ cðHðxðpðtÞÞÞ � Aðf ðpðtÞÞÞÞ�þc: ð22Þ

Here ‘‘bc” denotes the function floor with respect to c2. Wechoose such a discrete congestion price to facilitate the sta-bility analysis in the next subsection.

The above algorithm cannot be derived from the dualdecomposition of the problem (17)–(19). However, we willuse the problem (17)–(19) as a reference system, and char-acterize the performance of the above algorithm with re-spect to it.

5.2. Stochastic stability

Note that congestion price p(t) takes discrete values.Thus, congestion price p(t) evolves according to a discrete-time, discrete-space Markov chain. We need to show thatthis markov chain is stable, i.e., the congestion price processreaches a steady state and does not become unbounded. It iseasy to check that the Markov chain has a countable statespace, but is not necessarily irreducible. In such a generalcase, the state space is partitioned in transient set T and dif-ferent recurrent classes Ri. We define the system to be stableif all recurrent states are positive recurrent and the Markovprocess hits the recurrent states with probability one [38].This will guarantee that the Markov chain will be ab-sorbed/reduced into some recurrent class, and the positiverecurrence ensures the ergodicity of the Markov chain overthis class. We have the following result.

Theorem 4. The Markov chain described by Eq. (22) is stable.


Proof. Denote the dual function of the problem (17)–(19)by DðpÞ with an optimal price p* and subgradient �gðpÞ,i.e., �gðpÞ ¼ Að�f ðpÞÞ � HðxðpÞÞ with �f ðpÞ 2 arg maxf2PpT Aðf Þ.Consider the Lyapunov function VðpÞ ¼ kp� p�k2

2, we have

E½DVtðpÞjp� ¼ E½Vðpðtþ1ÞÞ�VðpðtÞÞjpðtÞ ¼ p�

¼ E½Vðb½pðtÞ�cgðpðtÞÞ�þcÞ�VðpðtÞÞjpðtÞ ¼ p�

¼ E½Vð½pðtÞ�cgðpðtÞÞ�þ � �Þ�VðpðtÞÞjpðtÞ ¼ p�6 E½VðpðtÞ�cgðpðtÞÞÞ�VðpðtÞÞjpðtÞ ¼ p�

þE½�2� � ð½pðtÞ�cgðpðtÞÞ�þ �p�Þþk�k22jpðtÞ ¼ p�

6 E½VðpðtÞ�cgðpðtÞÞÞ�VðpðtÞÞjpðtÞ ¼ p�

þ2� �p� þk�k22;

where � = [p(t) � cg(p(t))]+ � b[p(t) � c g(p(t))]+c. Note that0 6 � < c21, with 0 denoting the zero vector and 1 the vec-tor with every component being 1. Thus, 2� � p� þ k�k2

2 <

2c2kp�k1 þ c4k1k22. Let D ¼ 2kp�k1 þ c2k1k2

2, we have

E½DVtðpÞjp�6 E½VðpðtÞ� cgðpðtÞÞÞ �VðpðtÞÞjpðtÞ ¼ p� þ c2D

¼ E½�cgðpðtÞÞTð2ðpðtÞ � p�Þ� cgðpðtÞÞÞjpðtÞ ¼ p� þ c2D

¼ 2c�gðpÞTðp� � pÞ þ c2E½kgðpðtÞÞk22jpðtÞ ¼ p� þ c2D

6 2c�gðpÞTðp� � pÞþ c2ðG2 þDÞ;

where we again use the assumption that the norm ofg(p(t)) is bounded above by G. By the definition of subgra-dient, we further get

E½DVtðpÞjp� 6 2cðDðp�Þ � DðpÞÞ þ c2ðG2 þ DÞ:

Let

d ¼ maxDðpÞ�Dðp�Þ6cðG2þDÞ

kp� p�k2

and define A ¼ fp : kp� p�k2 6 dg. We obtain

E½DVtðpÞjp� 6 �c2ðG2 þ DÞIp2Ac þ c2ðG2 þ DÞIp2A;

where I is the index function. Thus, by Theorem 3.1 in[38], which is an extension of Foster’s criterion [1], theMarkov chain p(t) is stable. h

The above proof shows that the distance to the optimalp* has negative conditional mean drift for all prices thathave sufficiently large distance to p*, and implies that thecongestion price will stay near p* when c is small enough.

5.3. Performance evaluation

We now characterize the performance of the algorithm(20)–(22) in terms of the dual and primal objective func-tions of the reference system problem (17)–(19).

Theorem 5. The algorithm (20)–(22) converges statisticallyto within a small neighborhood of the optimal value Dðp�Þ, i.e.,

Dðp�Þ 6 DðE½pð1Þ�Þ 6 Dðp�Þ þ cðG2 þ DÞ2

; ð23Þ

where p(1) denotes the state of the Markov chain p(t) in thesteady state, and D ¼ 2kp�k1 þ c2k1k2

2.

Proof. The first inequality Dðp�Þ 6 DðE½pð1Þ�Þ alwaysholds, since Dðp�Þ is the minimum of the dual functionDðpÞ. Now we prove the second inequality. From the proofof Theorem 4, we have

E½DVtðpÞjp� ¼ E½Vðpðt þ 1ÞÞ � VðpðtÞÞjpðtÞ ¼ p�

6 2cðDðp�Þ � DðpÞÞ þ c2ðG2 þ DÞ:

Taking expectation over p, we get

E½DVtðpÞ� ¼ E½Vðpðt þ 1ÞÞ � VðpðtÞÞ�

6 2cðDðp�Þ � E½DðpÞ�Þ þ c2ðG2 þ DÞ:

Taking summation from s = 0 to s = t � 1, we obtain

E½VðpðtÞÞ� 6 E½Vðpð0ÞÞ� � 2cXt�1

s¼0

E½DðpðsÞÞ� þ 2ctDðp�Þ

þ tc2ðG2 þ DÞ:

Since E[V(p(t))] P 0, we have

2cXt�1

s¼0

E½DðpðsÞÞ� � 2ctDðp�Þ 6 E½Vðpð0ÞÞ� þ tc2ðG2 þ DÞ:


1t

Xt�1

s¼0

E½DðpðsÞÞ� � Dðp�Þ 6 E½Vðpð0ÞÞ� þ tc2ðG2 þ DÞ2tc

:

Note that p(t) is stationary and ergodic in some steadystate by Theorem 4, and so is DðpðtÞÞ. Thus,

limt!1

1t

Xt�1

s¼0

E½DðpðsÞÞ� ¼ E½Dðpð1ÞÞ�:

So,

E½Dðpð1ÞÞ� � Dðp�Þ 6 cðG2 þ DÞ2

:

Since DðpÞ is a convex function, by Jensen’s inequality,

DðE½pð1Þ�Þ � Dðp�Þ 6 cðG2 þ DÞ2

;

i.e., the algorithm converges statistically to within c(G2 + D)/2 of the optimal value Dðp�Þ. h

Since DðpÞ is a continuous function, Theorem 5 impliesthat the congestion price p approaches p* statisticallywhen c is small enough.

Corollary 6. x(t) is a stable Markov chain. Moreover, theaverage arrival rates E½xð1Þ� 2 P, where x(1) denotes thestate of the process x(t) in the steady state.

Proof. x(t) is a deterministic, finite-value function of p(t).x(t) is a stable Markov chain, since p(t) is E½xð1Þ� 2 P,otherwise the average congestion price E[p(1)] will beunbounded, which contradicts Theorem 4. h

Theorem 7. Let PðxÞ be the primal function and x* be theoptimal source rates of the reference system problem (17)–(19). The algorithm (20)–(22) converges statistically towithin a small neighborhood of the optimal value Pðx�Þ, i.e.,

Pðx�ÞP PðE½xð1Þ�ÞP Pðx�Þ � cðG2 þ DÞ2

: ð24Þ

A D B E FC

G H

Fig. 2. An example of a simple multihop wireless network.


Proof. The proof for the theorem is a straightforwardextension of the proof of Theorem 3, following similar pro-cedure as in the proof of Theorem 5. We skip the detailshere. h

Since PðxÞ is a continuous function and has a uniqueoptimal, Theorem 7 implies that the average source rateapproaches the optimal of the ideal reference system(17)–(19) when stepsize c is small enough. Theorems 5and 7 show that, surprisingly, the algorithm (20)–(22)can be seen as a distributed algorithm to approximatelysolve the ideal reference system problem that is not readilysolvable due to stochastic channel variations.

Our proofs for stability and performance bounds arerather general. They only use general properties of convex-ity and Markovity and the definition of subgradients. Wethus have presented a general technique and results regard-ing the stability and optimality of dual algorithm for convexoptimization in face of time-varying parameters. As theflow contention graph is a rather general construct andcan be used to capture the interdependence or contentionamong parallel servers of any queueing networks, the afore-mentioned results are applicable to any systems that can bemodelled by a general model of queueing network that isserved by a set of interdependent parallel servers withtime-varying service capabilities. In the next two sections,we will discuss two such applications. Other examples in-clude fair scheduling in a generalized switch [34], and TCP[22] with time-varying capacity as in last-hop wireless net-works. It can include power control as well [5], as powerdoes not change convexity of the feasible rate region.

As specific cases of the general model and algorithmpresented in Sections 4 and 5, we will discuss joint conges-tion control and medium access control design in multihopwireless networks with single-path routing and cross-layercongestion control, routing and scheduling design in thenetwork without prespecified paths in the next two sec-tions, respectively.

6. Joint congestion control and media access controldesign

TCP was originally designed for wireline networks,where links are assumed to have fixed capacities. However,as wireless channel is a shared medium and interference-limited, wireless links are ‘‘elastic” and the capacities theyobtain depend on the bandwidth sharing mechanism usedat the link layer. This may result in various TCP perfor-mance problems in wireless networks.

One such problem is TCP unfairness over multihop wire-less networks. Many existing wireless MAC protocols, suchas DCF specified in IEEE 802.11 standard [13], are trafficindependent and do not consider the actual requirements

of the flows competing for the channel. These MAC proto-cols suffer from the unfairness problem, caused by the loca-tion dependency of the contentions, and exacerbated by thecontention resolution mechanisms such as the binary expo-nential backoff algorithm adopted in DCF. When they inter-act with TCP, TCP will further penalize these flows withmore contention. This will result in significant TCP unfair-ness in multihop wireless networks [10,37,45–47]. To illus-trate this, consider the example in Fig. 2, and assume thereare four network-layer flows A ? B, C ? D, E ? F andG ? H. The flow C ? D experiences more contention andwill build up a queue faster than the other three flows.TCP will further penalize it by reducing the congestion win-dow more aggressively, and the resulting throughput offlow C ? D will be much less than that of the other flows.

In addition to the location dependency of contentions,correlation among links is also the key to understand theinteraction between transport and MAC layers. In wirelinenetworks, link bandwidth is well defined and links are dis-joint resources. But in wireless networks, as we mentionedabove, links are correlated due to interference, and net-work-layer flows that do not transverse a common linkmay still compete with each other. Thus, congestion is lo-cated at some spatial contention region [47]. Consider againthe example in Fig. 2, and assume there are two network-layer flows A ? F and G ? H. Link-layer flows BC, CD, DEand GH contend with each other, and congestion is locatedin the spatial contention region denoted by the rectangle.So, unlike wireline networks where link capacities provideconstraints for resource allocation, in multihop wirelessnetworks the contention relations between link-layer flowsprovide fundamental constraints for resource allocation.We need to exploit the interaction between transport andlink (MAC) layers, in order to improve the performance.

Eqs. (2) and (3) capture the constraints that arise fromchannel contention among wireless links. We model theresource allocation for multihop wireless networks as autility maximization problem with these constraints,

maxx;f

Xs

UsðxsÞ ð25Þ

subject to Rx 6 f ; ð26Þf 2 P; ð27Þ

which is a special case of the system problem (7)–(9). Withthis formulation, we can explicitly exploit the interactionbetween transport and MAC layers, and systematically car-ry out joint design of congestion and media access control.In the next subsection, a dual algorithm solving the systemproblem (25)–(27) is derived by applying the algorithm(10)–(13). The algorithm motivates a scheme for media ac-cess control in which link-layer flows are scheduledaccording to congestion prices.



Consider the Lagrangian of the problem (25)–(27) withrespect to the rate constraint

Lðp; x; f Þ ¼X

s

UsðxsÞ � pT Rx� fð Þ: ð28Þ

Interpreting pl as the congestion price at link l, we can usethe algorithm (10)–(13) to solve the problem (25)–(27) andits dual.

Rate control: At time t, given congestion price p(t),source s adjusts its sending rate xs according to the aggre-gate congestion price

PlRlspl along its path

xsðtÞ ¼ U0�1s ðX

l

RlsplðtÞÞ: ð29Þ

Scheduling: Over link l, send an amount of data for eachflow according to the rate f such that

f ðtÞ ¼ f ðpðtÞÞ 2 arg maxf2P

pT f : ð30Þ

If the network with time-varying channel is considered,each node monitors channel state h(t) and over link l sendsan amount of data for each flow according to the rate f suchthat

f ðtÞ ¼ f ðpðtÞÞ 2 arg maxf2PðhðtÞÞ

pT f : ð31Þ

Congestion price update: Each link l updates its price,according to

plðt þ 1Þ ¼ plðtÞ þ ct

Xs

RlsxsðpðtÞÞ � flðpðtÞÞ !" #þ

: ð32Þ

The above algorithm motivates a joint design schemewhere the link layer flows are scheduled according to con-gestion prices of the links. Also, note that Eqs. (29) and (32)are completely distributed and can be implemented atindividual sources and links using only local information.We will discuss the distributed solution to the schedulingproblem (30) in the next subsection.

6.2. Scheduling over multihop networks

We now come to the scheduling problem (30), whichwill also show up in the next section. Scheduling over mul-tihop networks is a difficult problem and in general NP-hard. To see this, note that problem (30) is equivalent toa maximum weight independent set problem over the flowcontention graph, which is NP-hard for general graphs. It iseasy to design some heuristic algorithm but is hard tobound its performance.

With the primary interference model, the schedulingproblem (30) is equivalent to the maximum weightedmatching problem3 over the connectivity graph {N,L} ofthe network. Maximum weighted matching problem canbe computed in polynomial time (see, e.g., [29]), but this re-

3 A matching in a graph is a subset of links, no two of which share acommon node. The weight of a matching is the total weight of all its links. Amaximum weighted matching in a graph is a matching whose weight ismaximized over all matchings of the graph.

quires centralized implementation. If implemented over amultihop network, each node needs to notify the centralnode of its weight and local connectivity information suchthat the central node can reconstruct the network topologyas a weighted graph. This will lead to an immense commu-nication overhead which is expensive in time and resources.There also exist simpler greedy sequential algorithms tocompute a weighted matching at most a factor of 2 awayfrom the maximum; see, e.g., [30]. But they also require cen-tralized implementation. We seek a distributed algorithmwhere each node participates in the computation itself usingonly local information.

A few distributed approximation algorithms exist formaximum weighted matching problem; see, e.g.,[40,42,12]. In [12], the author presents a simple distributedalgorithm to compute a weighted matching at most a factorof 2 away from the maximum in linear running time O(jLj).This algorithm is a distributed variant of the sequentialgreedy algorithm presented in [30]. We have utilized thisalgorithm to solve the scheduling problem (30) distributed-ly, see [4] for details. The resulting scheduling algorithm formultihop wireless networks is one of the best distributedalgorithms in terms of computational complexity andperformance bound. It has a linear complexity O(jLj). Sucha low complexity is important for the scalability andefficiency of multihop wireless networks. It achieves a per-formance of 1/2 of the maximum weight in the worst case,and in practice, numerical simulations show it typicallyachieves a performance within about 4/5 of the maximumweight. There also exist few other distributed approxima-tion algorithms; see, e.g., [21,43,24,14,32,28]. Especially,in [14,32,28] the authors present distributed random accessalgorithms that achieve nearly 100% throughput.

As for the overall performance of our cross-layer designwith approximate scheduling, we can extend the result in[21] to show that the performance is no worse than thatachieved by an exact design with a feasible rate region12 P at the link layer. Moreover, in [4] we also see that thisdistributed scheduling algorithm only results in a verysmall degradation in the performance of the cross-layerdesign for the network with time-varying channel, sincein this situation the exact solution of the scheduling is notas important and reasonable approximations work well.

6.3. Numerical examples

In this subsection, we provide numerical examples tocomplement the analysis in previous subsections. We con-sider a simple network with secondary interference asshown in Fig. 2, and assume that there are three networklayer flows G ? H, A ? F and D ? F with the same utilityfunction Us(xs) = logxs. We have chosen a simple topologyto facilitate discussion.

The network with fixed channel and single-rate devicesWe first consider a network with fixed link capacities.

For simplicity, we assume that all the links have one unitof capacity when active. Fig. 3 shows the evolution ofsource rates and their averages with the joint algorithm(29), (30), (32) with stepsize c = 0.2. We see that the sourcerates converge quickly to a neighborhood of the optimal

0 10 20 30 40 50 60 70 80 90 1000

0.2

0.4

0.6

0.8

1

Normalized Time

Nor

mal

ized

Sou

rce

Rat

es

Flow GHFlow AFFlow DF

0 10 20 30 40 50 60 70 80 90 1000

0.2

0.4

0.6

0.8

1

Normalized Time

Aver

age

Sour

ce R

ates


Fig. 3. The evolution of source rates in the network with fixed link capacities.

0 10 20 30 40 50 60 70 80 90 1000

2

4

6

8

10

Normalized Time

Nor

mal

ized

Con

gest

ion

Pric

es


0 10 20 30 40 50 60 70 80 90 1000

2

4

6

8

10

Normalized Time

Aver

age

Con

gest

ion

Pric

es


Fig. 4. The evolution of congestion prices in the network with fixed link capacities.


and oscillate around the optimal. This oscillating behaviormathematically results from the non-differentiability ofthe dual function and physically can be interpreted asdue to the scheduling process. However, the average

source rates are smooth and approach the optimum mono-tonically. Fig. 4 shows the evolution of the correspondingend-to-end congestion prices and the averages of the threeflows. Similarly, the congestion prices approach the opti-

0 10 20 30 40 50 60 70 80 90 1000

0.2

0.4

0.6

0.8

1

Normalized Time

Nor

mal

ized

Sou

rce

Rat

es


0 10 20 30 40 50 60 70 80 90 1000

0.2

0.4

0.6

0.8

1

Normalized Time

Aver

age

Sour

ce R

ates


Fig. 5. The evolution of source rates in the network with time-varying link capacities.


mum quickly. We also note that the performance of thealgorithm is much better than the bound of cG2/2 specifiedin Theorems 2 and 3.

The choice of the stepsize c is important. It character-izes the ‘‘optimality” of the algorithm, as shown in Theo-rems 2 and 3 (and also in Theorems 5 and 7). It alsoaffects the convergence speed. In oder to study the impactof different choices of the stepsize on the performance ofthe algorithm, we have run simulations with differentstepsizes. We found that the smaller the stepsize, theslower the convergence and the closer to the optimal,which is a general characteristic of any gradient basedalgorithm. So, there is a tradeoff between convergencespeed and optimality. In practice, the end user can firstchoose large stepsizes to ensure fast convergence, and sub-sequently, the stepsizes can be reduced once the sourcerate starts oscillating around some mean value.

The network with time-varying channel and multirate devicesWe now consider a network with time-varying link

capacities. For simplicity, we assume that the capacitiesof all links are identically, uniformly distributed over 0.5,1 and 1.5 units. Thus, the average capacity for each linkwhen active is the same as that in the example with fixedlink capacities.

Figs. 5 and 6 show the evolution of source rates, conges-tion prices and their averages with the same stepsizec = 0.2. The source rates and congestion prices have muchlarger variations than those with fixed channel, due tothe channel variations. But the average source rates andcongestion prices are still smooth, and converge quicklyand monotonically to optimal values. Our simulation re-sults have confirmed the conclusions from Theorems 5

and 7, which say that the average source rates and conges-tion prices approach the optimum of an ideal system withthe best feasible rate region at the link layer, and that algo-rithm (29), (31) and (32) can been seen as a distributedalgorithm to solve this ideal system problem. Also notethat, although the average link capacities when active arethe same as those in fixed channel, each flow achieves lar-ger sending rate. This is due to multi-user diversity: our‘‘optimal” scheduling (31) has implicitly considered mul-ti-user diversity.

6.4. Summary

We have presented a model for the joint design of con-gestion control and media access control for multihopwireless networks, where the resulting dual algorithm isto solve a utility maximization problem with constraintsthat arise from contention for the wireless channel. Thisalgorithm motivates a joint design where link-layer flowsare scheduled according to the congestion prices of thelinks.

There exist other ways to solve the resource allocationproblem (25)–(27). In [3], we also derive a primalalgorithm by solving the relaxation of the system problem(25)–(27). Based on the algorithm, we propose a traffic-dependent scheme for contention-based medium accesscontrol and generate congestion price directly fromthe MAC layer. As scheduling in multihop wireless net-works is an intrinsically hard problem, contention-basedmedium access seems a must. To further integratecongestion control and contention-based medium accessin the utility maximization framework will be a futureresearch step.

0 10 20 30 40 50 60 70 80 90 1000

1

2

3

4

5

6

7

Normalized Time

Nor

mal

ized

Con

gest

ion

Pric

es


0 10 20 30 40 50 60 70 80 90 1000

1

2

3

4

5

6

Normalized Time

Aver

age

Con

gest

ion

Pric

es


Fig. 6. The evolution of congestion prices in the network with time-varying link capacities.


7. Joint congestion control, routing and schedulingdesign

In the last section, we have discussed the resource allo-cation in multihop wireless networks where the path foreach network layer flow is given. However, as wirelessspectrum is a scarce resource, it may be costly to maintainend-to-end paths, and congestion control based on end-to-end feedback may consume too much bandwidth insignalling. Moreover, most routing schemes for multihopnetworks select paths that minimize hop count; see, e.g.,[15,31]. This implicitly predefines a path for any source–destination pair, independent of the pattern of trafficdemand and interference/contention among links. Thismay result in congestion at some region while other re-gions are underutilized. In order to achieve high end-to-end throughput and efficient resource allocation, the pathsshould not be decided exogenously but jointly optimizedwith congestion control and scheduling.

Since the actual paths that will be used are not specifieda priori, we will use a multicommodity flow model forrouting and model the resource allocation as a utility max-imization problem with the constraints (2) and (5),

maxx;f

Xs

UsðxsÞ ð33Þ

subject to xki 6

Xj:ði;jÞ2L

f ki;j �

Xj:ðj;iÞ2L

f kj;i; ð34Þ

f 2 P; ð35Þ

where i 2 N, k 2 D, i – k, and xki ¼ 0 if [i,k] R S � D. Again, this

problem is a special case of the system problem (7)–(9). Inthe next subsection, we apply the algorithm (10)–(13) toobtain a distributed subgradient algorithm for joint

congestion control, routing and scheduling. This algorithmmotivates a joint design where the source adjusts its send-ing rate according to the congestion price generated locallyat the source node, and backpressure from the differentialprice of neighboring nodes is used for optimal schedulingand routing.


Consider the Lagrangian of the problem (33)–(35) withrespect to the rate constraint

Lðp;x; f Þ ¼X

s

UsðxsÞ �X

i2N;k2D;i–k

pki xk

i �X

j:ði;jÞ2L

f ki;j þ

Xj:ðj;iÞ2L

f kj;i

!:

ð36Þ

Interpreting pkl as the congestion price at node i for the

flows to destination k, we can use the algorithm (10)–(13)to solve the problem (33)–(35) and its dual.

Rate control: At time t, given congestion price p(t), thesource s adjusts its sending rate xs according to the localcongestion price at the source node

xsðpÞ ¼ U0s�1ðpsÞ; ð37Þ

where ps ¼ pik for s = [i,k] 2 S � D. In contrast to traditional

TCP congestion control where the source adjusts its send-ing rate according to the aggregate price along its path,in this algorithm the congestion price is generated locallyat the source node.

Note that, since

Xi;k

pki

Xj

f ki;j �

Xj

f kj;i

!¼Xi;j;k

f ki;j pk

i � pkj

� �;


the scheduling problem is equivalent to the followingproblem

maxf2P

Xi;j

fi;j maxk

pki � pk

j

� �: ð38Þ

This motivates the following joint scheduling and routingalgorithm:

Scheduling: Each node i collects congestion price infor-mation from its neighbor j, finds destination k(t) such that

kðtÞ 2 arg maxk pki ðtÞ � pk

j ðtÞ� �

, and calculates differential

price wi;jðtÞ ¼ pkðtÞi ðtÞ � pkðtÞ

j ðtÞ and passes this informationto its neighbors. Allocate capacities ~f i;jðtÞ over links (i, j)such that~f ðtÞ 2 arg max

f2P

Xði;jÞ2L

wi;jðtÞfi;j: ð39Þ

If the network with time-varying channel is considered,each node monitors the channel state h(t) and allocatescapacities ~f i;jðtÞ over links (i, j) such that~f ðtÞ 2 arg max

f2PðhðtÞÞ

Xði;jÞ2L

wi;jðtÞfi;j: ð40Þ

A D

E

F

C

B

Fig. 7. A simple network with two network layer flows. All links arebidirectional.

0 10 20 30 40 50.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Normaliz

Nor

mal

ized

Sou

rce

Rat

es

0 10 20 30 40 51

1.5

2

2.5

3

Normaliz

Nor

mal

ized

Con

gest

ion

Pric

es

Fig. 8. Source rates and congestion prices in

Routing: Over link (i, j), send a number of bits for destinationk(t) according to the rate determined by the scheduling.

The wi,j values represent the maximum differential con-gestion price of destination k flows between nodes i and j.The above algorithm uses backpressure to do optimalscheduling and find optimal routing. Also note that thescheduling problem is solved by the following assignment,

f ki;jðtÞ ¼

~f i;jðtÞ if k ¼ kðtÞ;0 if k – kðtÞ:

(

Congestion price update: Each node i updates its price withrespect to destination k, according to

pki ðt þ 1Þ ¼ pk

i ðtÞ þ ct xki ðpðtÞÞ �

Xj:ði;jÞ2L

f ki;jðpðtÞÞ

"

�X

j:ðj;iÞ2L

f kj;iðpðtÞÞ

!!#þ; ð41Þ

and passes the price pki to its neighbors. Note that pk

i ðtÞ isinterpreted as congestion price at the beginning of timeslot t.

The above dual algorithm motivates a joint congestioncontrol, routing and scheduling design where at the trans-port layer sources s individually adjust their rates accord-ing to the local congestion price at the source nodes, andnodes i individually update their prices according to (41),and at the network/link layer nodes i solve the scheduling(39) and route data flows accordingly. Also, note that thecongestion control is not an end-to-end scheme. There isno need to maintain end-to-end paths and no communica-tion overhead for congestion control.

0 60 70 80 90 100ed Time

Flow AFFlow BE

0 60 70 80 90 100ed Time

Flow AFFlow BE

the network with fixed link capacities.

0 10 20 30 40 50 60 70 80 90 1000.3

0.4

0.5

0.6

0.7

0.8

Normalized Time

Aver

age

Sour

ce R

ates

Flow AFFlow BE

0 10 20 30 40 50 60 70 80 90 1001

1.5

2

2.5

3

Normalized Time

Aver

age

Con

gest

ion

Pric

es

Flow AFFlow BE

Fig. 9. The average source rates and congestion prices in the network with fixed link capacities.

0 10 20 30 40 50 60 70 80 90 1000

0.5

1

1.5

2

Normalized Time

Nor

mal

ized

Sou

rce

Rat

es

Flow AFFlow BE

0 10 20 30 40 50 60 70 80 90 1000.5

1

1.5

2

2.5

3

Normalized Time

Nor

mal

ized

Con

gest

ion

Pric

es Flow AFFlow BE

Fig. 10. Source rates and congestion prices in the network with time-varying link capacities.


7.2. Numerical examples

In this subsection, we provide numerical examples tocomplement the analysis in the previous subsections. We

consider a simple multihop network with primary interfer-ence as shown in Fig. 7, and assume that there are two net-work layer flows A ? F and B ? E with the same utilityUs(xs) = logxs.

0 10 20 30 40 50 60 70 80 90 1000.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Normalized Time

Aver

age

Sour

ce R

ates

Flow AFFlow BE

0 10 20 30 40 50 60 70 80 90 1001

1.5

2

2.5

3

Normalized Time

Aver

age

Con

gest

ion

Pric

es

Flow AFFlow BE

Fig. 11. The average source rates and congestion prices in the network with time-varying link capacities.

Links to Transmit

Utility Functions

Congestion Control

Routing

Scheduling

Con

gest

ion

Pric

e

Fig. 12. Layering as dual decomposition.


The network with fixed channel and single-rate devicesWe consider first the network with fixed link capacities.

For simplicity, we assume that links CE, EC, BF and FB haveone unit of capacity and all other links have 2 units ofcapacity when active. Fig. 8 shows the evolution of sourcerate and congestion price of each flow with the joint algo-rithm (37), (39) and (41) with stepsize c = 0.2. We see thatthey converge quickly to a neighborhood of the optimaland oscillate around the optimal. However, Fig. 9 showsthat the average source rates and congestion prices aresmooth and approach the optimum monotonically. Weagain note that the performance of the algorithm is muchbetter than the bound of cG2/2 specified in Theorems 2and 3.

The network with time-varying channel and multirate devicesWe now consider the network with time-varying link

capacities. For simplicity, we assume that links CE, EC, BFand FB’s capacities are identically, uniformly distributedover 0.5, 1 and 1.5 units, while other links’ capacities areidentically, uniformly distributed over 1, 2 and 3 units.Thus, the average capacity for each link when active isthe same as that in the example with fixed link capacities.

Figs. 10 and 11 show the evolution of source rates, con-gestion prices and their averages with the same stepsizec = 0.2. The source rates and congestion prices have muchlarger variations than those with fixed channel, due tothe channel variations. But the average source rates andcongestion prices are still smooth, and converge quicklyand monotonically to optimal values. Note that, althoughthe average link capacity when active is the same as thatin fixed channel, each flow achieves larger sending rates.

This is again due to multi-user diversity that we exploitwhen doing scheduling. Also note that the increase insending rate of flow BE is much more notable. This is be-cause node B has more neighbors and thus a much largermulti-user diversity.


7.3. Summary

We have presented a model for the joint design of con-gestion control, routing and scheduling for multihop wire-less networks. The resulting dual algorithm motivates ajoint design where at the transport layer, sources s adjusttheir rates according to the local congestion price at thesource nodes, and at the network/link layer nodes solvethe scheduling and route data flows according to backpres-sure in congestion between neighboring nodes. As our de-sign only requires nodes exchanging local informationwith their neighbors and does not need to maintain end-to-end paths, it has a very low communication overheadand can adapt to changing topologies such as those in mo-bile multihop networks.

8. Conclusions

We have seen in this paper that, by formulating a gen-eral utility maximization problem for the network design,duality theory leads to a natural ‘‘vertical” decompositioninto functional modules of various layers of the protocolstack and ‘‘horizontal” decomposition into distributedcomputation across various network nodes or links. Asshown in Fig. 12, our current theory integrates three func-tions—congestion control, routing and scheduling—intransport, network and link layers into a coherent frame-work. With this layering scheme, the dual variables ofthe utility maximization problem capture the networkstate information and are the information that is passedacross the interfaces among different layers. These layersinteract through and are coordinated by the dual variables,i.e., congestion prices, so as to achieve global optimality.Even though this framework does not provide all the de-sign and implementation details (such as the implementa-tion of congestion prices and signalling mechanism), ithelps us understand issues, clarify ideas, and suggestsdirections, leading to better and more robust designs formultihop wireless networks.

This framework—layering as dual decomposition in par-ticular and layering as optimization decomposition in gen-eral—holds promise for being extended to provide amathematical theory for network architecture, and to al-low us to systematically derive the layering structure ofthe various mechanisms of different protocol layers, theirinterfaces, and the control information that must crossthese interfaces to achieve a certain performance androbustness. In this general framework, application needs(possibly, plus other performance metrics such as networkcost) form the objective function (i.e., network utility to bemaximized) and the restrictions in resource provisioningare translated into the constraints of the generalized net-work utility maximization problem. By choosing differentobjective functions and having different sets of decisionvariables involved, we can explicitly characterize and tradeoff different design objectives such as performance, scala-bility and robustness.

There exist, however, some challenging issues with thisframework. First, utility design, i.e., how to model the useror application needs, is not an easy task, especially for real-

time applications. Second, the general utility maximizationproblems may be nonlinear, nonconvex optimization withinteger constraints. Third, this framework only involvesthe functionalities of the data plane of the network, butleaves out the issues related to the control plane such asimplementation and management complexity. Addressingthese issues will be a future research step.

Acknowledgments

The authors thank Mung Chiang for collaborations onthe topic. This work is partially supported by Army Insti-tute for Collaborative Biotechnologies, AFOSR awardFA9550-05-1-0032, ARO grant DAAD19-02-1-0283, NSFgrant CNS-0435520, NSF NetSE grant CNS-0911041, AROMURI grant W911NF-08-1-0233, ONR MURI grantN000140810747, and the Caltech Lee Center for AdvancedNetworking.

References

[1] S. Asmussen, Applied Probability and Queues, second ed., Springer,2003.

[2] D. Bertsekas, Nonlinear Programming, second ed., Athena Scientific,1999.

[3] L. Chen, S. Low, J. Doyle, Joint congestion control and media accesscontrol design for ad hoc wireless networks, in: Proceedings of IEEEConference on Computer Communications (Infocom), March 2005.

[4] L. Chen, S. Low, M. Chiang, J. Doyle, Cross-layer congestion control,routing and scheduling design in ad hoc wireless networks, in:Proceedings of IEEE Conference on Computer Communications(Infocom), April 2006.

[5] M. Chiang, Balancing transport and physical layers in wirelessmultihop networks: jointly optimal congestion control and powercontrol, IEEE Journal on Selected Areas in Communications 23 (1)(2005) 104–116.

[6] M. Chiang, S.H. Low, R.A. Calderbank, J.C. Doyle, Layering asoptimization decomposition, Proceedings of the IEEE (2007).

[7] R. Diestel, Graph Theory, Springer-Verlag, 1997.[8] A. Eryilmaz, R. Srikant, Fair resource allocation in wireless networks

using queue-length-based scheduling and congestion control, in:Proceedings of IEEE Conference on Computer Communications(Infocom), March 2005.

[9] Z. Fang, B. Bensaou, Fair bandwidth sharing algorithms based ongame theory frameworks for wireless ad hoc networks, in:Proceedings of IEEE Conference on Computer Communications(Infocom), March 2004.

[10] M. Gerla, R. Bagrodia, L. Zhang, K. Tang, L. Wang, TCP over wirelessmultihop protocols: simulation and experiments, in: Proceedings ofIEEE International Conference on Communications (ICC), June 1999.

[11] B. Hajeck, G. Sasaki, Link scheduling in polynomial time, IEEETransactions on Information Theory 34 (5) (1988) 901–917.

[12] J. Hoepman, Simple distribute weighted matchings, preprint,October 2004, Available from http://arxiv.org/abs/cs/0410047.

[13] IEEE, Wireless LAN media access control (MAC) and physical layer(PHY) specifications, IEEE Standard 802.11, June 1999.

[14] L. Jiang, J. Walrand, A distributed CSMA algorithm for throughputand utility maximization in wireless networks, Allerton Conferenceon Communication, Control, and Computing (2008).

[15] D.B. Johnson, D.A. Maltz, Dynamic source routing in ad-hoc wirelessnetworks, in: T. Imielinski, H. Korth (Eds.), Mobile Computing,Kluwer Academic Publishers, 1996.

[16] F.P. Kelly, A.K. Maulloo, D.K.H. Tan, Rate control for communicationnetworks: shadow prices, proportional fairness and stability, Journalof Operations Research Society 49 (3) (1998) 237–252.

[17] M. Kodialam, T. Nandagopal, Characterizing achievable rates inmulti-hop wireless networks: the joint routing and schedulingproblem, in: Proceedings of ACM International Conference on MobileComputing and Networking (MobiCom), September 2003.

[18] S. Kunniyur, R. Srikant, End-to-end congestion control schemes:utility functions, random losses and ECN marks, IEEE/ACMTransactions on Networking 11 (5) (2003) 689–702.

http://arxiv.org/abs/cs/0410047


[19] H.J. Kushner, P.A. Whiting, Convergence of proportional-fair sharingalgorithms under general conditions, IEEE/ACM Transactions onWireless Communications 3 (4) (2004) 1250–1259.

[20] X. Lin, N. Shroff, Joint rate control and scheduling in multihopwireless networks, in: Proceedings of IEEE Conference on Decisionand Control (CDC), December 2004.

[21] X. Lin, N. Shroff, The impact of imperfect scheduling on cross-layerrate control in multihop wireless networks, in: Proceedings of IEEEConference on Computer Communications (Infocom), March 2005.

[22] S.H. Low, D.E. Lapsley, Optimal flow control, I: basic algorithm andconvergence, IEEE/ACM Transactions on Networking 7 (6) (1999)861–874.

[23] S.H. Low, A duality model of TCP and queue managementalgorithms, IEEE/ACM Transactions on Networking 11 (4) (2003)525–536.

[24] E. Modiano, D. Shah, G. Zussman, Maximizing throughput in wirelessnetworks via gossiping, in: Proceedings of ACM InternationalConference on Measurement and Modeling of Computer Systems(Sigmetrics), June 2006.

[25] T. Nandagopal, T.E. Kim, X. Gao, V. Bharghhavan, Achieving MAClayer fairness in wireless packet networks, in: Proceedings of ACMInternational Conference on Mobile Computing and Networking(MobiCom), August 2000.

[26] M. Neely, E. Modiano, C. Rohrs, Dynamic power allocation androuting for time varying wireless networks, in: Proceedings of IEEEConference on Computer Communications (Infocom), April 2003.Journal version, IEEE Journal on Selected Areas in Communications,23(1) 89–103, January 2005.

[27] M. Neely, E. Modiano, C.P. Li, Fairness and optimal stochastic controlfor heterogeneous netwroks, in: Proceedings of IEEE Conference onComputer Communications (Infocom), March 2005.

[28] J. Ni, B. Tan, R. Srikant, Q-CSMA: queue length-based CSMA/CAalgorithms for achieving maximum throughput and low delay inwireless networks, Proceedings of IEEE INFOCOM Mini-Conference(2010).

[29] C.H. Papadimitriou, K. Steiglitz, Combinatorial Optimization:Algorithm and Complexity, Dover Publications, 1998.

[30] R. Preis, Linear time 1/2-approximation algorithm for maximumweighted matching in general graphs, in: Proceedings of the 16thAnnual Symposium on Theoretical Aspects of Computer Science(STACS), March 1999.

[31] C.E. Perkins, E.M. Royer, Ad-hoc on-demand distance vector routing,in: Proceedings of IEEE Workshop on Mobile Computing Systemsand Applications (WMCSA), Febuary 1999.

[32] J. Shin, D. Shah, S. Rajagopalan, Network adiabatic theorem: anefficient randomized protocol for contention resolution, Proceedingsof ACM SIGMETRICS (2009).

[33] N.Z. Shor, Minimization Methods for Non-Differentiable Functions,Springer-Verlag, 1985.

[34] A.L. Stolyar, MaxWeight scheduling in a generalized switch: statespace collapse and workload minimization in heavy traffic, Annualsof Applied Probability 14 (1) (2004) 1–53.

[35] A.L. Stolyar, On the asymptotic optimality of the gradient schedulingalgorithm for multiuser throughput allocation, Operation Research53 (1) (2005) 12–25.

[36] A.L. Stolyar, Maximizing queueing network utility subject tostability: Greedy primal–dual algorithm, Queueing Systems 50 (4)(2005) 401–457.

[37] K. Tang, M. Gerla, Fair sharing of MAC under TCP in wireless ad hocnetworks, in: Proceedings of IEEE Workshop on Multiaccess,Mobility and Teletraffic for Wireless Communications (MMT),October 1999.

[38] L. Tassiulas, A. Ephremides, Stability properties of constrainedqueueing systems and scheduling policies for maximumthroughput in multihop radio networks, IEEE Transactions onAutomatic Control 37 (12) (1992) 1936–1948.

[39] D.N. Tse, Multiuser Diversity in Wireless Networks, <http://www.eecs.berkly.edu/dtse/stanford416.ps>, April 2001.

[40] R. Uehara, Z. Chen, Parallel approximation algorithms for maximumweighted matching in general graphs, Information ProcessingLetters 76 (1–2) (2000) 13–17.

[41] X. Wang, K. Kar, Cross-layer rate control for end-to-end proportionalfairness in wireless networks with random access, in: Proceedings ofACM International Symposium on Mobile Ad Hoc Networking andComputing (MobiHoc), May 2005.

[42] M. Wattenhofer, R. Wattenhofer, Distributed weighted matching, in:Proceedings of the 18th Annual Conference on DistributedComputing (DISC), October 2004.

[43] X. Wu, R. Srikant, Regulated maximal matching: a distributedscheduling algorithm for multi-hop wireless networks with node-exclusive spectrum sharing, in: Proceedings of IEEE Conference onDecision and Control (CDC), December 2005.

[44] L. Xiao, M. Johnasson, S. Boyd, Simultaneous routing and resourceallocation for wireless networks, Proceedings of IEEE Conference onDecision and Control (CDC) (December) (2001).

[45] S. Xu, T. Saadawi, Does the IEEE 802.11 MAC protocol work well inmultihop wireless ad hoc networks?, IEEE CommunicationsMagazine 39 (6) (2001) 130–137

[46] K. Xu, S. Bae, S. Lee, M. Gerla, TCP behavior across multihop wirelessnetworks and the wired internet, in: Proceedings of ACMInternational Workshop on Wireless Mobile Multimedia(WoWMoM), September 2002.

[47] K. Xu, M. Gerla, L. Qi, Y. Shu, Enhance TCP fairness in ad hoc wirelessnetworks using neighborhood RED, in: Proceedings of ACMInternational Conference on Mobile Computing and Networking(MobiCom), September 2003.

[48] Y. Xue, B. Li, K. Nahrstedt, Price-based resource allocation in wirelessad hoc networks, in: Proceedings of IEEE/ACM InternationalWorkshop on Quality of Service (IWQoS), June 2003.

[49] Y. Yi, S. Shakkottai, Hop-by-hop congestion control over a wirelessmulti-hop network, in: Proceedings of IEEE Conference on ComputerCommunications (Infocom), March 2004.

Lijun Chen received the B.S. degree from the University of Science andTechnology of China; the M.S. degree from the Institute of TheoreticalPhysics, Chinese Academy of Sciences, and from the University of Mary-land at College Park; and the Ph.D. degree from California Institute ofTechnology. He is a Research Scientist with the Control and DynamicalSystems Department, California Institute of Technology. His currentresearch interests are in communication networks, optimization, gametheory, and their engineering application. Dr. Chen was a co-recipient ofthe Best Paper Award at the IEEE International Conference on Mobile Ad-hoc and Sensor Systems (MASS) in 2007.

Steven H. Low received the B.S. degree from Cornell University, and thePh.D. degree from the University of California, Berkeley, both in electricalengineering. He is a Professor with the Computer Science and ElectricalEngineering departments at California Institute of Technology, and anAdjunct Professor with Swinbourne University, Australia. Prior to that, hewas with AT&T Bell Laboratories, Murray Hill, NJ, and the University ofMelbourne, Melbourne, Australia. Dr. Low was a co-recipient of the IEEEBennett Prize Paper Award in 1997 and the 1996 R&D 100 Award. He wasa member of the Networking and Information Technology TechnicalAdvisory Group for the US President’s Council of Advisors on Science andTechnology (PCAST) from 2006 to 2007. He was on the Editorial Board ofthe IEEE/ACM TRANSACTIONS ON NETWORKING from 1997 to 2006 andthe Computer Networks Journal from 2003 to 2005. He is currently on theeditorial boards of Computing Surveys and Foundations and Trends inNetworking and is a Senior Editor of the IEEE JOURNAL ON SELECTEDAREAS IN COMMUNICATION.

John C. Doyle received the B.S. and M.S. degrees in electrical engineeringfrom Massachusetts Institute of Technology, and the Ph.D. degree in mathfrom the University of California, Berkeley. He is John G. Braun Professorof control and dynamical systems, electrical engineering, and bioengi-neering at California Institute of Technology. His group has contributed tomany software projects, including the Robust Control Toolbox (muTools),SOSTOOLS, Systems Biology Markup Language (SBML), and Fast AQM,Scalable TCP (FAST). His current research is in theoretical foundations forcomplex networks in engineering and biology, focusing on architecture,and for multiscale physics. His early work was in the mathematics ofrobust control, including recent extensions to nonlinear and hybrid sys-tems. Dr. Doyle has received the IEEE W. R. G. Baker Award, the IEEETRANSACTIONS ON AUTOMATIC CONTROL George S. Axelby Award(twice), and Best Paper awards at the ACM SIGCOMM and AACC AmericanControl conferences. His individual awards include the AACC EckmanAward, and the IEEE Control Systems Field Award, and the IEEE Centen-nial Outstanding Young Engineer Award.

http://www.eecs.berkly.edu/dtse/stanford416.ps

http://www.eecs.berkly.edu/dtse/stanford416.ps

Documents

Cross-layer design in multihop wireless networkslich1539/papers/Chen-2011-CLD...constraints at link layer for congestion control over multi-hop wireless networks. Xiao et al. [44]