Balancing Transport and Physical Layersin Wireless Multihop Networks: Jointly Optimal

Congestion Control and Power ControlMung Chiang

Electrical Engineering Department, Princeton University, NJ 08544

Abstract— In a wireless network with multihop transmissionsand interference-limited link rates, can we balance power controlin the physical layer and congestion control in the transport layerto enhance the overall network performance while maintaining thearchitectural modularity between the layers? We answer this ques-tion by presenting a distributive power control algorithm that cou-ples with existing TCP protocols to increase end-to-end throughputand energy efficiency of the network. Under the rigorous frame-work of nonlinearly constrained utility maximization, we provethe convergence of this coupled algorithm to the global optimumof joint power control and congestion control, for both synchro-nized and asynchronous implementations. The rate of convergenceis geometric and a desirable modularity between the transport andphysical layers is maintained. In particular, when congestion con-trol uses TCP Vegas, a simple utilization in the physical layer ofqueuing delay information suffices to achieve the joint optimum.Analytic results and simulations illustrate other desirable proper-ties of the proposed algorithm, including robustness to channel out-age and to path loss estimation errors, and flexibility in trading-offperformance optimality for implementation simplicity.

This paper presents a step towards a systematic understand-ing of ‘layering’ as ‘optimization decomposition’, where the overallcommunication network is modelled by a generalized network util-ity maximization problem, each layer corresponds to a decomposedsubproblem, and the interfaces among layers are quantified as theoptimization variables coordinating the subproblems. In the caseof the transport and physical layers, link congestion prices turn outto be the optimal ‘layering prices’.

Keywords: Congestion control, Convex optimization, Cross-layerdesign, Energy-aware protocols, Lagrange duality, Power control,Transmission Control Protocol, Utility maximization, Wireless ad hocnetworks.

I. I NTRODUCTION

We consider wireless networks with multihop transmissionsand interference-limited link rates. In order to achieve highend-to-end throughput in an energy efficient manner, congestioncontrol and power control need to be jointly designed and dis-tributively implemented. Congestion control mechanisms, suchas those in Transmission Control Protocol (TCP), regulate theallowed source rates so that the total traffic load on any linkdoes not exceed the available capacity. At the same time, the at-tainable data rates on wireless links depend on the interferencelevels, which in turn depend on the power control policy. Thispaper proposes and analyzes a distributed algorithm forjointlyoptimal end-to-end congestion control and per-link power con-trol. The algorithm utilizes the coupling between the transport

and physical layers to increases end-to-end throughput and toenhance energy efficiency in a wireless multihop network.

Congestion avoidance mechanisms in TCP variants have re-cently been shown to approximate distributed algorithms thatimplicitly solve network utility maximization problems. Tra-ditionally, this class of optimization problems are linearly con-strained by link capacities that are assumed to be fixed quanti-ties. However, network resources can sometimes be allocated tochange link capacities, therefore change TCP dynamics and theoptimal solution to network utility maximization. For example,in CDMA wireless networks, transmit powers can be controlledto induce different Signal to Interference Ratios (SIR) on thelinks, changing the attainable throughput on each link.

This formulation of network utility maximization with ‘elas-tic’ link capacities leads to a new approach of congestion avoid-ance in wireless multihop networks. The current approach ofcongestion control in the Internet is to avoid the developmentof a bottleneck link by reducing the allowed transmission ratesfrom all the sources using this link. Intuitively, an alternativeapproach is to build, in real time, a larger transmission ‘pipe’and ‘drain’ the queued packets faster on a bottleneck link. In-deed, a smart power control algorithm would allocate just the‘right’ amount of power to the ‘right’ nodes to alleviate the bot-tlenecks, which may then induce an increase in end-to-end TCPthroughput. But there are two major difficulties in making thisidea work: defining which link constitutes a ‘bottleneck’a pri-ori is infeasible, and changing the transmit power on one linkalso affects the data rates available on other links. Due to inter-ference in wireless CDMA networks, increasing the capacity onone link reduces those on other links. We need to find an algo-rithm that distributively and adaptively detects the ‘bottleneck’links and optimally ‘shuffles’ them around in the network.

This intuitive approach is made precise and rigorous in thispaper. After reviewing the background materials in Section IIand specifying the problem formulation in Section III, we pro-pose in Section IV a distributed power control algorithm thatcouples with existing TCP algorithms to solve the joint problemof congestion control and power control. The joint algorithmcan be distributively implemented on a multihop network, de-spite the fact that the data rate on a wireless link is a globalfunction of all the interfering powers. Interpretations in termsof data rate demand-supply coordination through shadow pricesare presented, as well as numerical examples illustrating thatend-to-end throughput and energy efficiency of the network can

indeed be significantly increased.

It is certainly not a surprise that performance can be enhancedthrough a cross-layer design. The more challenging task isto analyze the algorithm rigorously and to make it attractiveaccording to other important design criteria. In Section VI,we prove that, under very mild conditions, the proposed algo-rithm converges to the joint and global optimum of the nonlin-ear congestion-power control. In Subsection VII-A, we providethe sufficient conditions under which convergence to the globaloptimum is maintained despite errors in path loss estimation orpacket losses due to channel outage. Cross-layer designs usu-ally improve performance at the expense of higher complexityin communication and computation. In Subsection VII-B, wepropose a suite of simplified versions of the optimal algorithmto flexibly trade-off performance with complexity. In Subsec-tion VII-C, we prove that the algorithm converges under any fi-nite asynchronism in practical implementation, and characterizea condition under which asynchronous implementation does notinduce a reduction in convergence speed. In Subsection VII-D,we show that the rate of convergence of the algorithm is geo-metric, and provide a simple bound on convergence speed. Fur-ther suggestions on choosing algorithm parameters and achiev-ing convergence speedup are made in Subsection VII-E. Evenafter crossing the layers, architectural modularity is desirablefor practical implementation and future network evolution. Inthis paper, the desirable convergence is achieved as power con-trol uses the same link prices that are already generated by TCPfor regulating distributed users. Performance enhancement isachieved without modifying existing TCP protocol stack.

Assumptions behind the models and limitations on the resultsare clearly stated throughout the paper, while extensions areoutlined in Section V. This paper presents a step towards un-derstanding ‘layering’ as ‘optimization decomposition’, wherethe overall communication network is modelled by a general-ized utility maximization problem, each layer corresponds to adecomposed subproblem, and the interfaces among layers arequantified as the optimization variables coordinating the sub-problems. In the case of the transport and physical layers, linkcongestion prices turn out to be the optimal ‘layering prices’.Possible future research directions are discussed in Section VIII.

II. BACKGROUND AND RELATED WORK

Both power control in CDMA wireless networks and con-gestion control in the Internet are extensively researched topics.Many power control algorithms have been proposed in the liter-ature, but the effects of power control on source rate regulationthrough end-to-end congestion control have not been character-ized.

TCP is one of the two widely-used transport layer protocols inthe Internet. A main function performed by TCP is network con-gestion control and end-to-end rate allocation. Roughly speak-ing, there are two phases of TCP congestion control: slow startand congestion avoidance. Long-lived flows usually spend mostof the time in congestion avoidance. Similar to recent work onutility maximization models of TCP, we assume a deterministicflow model for the average equilibrium behavior of the conges-tion avoidance phase. TCP uses sliding windows to adjust the

allowed transmission rate in each source based on implicit orexplicit feedback of the congestion signals generated by ActiveQueue Management (AQM). Among the variants of TCP, suchas Tahoe, Reno, Vegas, and FAST, some use loss as conges-tion signal and others use delay. Most of this paper focuses ondelay-based congestion signals because of the nice propertieson convergence, stability, and fairness [21], and the simulationexamples use TCP Vegas [5] at the sources.

The basic rate allocation mechanism of TCP Vegas is as fol-lows. Letds be the propagation delay along the path originat-ing from sources, andDs be the propagation plus congestion-induced queuing delay. Obviouslyds = Ds when there is nocongestion on all the links used by sources. The window sizews is updated at each sources according to whether the differ-ence between the expected ratews

dsand the actual ratews

Ds, where

Ds is estimated by the timing of ACK packets, is smaller than aparameterαs:

ws(t + 1) =

ws(t) + 1Ds(t) if ws(t)

ds− ws(t)

Ds(t) < αs

ws(t)− 1Ds(t) if ws(t)

ds− ws(t)

Ds(t) > αs

ws(t) else.

The end-to-end throughput for each path is the allowed sourcerate xs, which is proportional to the window size:xs(t) =ws(t)Ds(t) .

Following the seminal work by Kellyet. al. [16], [17] thatanalyze network rate allocation as a distributed solution of util-ity maximization, TCP congestion control mechanisms have re-cently been analyzed as approximated distributed algorithmssolving appropriately formulated utility maximization problems(e.g., [18], [21], [22], [20], [24]). The key innovation in thisseries of work is to interpret source rates asprimal variables,link congestion measures asdual variables, and a TCP–AQMprotocol as a distributed algorithm over the Internet implicitlyto solve the following network utility maximization. Considera wired communication network withL links, each with afixedcapacity ofcl bps, andS sources, each transmitting at a sourcerate ofxs bps. Each source emits one flow, using a fixed setL(s) of links in its path, and has an increasing, strictly concave,and twice differentiable utility functionUs(xs). Network util-ity maximization is the problem of maximizing the total utility∑

s Us(xs) over the source ratesx, subject to linear flow con-straints

∑s:l∈L(s) xs ≤ cl for all links l:

maximize∑

s Us(xs)subject to

∑s:l∈L(s) xs ≤ cl, ∀l,

x º 0.(1)

Different TCP-AQM protocols solve for different strictly con-cave utility functions using different types of congestion signals.For example, TCP Vegas is shown [22] to be implicitly solving(1) for logarithmic utility functions:Us(xs) = αsds log xs, us-ing queuing delays as the dual variables. Although TCP andAQM protocols were designed and implemented without regardto utility maximization, now they can bereverse-engineeredto determine the underlying utility functions and to rigorouslycharacterize many important properties.

An underlying assumption in the utility maximization modelsof TCP is that each communication link is a fixed-size transmis-

sion ‘pipe’ provided by the physical layer. This assumption isinvalid when the sizes of the ‘pipes’ depend on time-varyingchannel conditions and adaptive physical layer resource allo-cation, such as transmit power control in interference-limitedwireless networks. Different cases of utility maximizationjointly over rates and powers have been studied for wireless cel-lular networks,e.g., in [9], [23], and, in general, optimization-theoretic or game-theoretic studies of wireless network resourceallocation using the utility framework have been reported,e.g.,in [19], [25], [26], [27], [31], [32]. This paper focuses on jointlyoptimal congestion control and power control in wireless multi-hop networks.

Augmenting the utility maximization framework to includelayers other than the transport layer may lead to a generalmethodology for cross-layer design. Cross-layer issues in com-munication networks have attracted the attention of many re-searchers, forming a literature that is too large to be exhaustivelyreviewed in one paragraph. Complementing these cross-layerinvestigations, we examine the balance between the transportand physical layers and provide a quantitative framework of co-design across layers 1 and 4, under which theorems of globalconvergence can be proved for nonlinearly coupled dynamics.This cross-layer issue is particularly interesting because con-gestion control is conducted end-to-end while power control islink-based. The resulting jointly optimal congestion control andpower control algorithm enhances end-to-end throughput andenergy efficiency in wireless multihop networks. Echoing someof the cautionary notes on cross-layer designs, we also put spe-cial emphasis on the practical implementation issues of robust-ness, asynchronism, complexity, and the rate of convergence.

We note that there are at least two possible interpretations ofthe phrase ‘balancing transport and physical layers in wirelessnetworks’:

• Characterize the impacts of physical layer resource alloca-tion on TCP throughput, which is the focus of this paper.

• Characterize the impacts of wireless channel variationson TCP throughput and try to distinguish between packetlosses due to congestion and those due to fading. Thisproblem, which has been actively researched in bothacademia and industry, is not the subject of this paper.However, we will investigate the robustness of our algo-rithm to fading. The nonlinear convex optimization meth-ods used here, as well as in [14], can also be used for powercontrol that guarantee certain levels of packet loss neces-sary to sustain a desired TCP throughput.

It should be noted that we do not consider joint optimizationover routing or medium-access control in this paper. However, ageneralized utility maximization problem is proposed at the endof the paper as a possible vehicle to rigorously and systemati-cally study ‘layering’ as ‘optimization decomposition’.

III. PROBLEM FORMULATION

Consider a wireless multihop network withN nodes and anestablished logical topology, where some nodes are sources oftransmission and some nodes act as ‘voluntary’ relay nodes. Asequence of connected linksl ∈ L(s) forms a route originating

from sources. Let xs be the transmission rate of sources, andcl be the ‘capacity’, in terms of the attainable data rate ratherthan the information-theoretic multi-terminal channel capacity,on logical link l. Note that each physical link may need to beregarded as multiple logical links. Source nodes are indexed bys and logical links byl.

Revisiting the utility maximization formulation (1), forwhich TCP congestion control solves, we observe that in aninterference-limited wireless network, data rates attainable onwireless links are not fixed numbersc as in (1), and instead canbe written, for a large family of modulations, as a global andnonlinear function of the transmit power vectorP and channelconditions:

cl(P) =1T

log(1 + KSIRl(P)).

Here constantT is the symbol period, which will be assumedto be one unit without loss of generality, and constantK =

−φ1

log(φ2BER)whereφ1, φ2 are constants depending on the mod-

ulation and BER is the required bit error rate [13]. The signal tointerference ratio for linkl defined as SIRl = PlGll∑

k 6=lPkGlk+nl

for a given set of path lossesGlk (from the transmitter on log-ical link k to the receiver on logical linkl) and a given set ofnoisesnl (for the receiver on logical linkl). The Glk factorsincorporate propagation loss, spreading gain, and other normal-ization constants. Notice thatGll is the path gain on linkl (fromthe transmitter on logical linkl to the intended receiver on thesame logical link). With reasonable spreading gain,Gll is muchlarger thanGlk, k 6= l, and assuming that not too many close-by nodes transmit at the same time,KSIR is much larger than1. In this case,cl can be approximated aslog(KSIRl).

This wireless channel model has several limitations. First,it assumes fixed target decoding error probabilities and codingmodulation schemes. Transmit power is the only resource thatis being adapted. Second, the assumption thatKSIR is muchlarger than 1 is not always true. With this assumption, it willbe shown that whilelog(KSIRl(P)) is a nonlinearnonconcavefunction ofP, it can be converted into a nonlinearconcavefunc-tion through a log transformation, leading to a critical convexityproperty that establishes the global optimality of the proposedalgorithm. The important role played by convexity in utilitymaximization will be further discussed in Section VIII. Last butnot least, simple decoding is not the only option for a wirelesschannel. Either multi-user decoding that does not treat all in-terferences as noise or simple ‘amplify-and-forward’ signallingstrategies will lead to different physical layer models.

The network model is also limited by the assumptions onfixed nodes, fixed single-path routing, and perfect CDMA-basedmedium access. In addition to rate and power controls, twoother mechanisms to reduce bottleneck congestion are schedul-ing over different time slots and routing through alternate paths.Indeed, adaptive routing for mobile networks, and scheduling orcontention-based medium access in broadcast wireless transmis-sions are important research topics in their own rights. Whileneither will be optimized jointly with the algorithm in this pa-per, a preliminary framework to incorporate these networkingaspects will be presented in Section VIII.

With the above assumptions, we have specified the following

network utility maximization with ‘elastic’ link capacities:

maximize∑

s Us(xs)subject to

∑s:l∈L(s) xs ≤ cl(P), ∀l,

x,P º 0(2)

where the optimization variables are both source ratesx andtransmit powersP. The key difference from the standard utilitymaximization (1) is that each link capacitycl is now a func-tion of the new optimization variables: the transmit powersP.The design space is enlarged fromx to bothx andP, whichare clearly coupled in (2). Linear flow constraints onx becomenonlinear constraints on(x,P). In practice, problem (2) is alsoconstrained by the maximum and minimum transmit powers al-lowed at each transmitter on linkl: Pl,min ≤ Pl ≤ Pl,max, ∀l.

The nonlinearly constrained optimization (2) may be solvedby centralized computation using the interior-point method forconvex optimization [4], after the log transformation that con-verts it into a convex optimization problem, as will be shownin Section VI. However, in the context of wireless ad hoc net-works, newdistributivealgorithms are needed to solve (2). Thusthe major challenges are the two global dependencies in (2):

• Source ratesx and link capacitiesc are globally coupledacross the network, as reflected in the range of summation{s : l ∈ L(S)} in the constraints in (2).

• Each link capacitycl(P), in terms of the attainable datarate under a given power vector, is a global function of allthe interfering powers.

Our primary goal in this paper is to distributively find the jointand globally optimal solution(x∗,P∗) to problem (2) by break-ing down these two global dependencies.

IV. OPTIMAL ALGORITHM, PRICING INTERPRETATION,AND NUMERICAL EXAMPLE

We present the following distributive algorithm and laterprove that it converges to the joint and global optimum of (2)and possesses several other desirable properties of a cross-layerdesign. We first present the ideal form of the algorithm, as-suming synchronized discrete time slots, no propagation delay,and full-scale message passing. Practical issues on asynchro-nism, propagation delay, complexity, robustness, and the rate ofconvergence will be investigated in Section VII. To make the al-gorithm and its analysis concrete, we will focus on delay-basedprice and TCP Vegas window update (as reflected in items 1 and2 in the algorithm, respectively) and the corresponding logarith-mic utility maximization over(x,P):

maximize∑

s

∑s αsds log xs

subject to∑

s:l∈L(s) xs ≤ cl(P), ∀l,x,P º 0

(3)

As for problem (2), in practice problem (3) is also constrainedby the maximum and minimum transmit powers allowed at eachtransmitter on linkl. Extensions to other TCP variants and con-gestion prices will be discussed in Section V.

Jointly Optimal Congestion-control and Power-control(JOCP) Algorithm

During each time slott, the following four updates are carriedout simultaneously until convergence:

1) At each intermediate node, a weighted queuing delayλl

is implicitly updated1, whereγ > 0 is a constant weight:

λl(t+1) =

λl(t) +

γ

cl(t)

∑

s:l∈L(s)

xs(t)− cl(t)

+

.

(4)2) At each source, total delayDs is measured and used to up-

date the TCP window sizews. Consequently, the sourceratexs is updated:

ws(t + 1) =

ws(t) + 1Ds(t) if ws(t)

ds− ws(t)

Ds(t) < αs

ws(t)− 1Ds(t) if ws(t)

ds− ws(t)

Ds(t) > αs

ws(t) else.(5)

xs(t + 1) =ws(t + 1)

Ds(t).

3) Each transmitterj calculates a messagemj(t) ∈ R+

based on locally measurable quantities, and passes themessage to all other transmitters by a flooding protocol:

mj(t) =λj(t)SIRj(t)

Pj(t)Gjj.

4) Each transmitter updates its power based on locally mea-surable quantities and the received messages, whereκ >0 is a constant:

Pl(t + 1) = Pl(t) +κλl(t)Pl(t)

− κ∑

j 6=l

Gljmj(t). (6)

With the maximum and minimum transmit power con-straint(Pl,max, Pl,min) on each transmitter, the updatedpower is projected onto the interval[Pl,max, Pl,min].

We first present some intuitive arguments on this algorithmbefore proving the convergence theorem and discussing thepractical implementation issues. Item 2 is simply the TCP Ve-gas window update [5]. Item 1 is a modified version of queuingdelay price update [22] (and the original update [5] is an ap-proximation of item 1). Items 3 and 4 describe a new powercontrol using message passing [7]. Taking in the current values

of λj(t)SIRj(t)Pj(t)Gjj

as the messages from other transmitters indexedby j, the transmitter on linkl adjusts its power level in the nexttime slot in two ways: first increase power directly proportionalto the current price (e.g., queuing delay in TCP Vegas) and in-versely proportional to the current power level, then decreasespower by a weighted sum of the messages from all other trans-mitters, where the weights are the path lossesGlj . Intuitively, ifthe local queuing delay is high, transmit power should increase,with more moderate increase when the current power level is al-ready high. If queuing delays on other links are high, transmitpower should decrease in order to reduce interference on thoselinks.

1This is using an average model for deterministic fluids. The difference be-tween the total ingress flow intensity and the egress link capacity, divided by theegress link capacity, gives the average time that a packet needs to wait beforebeing sent out on the egress link.

Note that to computemj , the values of queuing delayλj ,signal-interference-ratio SIRj , and received power levelPjGjj

can be directly measured by nodej locally. This algorithm onlyuses the resulting messagemj but not the individual values ofλj , SIRj , Pj andGjj . Each message is simply a real number.To conduct the power update,Glj factors are assumed to be es-timated through training sequences. In practical wireless ad hocnetworks,Glj are stochastic rather than deterministic and pathloss estimations can be inaccurate. The effects of the fluctua-tions ofGlj will be discussed in Subsection VII-A.

We also observe that the power control part of the joint algo-rithm can be interpreted as the selfish maximization of a localutility function of power by the transmitter of each linkl:

maximizePlUl(Pl)

whereUl(Pl) = λlcl − βlPl and βl =∑

j 6=l Gljmj . Thiscomplements the standard interpretation of congestion controlas the selfish maximization of a local utility functionUs(xs) byeach sources.

The known source algorithm (5) and queue algorithm (4) ofTCP-AQM, together with the new power control algorithm (6),form a set of distributed, joint congestion control and resourceallocation in wireless multihop networks. As the transmit pow-ers change, SIR and thus data rate also change on each link,which in turn change the congestion control dynamics. At thesame time, congestion control dynamics change the dual vari-ablesλ(t), which in turn change the transmit powers. Fig-ure 1 shows this nonlinear coupling of ‘supply’ (regulated bypower control) and ‘demand’ (regulated by congestion control),through the same shadow pricesλ that are currently used byTCP to regulate distributed demand. Nowλ serves the sec-ond function of cross-layer coordination in the JOCP Algorithm.Theorem 1 in Section VI proves that this globally coupled, non-linear dynamic converges to the jointly optimal(x∗,P∗).

Shadow Price

(Supply)(Demand)

Shadow Price Shadow Price

P

cx

x

Power ControlTransmit Node

TCP Source Node

Node QueueIntermediate

Fig. 1. Nonlinearly coupled dynamics of joint congestion and power control.

It is important to note that there is no need to change the exist-ing TCP congestion control and queue management algorithms.All that is needed to achieve the joint and global optimum of (3)is to utilize the values of weighted queuing delay in designingpower control algorithm in the physical layer. This approach iscomplementary to some recent suggestions in the Internet com-munity to pass physical layer information for a better control ofrouting and congestion in upper layers.

Much recent work has been done on opportunistic schedulingat the MAC layer based on the physical layer channel condi-tions. The JOCP Algorithm complements such work by consid-ering how can physical layer resource allocation be adapted toenhance the end-to-end utilities. Transport layer utilities guidehow power control should be conducted, using very little in-formation exchange across the layers and requiring no change

within the transport layer.

Using the JOCP Algorithm (4,5,6), we simulated the abovejoint power and congestion control for various wireless net-works with different topologies and fading environments. Theadvantage of such a joint control can be captured even in a smallillustrative example, where the logical topology and routes forfour multi-hop connections are shown in Figure 2. Sources ateach of the four flows use TCP Vegas window updates withαs

ranging from 3 to 5. The path lossesGij are determined by therelative physical distancesdij , which we vary in different exper-iments, byGij = d−4

ij . The target BER is10−3 on each logicallink.

4

32

1

Fig. 2. The logical topology and connections for an illustrative example.

Transmit powers, as regulated by the proposed distributedpower control, and source rates, as regulated through TCP Ve-gas window update, are shown in Figure 3. The initial condi-tions of the graphs are based on the equilibrium states of TCPVegas with fixed power levels of2.5mW . With power control,the transmit powersP distributively adapt to induce a ‘smart’capacityc and queuing delayλ configuration in the network,which in turn lead to increases in end-to-end throughput as indi-cated by the rise in all the allowed source rates. Notice that somelink capacities actually decrease while the capacities on the bot-tleneck links rise to maximize the total network utility. This isachieved through a distributive adaptation of power, which low-ers the power levels that cause most interference on the linksthat are becoming a bottleneck in the dynamic demand-supplymatching process. Confirming our intuition, such a ‘smart’ al-location of power tends to reduce the spread of queuing delays,thus preventing any link from becoming a bottleneck. Queuingdelays on the four links do not become the same though, due tothe asymmetry in traffic load on the links and different weightsin the logarithmic utility objective functions.

We indeed achieve the primary goal of this co-design acrossthe transport and physical layers. The end-to-end throughputper watt of power transmitted,i.e., the Throughput Power Ratio(TPR), is82% higher with power control. A series of simula-tions are conducted based on different fading environments andTCP Vegas parameter settings. Based on the resulting statisticsof TPR, We see that power control (6) increases TCP through-put and TPR in all experiments, and in78% of the instances, en-ergy efficiency rises by75% to 115%, compared to TCP withoutpower control. Power control and congestion control, each run-ning distributively and coordinated through the dual variables,work together to enhance the energy efficiency of multi-hoptransmissions across wireless multihop networks.

V. EXTENSIONS

Last section only describes the basic version of the JOCP al-gorithm. Many variations can be readily accommodated withoutsubstantial changes in the algorithm and its analysis.

For example, the source utilities can be any increasing,strictly concave functionsUs other than the logarithm function.

0 500 1000 1500 20001

1.5

2

2.5

3

3.5

4Transmit Powers

mW

0 500 1000 1500 20000

2

4

6

8

10Link Data Rates

kbps

0 500 1000 1500 20000

1

2

3

4

5

6Link Queuing Delays

Time

ms

0 500 1000 1500 20001

2

3

4

5End−To−End Throughputs

Time

kbps

Fig. 3. A typical numerical example of joint TCP Vegas congestion controland power control. The top left graph shows the primal variablesP. The lowerleft graph shows the dual variablesλ. The lower right graph shows the primalvariablesx, i.e., the end-to-end throughput. In order of their y-axis values afterconvergence, the curves in the top left, top right, and bottom left graphs areindexed by the third, first, second, and fourth links in Figure 2. The curves inthe bottom right graph are indexed by flows 1, 4, 3, 2.

Different utilities represent different types of TCP variants. Aswill be shown in the proof in Section VI, (5) and (6) are solvingtwo decomposed subproblems that are coordinated by the con-gestion pricesλ. Instead of updatingλ after moving only onestep along the solution path in these two subproblems, we couldhave waited for the convergence of the subproblems for a givenλ. In that case, each source would be solving the followingproblem: x∗s(λ) = U

′−1s (

∑l∈L(s) λl), and the power update

in (6) would be allowed to converge beforeλ are updated by(4) (which is more practical if the time scale of power update ismuch smaller than that of link price update). Convergence theo-rem in Section VI remains valid with the above generalizations.

If metrics other than queuing delay is used as congestionprice, e.g., packet loss in TCP Reno, then the price update (4)will look different. Any link prices with the following equilib-rium properties can act as the dual variables coordinating con-gestion control and power control:λ∗l (

∑s:l∈L(s) x∗s−c(P∗)) =

0, ∀l. However, depending on the specific price update equa-tion, convergence may not be guaranteed.

If energy efficiency is desired to be modelled explicitly in theobjective function, we can subtract a sum of increasing, con-vex power cost functions

∑l Vl(Pl) from the network utility∑

s Us(xs), and accordingly modify the power update equation.

In addition to end-to-end rate allocation over a fixed single-path route, multicommodity flow type of routing can easily bejointly optimized with power control (6). If the relay nodes re-quire incentives to help relay traffic originating from the sourcenodes, joint optimization over source rate, total relay rate, in-centive pricing, and transmit power can be conducted using asimilar message passing approach.

The balance between the transport and physical layers is im-portant not only for wireless ad hoc or cellular networks, it is

also relevant to the wired Internet. For example, physical layerresource allocation, in terms of adaptation of coding, modula-tion, and interleaving parameters, in DSL lines at the access partof the network can also be optimized based on TCP parametersand variables, in order to enhance the end-to-end performance.

VI. PERFORMANCEEVALUATION : CONVERGENCE

THEOREM AND EQUILIBRIUM STATE

It is not too surprising that allowing cross-layer interactionsimproves the performance of wireless multihop networks. Therest of this paper is devoted to the more interesting and challeng-ing task of proving that the JOCP Algorithm also has the follow-ing desirable properties: global convergence to the jointly opti-mal (x∗,P∗), robustness to parameter perturbation and asyn-chronism, graceful tradeoff between complexity and perfor-mance, and geometric rate of convergence.

We first show that convergence of the nonlinearly coupledsystem, formed by the JOCP Algorithm and shown in Figure 1,is guaranteed under two mild assumptions. First,Pl are withina range betweenPl,min > 0 andPl,max < ∞ for each linkl.Second, when link prices are high enough, source rates can bemade very small: for anyε > 0, there exists aλmax such that ifλl > λmax, thenxs(λ) < ε for all sourcess that use linkl. Tomake the analysis concrete, we again focus on the case of TCPVegas with logarithmic source utilities. But the proof techniqueis applicable to the interaction between other TCP sources withdifferent utilities and the power control algorithm (6), as long asthe congestion price update converges.

It is also interesting to note that the two decomposed prob-lems in the proof are both geometric programming, a class ofnonlinear optimization that was invented in 1960s [12] and re-cently found many applications in communication systems,e.g.,in [8], [10], [14], [15]. The JOCP Algorithm can be viewed as adistributed solution to a class of geometric programs.

Theorem 1:For small enough constantsγ and κ, the dis-tributed JOCP Algorithm (4,5,6) converges to the global opti-mum of the joint congestion control and power control problem(3).

Proof: We first associate a Lagrange multiplierλl foreach of the constraints

∑s:l∈L(s) xs ≤ cl(P). Using the KKT

optimality conditions for convex optimization [2], [4], solvingproblem (3) (or (2))is equivalent to satisfying the complemen-tary slackness condition and finding the stationary points of theLagrangian.

Complementary slackness condition states that at optimality,the product of the dual variable and the associated primal con-straint must be zero. This condition is satisfied since the equilib-rium queuing delay must be zero if the total equilibrium ingressrate at a router is strictly smaller than the egress link capacity.

We now find the stationary points of the Lagrangian:Isystem(x,P,λ) = (

∑s Us(xs) −

∑l λl

∑s:l∈L(s) xs) +

(∑

l λlcl(P)). By linearity of the differentiation operator, thiscan be decomposed into two separate maximization problems:

maximizexº0

∑s Us(xs)−

∑s

∑l∈L(s) λlxs,

maximizePº0 Ipower(P, λ) =∑

l λlcl(P).

The first maximization is already implicitly solved by the con-gestion control mechanism for differentUs (such as TCP Vegasfor Us(xs) = αsds log xs). But we still need to solve the secondmaximization, using the Lagrange multipliersλ as the shadowprices to allocate exactly the right power to each transmitter,thus increasing the link data rates and reducing congestion atthe network bottlenecks. For scalability in ad hoc networks, thispower control must also be implemented distributively, just likethe congestion control part. Since the data rate on each wirelesslink is a global function of all the transmit powers, the powercontrol problem cannot be nicely decoupled into local problemsfor each link as in [31]. However, we show that distributed so-lution is still feasible, as long as an appropriate set of limitedinformation is passed among the nodes.

But we first need to establish that, if the algorithm con-verges, the convergence is indeed toward the global optimum.We will establish that the partial Lagrangian to be maximizedIpower(P) =

∑l λl log(SIRl(P)) is a strictly concave func-

tion of a logarithmically transformed power vector. LetPl =log Pl, ∀l, we haveIpower(P) =

∑

l

λl logGlle

Pl

∑k GlkePk + nl

=∑

l

λl

[log(Glle

Pl)− log

(∑

k

GlkePk + nl

)]

=∑

l

λl

[log(Glle

Pl)− log

(∑

k

exp(Pk + log Glk) + nl

)].

The first term in the square bracket is linear inP, and the secondterm is concave inP because the log of a sum of exponentialsof linear functions ofP is convex, as verified below.

Taking the derivative ofIpower(P) with respect toPl, wehave

∇lIpower(P) = λl −∑

j 6=l

λjGjlePl

∑k 6=j GjkePk + nj

= λl − Pl

∑

j 6=l

λjGjl∑k 6=j GjkPk + nj

.

Taking derivatives again, for each of the nonlinear

−λl log(∑

k exp(Pk + log Glk) + nl

)terms in Ipower(P),

we obtain the Hessian:

Hl =−λl

(∑

k zlk + nl)2

((∑

k

zlk + nl

)diag(zl)− zlzT

l

)

where zlk = exp(Pk + log Glk) and zl is a column vector[zl1, zl2, . . . , zlN ]T .

Matrix Hl is indeed negative definite: for all vectorsv,

vT Hlv =−λl

((∑

k zlk + nl)(∑

k v2kzlk

)− (∑

k vkzlk)2)

(∑

k zlk + nl)2< 0.

(7)This is because of the Cauchy Schwarz inequality:(aT a)(bT b) ≥ (aT b)2 whereak = vk

√zlk andbk =

√zlk

and the fact thatnl > 0. Therefore,Ipower(P ) is a strictlyconcave function ofP, and its Hessian is a negative definiteblock diagonal matrixdiag(H1,H2, . . . ,HL). Interestingly,some of the statements in later propositions depend on theinvertibility of H, which are provided for by the nonzero noiseterms.

Coming back to theP solution space instead ofP, it is easyto verify that the derivative ofIpower(P) with respect toPl is

∇lIpower(P) =λl

Pl−

∑

j 6=l

λjGjl∑k 6=j GjkPk + nj

.

Therefore, the logarithmic change of variables simply scaleseach entry of the gradient byPl: ∇lIpower(P) =1Pl∇lIpower(P). Power update can be conducted in eitherP

or P domain.

We now use the gradient method [4], with a constant step sizeκ, to maximizeIpower(P):

Pl(t + 1) = Pl(t) + κ∇lIpower(P)

= Pl(t) + κ

λl(t)

Pl(t)−

∑

j 6=l

λj(t)Gjl∑k 6=j GjkPk(t) + nj

.

Simplifying the equation and using the definition of SIR, wecan write the gradient steps as the following distributed powercontrol algorithm with message passing:

Pl(t + 1) = Pl(t) +κλl(t)Pl(t)

− κ∑

j 6=l

Gljmj(t)

wheremj(t) are messages passed from nodej:

mj(t) =λj(t)SIRj(t)

Pj(t)Gjj.

These are exactly items 3 and 4 in the JOCP Algorithm.

It is known [2] that when the step size along the gradientdirection is optimized, the gradient-based iterations converge.Such an optimization of step sizeκ in (6) would require globalcoordination in a wireless ad hoc network, and is undesirable orinfeasible. However, in general gradient-based iterations with aconstant step size may not converge.

By the descent lemma [2], convergence of the gradient-basedoptimization of a functionf(x), with a constant step sizeκ,is guaranteed iff(x) has the Lipschitz continuity property:‖∇f(x1) − ∇f(x2)‖ ≤ L‖x1 − x2‖ for someL > 0, andthe step size is small enough:ε ≤ κ ≤ 2−ε

L for someε > 0. It isknown thatf(x) has the Lipschitz continuity property if it has aHessian bounded inl2 norm.

The HessianH of∑

l λlcl(P) can be verified to be

Hll =∑

j 6=l

λj

(Gjl∑

k 6=j GjkPk + nj

)2

− λl

P 2l

, (8)

Hli =∑

j 6=l,i

λjGjlGji(∑k 6=j GjkPk + nj

)2 , i 6= l. (9)

The second assumption for Theorem 1 leads to the conclusionthat λ are upper bounded [29], which, together with the firstassumption for Theorem 1, shows that‖H‖2 is upper bounded.The upper bound can be estimated by the following inequality:

‖H‖2 ≤√‖H‖1‖H‖∞

where‖H‖1 is the maximum column-sum matrix norm ofH,and‖H‖∞ is the maximum row-sum matrix norm.

Therefore, the power control part (6) converges for a smallenough step sizeκ:

ε ≤ κ ≤ 2− ε

L′

where

(L′)2 = maxi

(∑

l

∑j 6=l,i

λjGjlGji(∑k 6=j

GjkPk+nj

)2

+

∣∣∣∣∣∑

j 6=l λj

(Gjl∑

k 6=jGjkPk+nj

)2

− λl

P 2l

∣∣∣∣∣

)

×maxl

(∑

i

∑j 6=l,i

λjGjlGji(∑k 6=j

GjkPk+nj

)2

+

∣∣∣∣∣∑

j 6=l λj

(Gjl∑

k 6=jGjkPk+nj

)2

− λl

P 2l

∣∣∣∣∣

)

andε can be any small positive number≤ 21+L′ .

It is known [22] that TCP Vegas converges for a small enoughstep size0 < γ ≤ 2αmindmincmin

LmaxSmaxx2max

, whereαmin anddmin are thesmallest TCP source parametersαs andds among the sources,respectively,xmax is the largest possible source rates,cmin isthe smallest link data rate,Lmax is the largest number of linksany path has, andSmax is the largest number of sources sharinga link.

Convergence of TCP Vegas assumes thatcmin 6= 0. SinceSIRl is lower bounded by Pl,minGll∑

j 6=lPj,maxGlj+nl

, eachcl is lower

bounded by a strictly positive number. (In fact, the formulationin (2) assumes high SIR in the first place.) Consequently, TCPVegas (5,4) also converges. By the convergence result of simul-taneous gradient-method to the saddle point of minmax prob-lems [3], [28] (in this case, minimizing the Lagrangian over dualvariables and the maximizing it over the primal variables to thesaddle point of the Lagrangian, which is the optimal(x∗,P∗)),the JOCP Algorithm converges.

Sincecl can be turned into a concave function inP, eachconstraint

∑s:l∈L(s) xs − cl(P) ≤ 0 in (2) is an upper bound

constraint on a convex function in(x, P). So problem (2) canbe turned into maximizing a strictly concave objective functionover a convex constraint set. The established convergence isthus indeed toward a unique global optimum.

In addition to convergence guarantee, total network utility∑s Us(xs) with power control can never be smaller than that

without power control, because by allowing power adaptation,we are optimizing over a larger constraint set. Note that an in-crease in network utility

∑s Us(xs) is not equivalent to a higher

total throughput∑

s xs, since the utility functions are not iden-tity functions, but strictly concave functions. However, empiri-cal evidence from simulation suggests that at least in the loga-rithmic utility case of TCP Vegas, both throughput and energyefficiency will indeed rise significantly after power control (6)regulates data rate supply, and dual variablesλ coordinate datarate demand with supply.

VII. SOME PRACTICAL ISSUES: ROBUSTNESS,COMPLEXITY REDUCTION, ASYNCHRONOUS

IMPLEMENTATION, AND CONVERGENCESPEED

We use various tools from nonlinear optimization, distributedalgorithm, and linear algebra to rigorously study other importantproperties of the JOCP Algorithm. The proofs of the proposi-tions can be found in [6].

A. Robustness

Robustness is often as important as optimality of an algo-rithm. We focus on the following robustness properties of theJOCP Algorithm:

1) The effects of inaccurate estimations of the path losses atvarious nodes. Even with an accurate estimation, mobilityof the nodes and fast variation of the fading process maylead to a mismatch between theGij used in the powerupdate algorithm and theGij that actually appear in thelink data rate formula.

2) The effects of packet loss due to wireless channel outageduring deep fading.

First, it is assumed in the power control algorithm (6) that thepass loss factorsGij are perfectly estimated by the receivers. Itis useful to know how much error in the estimation ofGij can betolerated without losing the convergence of joint power controland TCP congestion control.

Denoting the error in the estimation ofGij at timet as ∆Gij(t), and suppressing the time index onλ(t),P(t), SIR(t), ∆Gij(t), we provide a sufficient con-dition in the following

Proposition 1: Convergence to the global optimum of (3) isachieved through the JOCP Algorithm (4,5,6) withGij estima-tion errors, if there exists aT such that for all timest ≥ T , thefollowing inequality holds:

∑l

∑j 6=l

∑k 6=l(GjlGkl −∆Gjl∆Gkl)

λjλkSIRjSIRk

PjPkGjjGkk

> 2∑

l

∑j 6=l

λlλjGjl

PlPjGjjSIRj − λ2

l

P 2l

.

While Proposition 1 gives an analytic condition of conver-gence with inaccurate estimations ofGij for any network, nu-merical experiments can be carried out in simulations where theGij factors in (6) are perturbed randomly within a range. Re-sults of one typical experiment is shown in the lower left graphin Figure 4, for the same network topology and logical connec-tions as in Figure 2. In this simulation, theGij factors are gen-erated at random between+25% and−25% of their true val-ues. The algorithm converge to the same global optimum aftera much longer and wider transient period.

0 100 200 300 4001

2

3

4

5

6Baseline Case

kbps

0 100 200 300 4001

2

3

4

5

6Larger Step Size Case

kbps

0 100 200 300 4001

2

3

4

5

6Wrong Fading Estimate Case

kbps

0 100 200 300 4001

2

3

4

5

6Packet Loss Case

kbps

Fig. 4. Robustness of joint power control and TCP Vegas. Top left case is thebaseline performance of the four end-to-end throughput (the same as in Figure3). Top right case shows that a larger step size in the algorithm acceleratesconvergence but also leads to larger variances. Bottom left case shows that thealgorithm is robust to wrong estimates of path losses. Bottom right case showsrobustness against packet losses on links with wireless channel outage.

Another peculiar feature of wireless transmissions is that dur-ing deep fading, SIR on a link may become too small for correctdecoding at the receiver. This channel outage induces packetlosses on the link. Consequently the queue buffer sizes becomesmaller than they should have been. Analysis of TCP in suchlossy environment has been carried out, for example in [1]. Inour framework of nonlinear optimization, since queuing delaysare implicitly used as the dual variablesλ in TCP Vegas, suchchannel variations lead to incorrect values of the dual variables.Sources will mistake the decreases in total queuing delay as in-dications of reduced congestion levels, and boost their sourcerates through TCP update accordingly. Having incorrect pric-ing on the wireless links may thus prevent the joint system fromconverging to the global optimum. We have the following suf-ficient condition for convergence, where outage-induced packetloss on linkl is denoted as∆yl:

Proposition 2: Convergence to the global optimum of (3)is achieved through the JOCP Algorithm (4,5,6) with packetlosses, if there exists aT such that for all timest ≥ T , thefollowing inequality holds:

∑l

[1

P 2l

(λ2

l −(

∆yl

cl

)2)

+∑

j 6=l

(GjlSIRj

GjjPj

)2(

λj −(

∆yj

cj

)2)]

> 2∑

l

∑j 6=l

(λjλl − ∆yl∆yj

clcj

)GjlSIRj

GjjPlPj.

Because the chance of having simultaneous channel outagesat all links is small, it is reasonable to expect that only few∆yl

are nonzero at any time. We again numerically experiment withchannel outage induced packet loss on various links, and a typ-ical result is shown in the lower right graph in Figure 4 wherethe underlying outage probability is20%. The convergence isslower but still maintained toward the optimal solution.

B. Complexity reduction

Another practical issue concerning the JOCP Algorithm isthe tradeoff between performance optimality and implementa-tion simplicity. The increases in TCP throughput and energyefficiency have been achieved with a rise in the communica-tion complexity of message passing. There can be many termsin the

∑j 6=l Gljmj(t) sum in (6) as the number of transmit-

ters increases. Fortunately, those transmitters farther away fromtransmitterl will have their messages be correspondingly multi-plied by a much smallerGlj ∝ d−α

lj , whereα ranges between 2and 6. Their messagesmj will therefore be given much smallerweights in the power update.

This leads to a simplified power control algorithm, whereeach transmitterl uses the path loss estimations to form a smallsetJl of neighbors whose messages will be needed and usedin the power update. Naturally, if there areV elements in setJl, they should correspond to the nodes with theV largestGlj

toward nodel. The power update equation becomes:

Pl(t + 1) = Pl(t) +κλl(t)Pl(t)

− κ∑

j∈Jl

Gljmj . (10)

The following sufficient condition of convergence with thesimplified algorithm can be shown:

Proposition 3: Convergence to the global optimum of (3) isachieved through the simplified version of the JOCP Algorithm(4,5,10), if there exists aT such that for all timet ≥ T , thefollowing inequality holds:

∑

l

∑

j∈Jl

(GjlλjSIRj

GjjPj

)2

> 2∑

l

∑

j 6=l

λlλjGjl

PlPjGjjSIRj − λ2

l

P 2l

.

(11)

The reduction in complexity can be measured by the ratio

∆COM =∑

l |Jl|M(M − 1)

whereM is the total number of transmitters in the network. Ob-viously, 0 ≤ ∆COM ≤ 1, and a smaller∆COM representsa simpler and less optimal message passing and power update.The effectiveness of complexity reduction through partial mes-sage passing depends on the path loss matrixG. While the in-tuition is clear: the reduced-complexity versions do not workwell for network topologies where nodes are evenly spread out,we do not yet have an analytic characterization on the trade-off between∆COM and energy efficiency enhancement or themaximized network utility.

C. Asynchronous implementation

The algorithmic analysis thus far has been limited to the casewhere propagation delay is insignificant and all the local clocksare synchronized, which is not practical in large wireless ad hocnetworks. In this Subsection, we investigate the convergence ofthe algorithm under asynchronous implementation, with vari-able propagation delays and clock asynchronism.

Suppose each source updatesxs and each transmitter updatesPl at asynchronous time slots, using possibly outdated variables,

such asλl andmj , in their update. At least one local update iscarried out sometime within a window ofD time slots, and thevariables used in the update can be delayed up toD time slots.We have the following

Proposition 4: The asynchronous JOCP Algorithm con-verges if and only ifD is finite.

This result shows that the proposed algorithm is able to sup-port asynchronous implementation as long as the constantsκ, γare small enough. An adverse effect of asynchronism is the re-duction of the maximum step sizes allowed for convergence tobe maintained, which reduces the convergence speed. However,in the case of sufficiently small asynchronism:

D ≤ min{

L′

2− ε,LmaxSmaxx2

max

2αmindmincmin

},

propagation delay, delay in message passing, and clock asyn-chronism in rate-power updates become the loose constraints onthe maximum step sizes, and do not have to cause a reduction inthe rate of convergence.

D. Rate of convergence

So far we have focused on the equilibrium behaviors of theJOCP Algorithm. In general, very little is understood on thetransient behaviors of the dynamics of JOCP Algorithm, or ofjust TCP congestion control alone. This Subsection providesa preliminary analysis on the rate of convergence of the powercontrol algorithm. Rate of convergence for any distributive algo-rithm on wireless ad hoc networks is particularly important be-cause the network topology are dynamic and source traffic mayexhibit a low degree of stability. A key question for practical im-plementation of the proposed cross-layer design is whether thecoupled nonlinear dynamics between TCP and power controlcan proceed reasonably close to the equilibrium before the net-work topology, routing, and source characteristics change dra-matically.

Convergence analysis for distributive nonlinear optimizationcan take several different approaches. We focus on the morepractical local analysis approach, which investigates the rate ofconvergence after the algorithm reaches a point reasonably closeto the optimum. Because our algorithm nonlinearly depends onthe path loss matrixG, exact and closed-form results on therate of convergence is very difficult to obtain. Nonetheless, thefollowing result on the geometric convergence property and aloose bound on the convergence speed can be proved.

LetU (k) be the network utility at thekth iteration of the JOCPAlgorithm, andU∗ be the maximized network utility. Lete(k) =|U (k)−U∗| be the error. LetP(k) be the power vector at thekthiteration, andP∗ be the optimizer. Denote byM (k) the largesteigenvalues of thekth iteration Hessian ofIpower(P(k)), andm(k) the smallest eigenvalue. LetM = lim supk→∞M (k) andm = lim supk→∞m(k), and assumeM,m ∈ R. Assume thatthe limit of the Hessian derived in (8) ask → ∞ exist and isdenoted byH = {Hij}.

Proposition 5: The JOCP Algorithm converges geometri-cally, i.e., there existq > 0 andβ ∈ (0, 1) such that for allk,e(k) ≤ qβk. With an appropriate constant parameterκ, the rate

of convergence (of the power control part) is at leastM ′−m′M ′+m′ ,

where

M ′ = maxi

Hii +

∑

j 6=i

|Hij |

m′ = mini

Hii −

∑

j 6=i

|Hij | .

A similar result holds for the rate of convergence of the con-gestion control part. However, we add the cautionary note thatthe above lower bound on the rate of convergence is based onthe worst case scenario and can be orders of magnitude loose.Depending on the path loss environment in the network, numer-ical simulations show that the actual convergence speed is oftenmuch faster than the bound in Proposition 5.

E. Further algorithmic enhancements

In concluding our performance analysis of the JOCP Algo-rithm, we briefly outline a couple of algorithmic enhancementsthat can be readily accomplished.

It is desirable to choose a constant step size that is neither solarge that the algorithm diverges (e.g., violating the conditionsin Section IV), nor so small that the convergence is too slow.One way to accomplish this is to let each source and each trans-mitter autonomously decrease the step sizes at each time slottaccording to the following rule:

γ(t) = κ(t) =ω

t, ω > 0.

Such a diminishing sequence of step sizes also makes the al-gorithm even more robust: errors in queuing delaysλ and pathlossesG that are proportional to the magnitudes ofλ andG canbe tolerated.

It is also possible to speed up the convergence of the algo-rithm by diagonally scaling the distributed gradient method:

Pl(t + 1) = Pl(t) + κW∇lIpower(P)

whereW ideally should be the inverse of the HessianH ofIpower(P). Since forming this inverse will require extensiveglobal coordination and centralized computation, we approxi-mate the inverse by letting

W = diag(H−1ii ).

Substituting the expression forHii in (8) and simplifying theexpressions, we arrive at the following accelerated algorithm:

Pl(t + 1) = Pl(t) + κ

λl(t)Pl(t)

−∑j 6=l Gljmj(t)

λl(t)P 2

l(t)−∑

j 6=l(Gljmj(t))2

λj(t)

. (12)

Therefore, by passing an additional message: the explicit valueof priceλj(t) from nodej, the jointly optimal congestion con-trol and power control algorithm can converge faster.

VIII. T O LAYER OR NOT TO LAYER: LAYERING AS

OPTIMIZATION DECOMPOSITION

Like [9], [26], [30], [31], this paper can be viewed as a casestudy of the ‘layering as optimization decomposition’ approach,which may allow us to integrate many layers in wired and wire-less networks, and to rigorously quantify the generalarchitec-tural principlesand inherent tradeoffs of layering. If a mappingcan be found from differentdecompositionsof a generalizedutility maximization problem to differentlayeringschemes, andfrom primal or Lagrange dualvariablescoordinating the sub-problems to theinterfacesamong the layers, then we can tacklethe question ‘how to and how not to layer’ by investigating thepros and cons of decomposition techniques. By comparing theobjective values under optimal decompositions, suboptimal de-compositions, and decompositions with some layering variablesfixed, we can seek ‘separation theorems’ among layers: condi-tions under which strict layering incurs no loss of optimality.Robustness of these separation theorems can be further charac-terized by sensitivity analysis in optimization theory: how muchwill the differences in the objective value (between differentlayering schemes) fluctuate as constant parameters in the util-ity maximization problem are perturbed. In addition to ‘verticaldecomposition’ across layers of functional modules, ‘horizontaldecomposition’ across geographically diverse nodes may alsobe conducted viafunctionsof layering variables.

If layering schemes are viewed as decompositions of someglobal optimization problems, the price of layering or re-layering, the price of no layering, and the price of cross-layeringmay be quantified in both reverse and forward engineering di-rections:

• Reverse engineering: given a layered protocol stack, whatis the optimization problem that it implicitly solves?

• Forward engineering: given a utility maximization formu-lation, how to decompose it into subproblems, solve eachsubproblem individually, then solve the overall problem?

In the case of wireless networks, the transmission mediumis an untethered, unshielded, broadcast one, with time-varyingchannels that have attenuation, shadowing, and fading. A wire-less network is essentially a space with electromagnetic energypropagating in it. There is no a priori definition of ‘link ca-pacity’ or even of ‘link’. All of the following issues compli-cate the utility maximization model substantially: signal inter-ference and power control, packet collision and medium accesscontrol, rate-reliability tradeoff and coding, spatial diversity andmultiple-antenna transmissions. Therefore, the following utilitymaximization problem and its decomposition and distributed al-gorithms needs to be studied:

maximize∑

s Us(xs) +∑

j Vj(wj)subject to Rx ¹ c(w,Pe),

x ∈ C1(Pe)⋂ C2(F),

R ∈ R, F ∈ F , w ∈ W.

(13)

Here,xs denotes the rate for sources andwj denotes the phys-ical layer resource at network elementj. The utility functionsUs andVj may be any nonlinear, monotonic functions.R isthe routing matrix andc are the logical link capacities as func-tions of both physical layer resourcesw and the desired decod-

ing error probabilitiesPe. The issue of signal interference andpower control can be captured in this functional dependency.The rates must also be constrained by the interplay betweenphysical layer decoding reliability and upper layer error controlmechanisms like ARQ in the link layer. This constraint set is de-noted asC1(Pe), and captures the issue of rate-reliability trade-off and coding. Constraint on the rates by the medium accesssuccess probabilities is represented by the constraint setC2(F)whereF is the contention matrix [25]. The issue of packet col-lision and medium access control is captured in this constraint.The set of possible physical layer resource allocation schemesis represented byW, that of possible scheduling or contentionbased medium access schemes byF , and that of single-path ormulti-path routing schemes byR. The optimization variablesarex,w,Pe,R,F.

Five layers in the current standard protocol stack are modelledin (13), although the decompositions of (13) donot have to bealong the lines dictated by the current layering structure:

• Application layer. Utility functions Ui andVj model theapplication needs.

• Transport layer.The end-to-end throughput is representedas the source ratexs for each end users.

• Network layer. The routing matrix can be designed byvaryingR within the constraint setR.

• Link layer. Through scheduling, antenna beamforming,and spreading code assignment, the contention matrixFcan be designed within the constraint setF . The rates arethen constrained by contention-free or contention-basedaccess schemes as described by the constraint setC2.

• Physical layer.Adaptive resource allocations,e.g., powercontrol, adaptive modulation, coding with embedded di-versity, will lead to different logical link capacitiesc asfunctions of decoding error probabilitiesPe. The rate-reliability tradeoff forms the constraint setC1.

The generic formulation (13) can be specialized in differentcases by specifying the constraint sets. The two most difficultissues are time-scale and nonzero duality gap. In this paper,we have assumed that the time scale of power control and con-gestion control is longer than the time scale needed for channelcoding to achievecl, and shorter than the time scale of dynamicchanges in network topology and routing. Using the approx-imation thatKSIR is much larger than1 and a log transfor-mation, we have turned problem (2) into a convex optimizationwith strictly feasible solutions, thus having zero duality gap.

Last but not least, this paper crosses the transport and physicallayers only from a network performance viewpoint. Other thanthe discussion on how the JOCP Algorithm maintains the mod-ularity between the two layers, this paper is not examining themost important reason for layering. Layering, like many othernetworking principles, isnot established only forefficiencyofperformance metrics in terms of throughput, latency, distortion,or energy efficiency, but also forrobustnessin terms of impor-tant X-ities: evolvability, scalability, verifiability, manageabil-ity, deployability, survivability, adaptability ... Compared withstandard performance metrics, these X-ities are much less well-understood, often without any theoretical foundations, quanti-tative frameworks, or even units of measurement. Yet X-ities

are crucial if we are to analyze current layering and design fu-ture ones properly. Initial results on quantifying some basic as-pects of evolvability have recently been obtained [11]. It will bemost interesting and challenging to investigate how the X-itiesaspects may be understood through the general framework of‘layering as optimization decomposition’.

IX. CONCLUSION

We present a distributed power control algorithm that cou-ples with existing TCP congestion control algorithms to in-crease end-to-end throughput and energy efficiency of multi-hop transmissions in wireless multihop networks. No modifica-tion to TCP is needed to achieve the optimal balancing betweendata rate demand (regulated through TCP) and supply (regulatedthrough power control). We prove that the nonlinearly coupledsystem converges to the global optimum of the joint congestioncontrol and power control problem. The convergence is geo-metric and can be maintained under finite asynchronism. Theproposed algorithm is robust to wireless channel variations andpath loss estimation errors. Suboptimal but much simplified ver-sions of the algorithm are presented for scalable architectures.

As a step towards ‘layering as optimization decomposition’,this paper expands the scope of the network utility maximiza-tion methodology to handle nonlinear, elastic link capacities.This extension enables us to rigorously prove that the proposedJOCP Algorithm has the above desirable properties in achievingthe optimal balance between the transport and physical layers inwireless multihop networks.

ACKNOWLEDGEMENT

The author is grateful for many helpful discussions on thesubject with Nicholas Bambos, Stephen Boyd, Rob Calderbank,Vincent Chan, Lijun Chen, John Doyle, Steven Low, DanielO’Neill, Daniel Palomar, and Lin Xiao.

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PLACEPHOTOHERE

Mung Chiang is an Assistant Professor of ElectricalEngineering at Princeton University. He received theB.S. (Honors) in Electrical Engineering and Mathe-matics, M.S. and Ph.D. degrees in Electrical Engineer-ing from Stanford University in 1999, 2000, and 2003,respectively. Professor Chiang conducts research inthe areas of nonlinear optimization of communica-tion systems, network resource allocation and conges-tion control algorithms, and information theory andstochastic analysis of communication systems. Hehas been awarded as a Hertz Foundation Fellow, Stan-

ford Graduate Fellow, NSF Graduate Fellow, and received Stanford UniversitySchool of Engineering Terman Award and SBC Communications New Technol-ogy Introduction Contribution Award. He is the Lead Guest Editor of the IEEEJSAC Special Issue on ‘Nonlinear Optimization of Communication Systems’, aGuest Editor of the IEEE Trans. Inform. Theory Special Issue on ‘Networkingand Information Theory’, and the Program Co-Chair of the 38th Conference onInformation Sciences and Systems.