air pollutant & control device characteristics effects on emission

IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 23, NO. 1, JANUARY 2005 89

Dynamic Power Allocation and Routing forTime-Varying Wireless Networks

Michael J. Neely, Eytan Modiano, Senior Member, IEEE, and Charles E. Rohrs

Abstract—We consider dynamic routing and power allocationfor a wireless network with time-varying channels. The networkconsists of power constrained nodes that transmit over wirelesslinks with adaptive transmission rates. Packets randomly enter thesystem at each node and wait in output queues to be transmittedthrough the network to their destinations. We establish the ca-pacity region of all rate matrices ( ) that the system can stablysupport—where represents the rate of traffic originating atnode and destined for node . A joint routing and power allo-cation policy is developed that stabilizes the system and providesbounded average delay guarantees whenever the input rates arewithin this capacity region. Such performance holds for generalarrival and channel state processes, even if these processes are un-known to the network controller. We then apply this control algo-rithm to an ad hoc wireless network, where channel variations aredue to user mobility. Centralized and decentralized implementa-tions are compared, and the stability region of the decentralizedalgorithm is shown to contain that of the mobile relay strategy de-veloped by Grossglauser and Tse (2002).

Index Terms—Capacity, control, optimization, queueing.

I. INTRODUCTION

WIRELESS systems have emerged as a ubiquitous partof modern data communication networks. Demand for

these systems continues to grow as applications involving bothvoice and data expand beyond their traditional wireline servicerequirements. In order to meet the increasing demand in datarates that are currently being supported by high-speed wired net-works composed of electrical cables and optical links, it is im-portant to fully utilize the capacity available in wireless systems,as well as to develop robust strategies for integrating these sys-tems into a large scale, heterogeneous data network. Emergingmicroprocessor technologies are enabling wireless units to beequipped with the processing power needed to implement adap-tive coding techniques and to make intelligent decisions aboutpacket routing and resource management. It is expedient to takefull advantage of these capabilities by designing efficient net-work control algorithms.

In this paper, we develop algorithms for dynamic routingand power allocation in a wireless network consisting of

Manuscript received October 15, 2003; revised August 2, 2004. This workwas supported in part by the National Science Foundation (NSF) ITR underGrant CCR-0325401 and in part by the Defense Advanced Research ProjectsAgency/Air Force Office of Scientific Research (DARPA/AFOSR) through theUniversity of Illinois under Grant F49620-02-1-0325 for the project entitled“Cooperative Networked Control of Dynamical Peer-to-Peer Vehicle Systems.”

M. J. Neely is with the Department of Electrical Engineering, University ofSouthern California, Los Angeles, CA 90089 USA (e-mail: [email protected]).

E. Modiano and C. E. Rohrs are with the Laboratory for Information and De-cision Systems, Massachusetts Institute of Technology, Cambridge, MA 02139USA (e-mail: [email protected]; [email protected]).

Digital Object Identifier 10.1109/JSAC.2004.837349

Fig. 1. (a) Wireless network with multiple input streams. (b) Close-up of onenode, illustrating the internal queues.

power constrained nodes. Time is slotted, and in every timeslot, the channel conditions of each link randomly change (dueto external effects such as fading, user mobility, and/or time-varying weather conditions). Multiple data streams ran-domly enter the system, where represents an exogenousprocess of packets arriving to node destined for node . Packetsare dynamically routed from node to node over multihop pathsusing wireless data links.

Nodes can transmit data over multiple links simultaneouslyby assigning power to the link for each node pair ac-cording to a power matrix , subject to a totalpower constraint at each node. Transmission rates over all linksare determined by the power allocation matrix and the cur-rent channel state according to a rate-power curve .Each node contains internal queues for storing data ac-cording to its destination (Fig. 1). A controller allocates powerand schedules the data to be routed over the links in reaction tochannel state and queue backlog information. The goal of thecontroller is to stabilize the system and thereby achieve max-imum throughput and maintain acceptably low network delay.

We establish the network capacity region: The set of all inputrate matrices that the system can stably support (where

represents the rate of data entering node destined for node). This region is determined by considering all possible routing

and power allocation strategies, and can be expressed in termsof the steady-state channel probabilities, the node power con-straints, and the rate-power function . We emphasizethat this is a network layer notion of capacity, where isa general function representing the rate achievable on the wire-less links under a given physical layer modulation and codingstrategy. This is distinct from the information theoretic capacityof the wireless network, which includes optimization over all

0733-8716/$20.00 © 2005 IEEE

90 IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 23, NO. 1, JANUARY 2005

possible modulation and coding schemes, and involves many ofthe unsolved problems of network information theory. We donot address the information theoretic capacity in this work, anduse the term capacity to represent network layer capacity.

We present a joint routing and power allocation policy thatstabilizes the system and provides bounded average delay guar-antees whenever the input rates are strictly inside the networkcapacity region. Such performance holds for general ergodic ar-rival and channel state processes, even if the specific channelprobabilities and packet arrival rates are unknown to the net-work controller. The strategy involves maximizing a stochasticdrift metric every time slot. We implement centralized and de-centralized versions of the algorithm for an ad hoc wireless net-work, where channel variations are due to user mobility. Further,we show that this dynamic control strategy can be viewed as ageneralization of a corresponding static optimization procedure,establishing a fundamental relationship between network opti-mization and stochastic control.

Previous work on capacity, optimization, and control of wire-less networks is found in [1]–[27]. Connectivity and asymptoticcapacity analysis for large static networks is presented in [4] and[5], and for mobile networks in [6]. The exact capacity of a wire-less uplink and downlink with multiple users is developed in[7]–[9], where it is assumed that all users have infinite backlog.

Optimization approaches to network resource allocation aredeveloped in [10]–[14] and [29]–[32]. In [10], static routing andpower allocation is treated using convex optimization. In [11],various cost metrics are formulated as geometric programs toaddress resource allocation and quality-of-service. Game theo-retic approaches for wireless downlinks are developed in [12]and [13], and for flow networks in [29]–[31], where pricingschemes are considered for achieving a static equilibrium withrespect to some utility metric. The equilibrium computed in[31] is shown to be within a constant factor of the maximumutility. Similar constant factor bounds are developed in [32] forshortest-path routing in static networks with simplified modelsof network delay. Optimal power allocation for minimizing en-ergy expenditure in a network with given rate requirements isconsidered in [14] under the assumption that transmission ratesare linear functions of the signal to interference ratio on eachlink. In this case, although the network channels and rate re-quirements are constant, the optimal solution is not static butrequires the computation of a periodic transmission scheduleto achieve optimality. Simple approximations to optimal sched-uling are developed in [15], and schedules for one-dimensionalnetworks are developed in [16].

These optimization approaches do not consider the practicalissues of network control, where queue management, sched-uling, and resource allocation decisions must be made in thepresence of stochastic packet arrivals and time-varying channelconditions. Control problems are addressed in [3] and [18]–[27].In [18], a stabilizing power allocation and routing strategy isdeveloped for a multibeam satellite downlink with random in-puts and time-varying channels. Related problems of schedulingusers over a single server downlink are considered in [19]–[23].In [24], a delay optimal strategy is developed for a multiaccessuplink in systems with symmetric user parameters. Asymptoti-cally optimal strategies using heavy traffic limits are developed

Fig. 2. (a) Set of rate-power curves for improving channel conditionsS ; S ; S . (b) Curve restricted to a finite set of operating points correspondingto full packet transmissions. Curves illustrate behavior on link (a; b) when thesingle power parameter P is increased, in which case the concave increasingprofiles are typical.

in [25] and [28] for scheduling multiple users over a shared re-source. Transmitter scheduling and power control for one-hopstatic networks are considered in [26], and one-hop networkswith time-varying topology are considered in [27]. Our work isinspired by the approach of Tassiulas in [3], where a Lyapunovdrift technique is used to develop a throughput optimal linkscheduling policy for a multihop packet radio network. Furtherwork on Lyapunov analysis is found in the switching and sched-uling literature [35]–[37] and in [38], [39].

The main contributions in this paper are the formulation ofa general power control problem for time-varying wireless net-works, the characterization of the network layer capacity region,and the development of capacity achieving routing and powerallocation algorithms that offer delay guarantees and considerthe full effects of queueing. These algorithms hold for systemswith general arrival and channel processes, including ad hoc net-works with mobility. This work unifies notions of network ca-pacity, network optimization, and network control.

In the next section, we introduce the power allocationproblem for wireless networks. In Section III, we define net-work stability and establish the capacity region for wirelessnetworks. In Sections IV and V, a dynamic control strategyis developed and shown to stabilize the network and provideaverage delay guarantees. Distributed implementations areconsidered in Section VI, where optimal distributed controlis established for networks with independent channels, and adistributed approximation algorithm is developed for networkswith interference. We implement the distributed approximationfor an ad hoc mobile network and show through analysisand simulation that the algorithm offers higher data rates andlower delay than the Grossglauser–Tse relay algorithm of [6].Finally, in Section VII, a perspective on dynamic optimizationis provided by relating the optimal network control algorithmsdeveloped here to a classic iterative solution of a static convexprogram.

II. SYSTEM MODEL

Consider an node network with time-varying channels,as in Fig. 1. Let denote the matrix processof channel states, where represents the current state ofchannel (representing, for example, attenuation valuesand/or noise levels). Time is slotted with slots normalizedto integral units . We assume that channels

NEELY et al.: DYNAMIC POWER ALLOCATION AND ROUTING FOR TIME-VARYING WIRELESS NETWORKS 91

hold their state for the duration of a time slot, and are knownto the network controller at the beginning of each slot. Suchinformation can be obtained either through direct measurement(where time slots are assumed to be long in comparison tothe required measurement time) or through a combination ofmeasurement and channel prediction. The channel processtakes values on a finite-state space, and is ergodic with timeaverage probabilities for each state .

Every time slot, a controller determines transmission rateson each link by allocating a power matrixsubject to a total power constraint for allnodes . Additional power constraints can be introduced, such asconstraints on the number of outgoing links that can be allocatedpower when a node is transmitting or receiving. It is, therefore,useful to represent the power constraint in the form ,where is a compact set of acceptable power allocations thatincludes the power limits for each node.

Link rates are determined by a corresponding rate-powercurve (see Fig. 2). It is assumed thatdata can be split continuously, so that each time slot the trans-mission rate determines the number of bits that can betransferred over the wireless link . Such an assumption isvalid if variable length packets can be split and repackaged withnew headers for resequencing at the destination (we neglectthe extra bits due to such headers in this analysis). Alternately,splitting and relabeling can be avoided altogether if all packetshave fixed lengths and the transmission rates are restricted tointegral multiples of the packet-length/time slot quotient [seeFig. 2(b)].

Note that, in general, the transmission rate over a linkof the network depends on the full matrix of power allocationdecisions. This is because communication rates over the linkmay be influenced by interference from other channels. For ex-ample, achievable data rates could be approximated by using thesignal-to-interference ratio (SIR) in the capacity formula for awhite Gaussian noise channel, where the SIR over link isdefined as the attenuated signal power divided by the total inter-ference at node .

Example: Rate-Power Curve:

(1)

where and represent noise and fading coefficients asso-ciated with the particular channel state .

Alternatively, the curves could represent rates for aspecific set of coding schemes designed to achieve a sufficientlylow probability of error. Note that practical systems rely on a fi-nite databank of codes and, hence, may be restricted to a finiteset of feasible operating points. In this case, rate-power curvesare piecewise constant [see Fig. 2(b)]. In general, we assumeonly that is upper semicontinuous1 in the power matrix

for all states , and hence at points of discontinuity the func-tion takes its largest limiting value [40].

1That is, lim sup � (P ; S) � � (P ; S) for all (a; b); P ; S, andany sequence fP g such that P ! P .

A. Control Decision Variables and the Queueing Equation

Each network node maintains a set of output queues forstoring data according to its destination. For convenience, weclassify all data flowing through the network as belonging to aparticular commodity , representing the desti-nation node for the data. A network control algorithm makesdecisions about power allocation, routing, and scheduling. As ageneral algorithm might schedule multiple commodities to flowover the same link on a given slot, we define as the rateoffered to commodity traffic along link during time slot. The controller, thus, makes the following decisions.

Power Allocation: Choose such that .Routing/Scheduling: Choose such that

(2)

Note that in the special case where there is no power alloca-tion, the process is purely determined by the dynamicchannel states of the network, and any network control algo-rithm reduces to pure routing and scheduling.

Let represent the amount of commodity bits thatarrive exogenously to the network at node during slot . Let

represent the current backlog of bits in node destinedfor node . The processes evolve according to the fol-lowing queueing dynamics:

(3)

The expression above is an inequality instead of an equality be-cause endogenous arrivals may be less than if neigh-boring nodes have little or no commodity data to transmit. Thegoal of the controller is to maintain low backlog and thereby sta-bilize the system. For the first part of this paper, we assume thatcentralized control is possible, so that the network controller hasaccess to the full backlog and channel state matrices and

every time slot. Distributed implementations where eachnode makes independent control decisions using only local in-formation are considered in Sections VI and VII.

III. STABILITY AND THE NETWORK CAPACITY REGION

Here, we develop the capacity region of all data rates stabiliz-able by a wireless network. We begin with a precise definitionof stability.

A. Stability of Queueing Systems

Consider a single queue in isolation, with an input processand a time-varying server process . Let the unfinished

work function represent the amount of unprocessed bitsremaining in the queue at time , which is determined by thestochastics of the input and server processes. As a measure of


the fraction of time the unfinished work is above a certain value, we define the following “overflow” function :

The above limit2 always exists, so that .Definition 1: A single server queueing system is stable if

as . A network of queues is stable if allindividual queues are stable.

B. Network Capacity

Assume input processes are stationary and ergodicwith rates , so that withprobability 1. Let represent the correspondingrate matrix, having all diagonal entries equal to zero. As-sume the channel process is stationary and ergodic withsteady-state channel probabilities .

Definition 2: The network capacity region is the closureof the set of all rate matrices that can be stably supportedover the network, considering all possible algorithms (possiblythose with full knowledge of future events).

Remarkably, we show that stabilizing policies do not requireknowledge of future events and, hence, such knowledge does notexpand the region of stabilizable rates. To build intuition aboutthe set , we first consider the capacity region of a traditionalwireline network with no time variation, defined on a weightedgraph with nodes, edges, and edge capacities given by alink matrix . The link matrix describes the rate at whichnode can deliver data to node , so that if there isno directed edge from node to node , and is equal to thepositive transmission rate for that link, otherwise. The networkcapacity region is described implicitly as the set of all arrivalrate matrices for which there exist multicommodity flowvariables (for ) that satisfy a set of flowconservation equations, and which additionally satisfy the linkconstraint for all links . This constraintensures that the total flow over any link does not exceed the linkcapacity.

The capacity region of a wireless network differs from that ofa wireline network only in the description of the link constraint.Indeed, first note that the time-varying channel conditions of awireless network require link capacities to be defined in a timeaverage sense, where the resulting transmission rate over a givenlink is averaged over all possible channel states. Second,the resulting time average link rates are not fixed, but dependon the (potentially nonergodic) power allocation policy. Thus,instead of describing the network as a single weighted graph

of link rates, the network is described by a collection ofgraphs, or a graph family . We define the graph family as thefollowing set of node-to-node transmission rate matrices:

Convex Hull (4)

2Where the lim sup of a function f(t) always exists, and is defined:lim sup f(t) = lim [sup f(�)].

where addition and scalar multiplication of sets is used,3 andthe convex hull of a set is defined as the set of all convexcombinations of elements(where are probabilities summing to 1).

Thus, a transmission rate matrix is in graphfamily if and only if can be represented asfor some set of matrices , each one being inside the convexhull of the set of node-to-node transmission rates achievable bypower allocation under channel state . In the following the-orem, it is shown that graph family can be viewed as the setof all long-term transmission rates that the network canbe configured to support on the single-hop wireless links con-necting node pairs .

Theorem 1: Capacity Region for a Wireless Network—Thecapacity region is the set of all input rate matricesfor which there exist multicommodity flow variablessatisfying

(5)

where (6)

and for some (7)

Thus, a rate matrix is in the capacity region if thereexists a matrix that defines link capacities in a tra-ditional graph network, such that there exist multicommodityflow variables , which support the rates with respectto this graph. Note that inequalities (5) constrain flow variablesto be nonnegative and to be “efficient,” in that no node routesdata to itself, and no node reinjects delivered data back into thenetwork. Inequality (6) is a conservation constraint that ensuresthe total flow of commodity data into a given node is less thanor equal to the total flow out of that node, provided that nodeis not the destination.

The proof of Theorem 1 involves showing that isnecessary for stability, and that interior to is sufficient.The necessary condition is proven in Appendix A, where it isshown that no control algorithm can achieve stability beyondthe set , even if the entire set of future events is known in ad-vance. Sufficiency is shown in the next section, where a stabi-lizing control policy is constructed.

IV. DYNAMIC CONTROL POLICY

Here, we develop a control policy that stabilizes the networkwhenever the input rate matrix is interior to the capacity re-gion . The policy is inspired by the maximum differentialbacklog algorithms developed by Tassiulas and Ephremides in[3] for stable server scheduling in a multihop radio network,and generalizes the Tassiulas–Ephremides algorithm by consid-ering power allocation and addressing networks with general in-terference properties and time-varying channel characteristics.We then develop a bound on end-to-end delay by relating per-formance to that of a stationary policy based on the multicom-modity flow variables of Theorem 1.

3For sets A;B and scalars �; �, the set �A + �B is defined as f j =�a + �b for some a 2 A; b 2 Bg.


Every time slot the network controller observes the channelstate and the matrix of queue backlogsand performs routing and power control as follows.

Dynamic Routing and Power Control (DRPC) Policy:

1) For all links , find commodity such that

and define

(8)

2) Power Allocation: Choose a matrix such that

(9)

3) Routing: Define transmission rates as follows:

if andotherwise

(10)

For each link , transmit commodity data accordingto the rate offered by the power allocation. If any node doesnot have enough bits of a particular commodity to send overall its outgoing links requesting that commodity, null bits aredelivered.Note that the values represent the maximum differential

backlog between nodes and (maximized over all commodi-ties ). The policy thus uses backpressure in an effort to equalizedifferential backlog. This strategy is most effective when poweris allocated to maximize the rate-backlog product in (9). Weemphasize that this scheme does not require knowledge of thearrival rates or channel statistics, and does not use any pre-specified set of routes. The route for each unit of data is founddynamically.

A. Network Parameters

In order to analyze the stability and delay properties of theDRPC algorithm, we first specify the following network param-eters. Define and as the maximum transmission rateout of any node and into any node, respectively, under the bestchannel condtions

Further assume that the second moment of exogenous arrivalsto any node is bounded every time slot by some finite max-imum value , regardless of past history. Specifically, de-fine . Then, for all nodes and all ,we have

Note that for any random variable , sothat we also have .

Define as the set of time slots at whichduring the interval , and defineas the total number of such slots. For a given value , wedefine the convergence interval to be the smallest number oftime slots such that for any , any , and regardless of pasthistory, we have

(11)

(12)

Such a value must exist for any stationary and ergodicchannel and arrival processes with arrival rates andchannel probabilities , respectively. This convergence in-terval represents the time period over which the network isexpected to reach steady-state, regardless of past history. Moregenerally, we assume a finite interval size exists for anygiven , and define arrival processes and channel processesfor which this assumption holds to be rate convergent andchannel convergent, respectively. We note that for systems withindependent identically distributed (i.i.d.) arrivals and channelstates, steady-state is exactly achieved every time slot, so that

even when . Below, we develop a bound on theend-to-end delay of the DRPC policy in terms of the parameters

, and .

B. Stability and Delay Performance

Theorem 2: Stability of DRPC—Suppose an -node wire-less network has capacity region and rate matrix suchthat for some . Then, jointly routing and allo-cating power according to the above DRPC policy stabilizes thesystem and guarantees bounded average congestion satisfying

(13)

for corresponding to in (11) and (12), and where

(14)

(15)

and the overbar notation of (13) is defined

The theorem is proven in Section V. Note that the averagecongestion bound grows asymptotically like as the data ratesare increased, where can be viewed as the “distance” mea-sure of the rate matrix to the boundary of the capacity region.Consider now an input matrix , where each user sends attotal rate , so that . Suppose the matrix satisfies


, and define as the row sum of thematrix. Let represent an effective loading on

each user (assumed independent of ). Note that , and. From Little’s Theorem, the average bit delay

satisfies and, hence

If (as in static Gupta–Kumar networks [4], [5]),then . If (as in mobileGrossglauser–Tse networks [6]), then .

Note that when the network is lightly loaded, there is verylittle information contained in the differential backlog values.Hence, packets might take many false turns, which could leadto significant delay for large networks (consider the above delaybound for small). Performance can often be improved by usingthe DRPC algorithm with a restricted set of desirable routes foreach commodity. However, restricting the routes may reducenetwork capacity, and may be harmful in time-varying situa-tions, where networks change and links fail. Alternatively, wecan keep the full set of routes, but program a bias into the DRPCalgorithm so that in low loading situations, nodes are inclinedto route packets in the direction of their destinations. We usethis idea in the following enhanced DRPC algorithm, defined interms of constants and .

Enhanced DRPC Algorithm: For all links , and all com-modities , define

, and define as the maximizer of over all. Power allocation and routing is done as before,

solving the optimization (9) with respect to .The parameters can be chosen as scaled hop count esti-

mates between nodes and , so that, in the absence of backloginformation, data is routed to reduce the remaining distance tothe destination. The values are any weights for prioritizingcommodity service in node . This enhanced DRPC algorithmcan be shown to stabilize the network for any constantsand . We note that the weight can be usedin the same manner as a routing table, and the unfinished workquantities can be updated each time slot by having neighboringnodes transmit their backlog changes over a low bandwidth con-trol channel. As each wireless link transmits only a single com-modity every time slot, the number of such backlog incrementsrequired to be transmitted over the control channel by any useris on the order of the number of neighboring nodes.

V. PERFORMANCE ANALYSIS

We analyze the performance of the DRPC algorithm by com-paring it to a stationary algorithm that makes scheduling deci-sions according to the multicommodity flow variablesof Theorem 1. Indeed, suppose the rate matrix and thechannel probabilities were known in advance, and supposethere is a value such that . Then, a setof multicommodity flow variables and a link rate matrix

must exist that satisfy the constraints (5)–(7) withrespect to the rate matrix . In particular

for (16)

(17)

Let represent the transmission rateoffered over link on slot . We first show that it is possibleto allocate power in reaction to the current channel state so thatthe time average of converges to .

Lemma 1: Graph Family Achievability—A stationary ran-domized power allocation policy can be implementedso that the resulting process satisfies

with probability 1 for all . The structure of the policy is asfollows: Every time slot in which the channel state isobserved, the power matrix is chosen randomly froma finite set of allocations according to a set ofprobabilities .

Proof: Note that . The proof followsby expressing each matrix as a convex combination ofmatrices in according to Caratheodory’sTheorem [40], and defining the probabilities of the stationaryrandomized scheme according to the weights of each convexcombination. A full proof is given in [1].

Note that the above lemma is an existential result and is notoffered as a practical means of allocating power. We now de-fine a stationary algorithm STAT that uses the above power al-location result to stabilize the network. While this policy cannotbe implemented without complete knowledge of the input rates

, channel probabilities , and flow variables , it yieldsa simple delay bound.

Stationary Randomized Policy (STAT)—Power Allocation:Every time slot, observe the channel state and allocate poweraccording to the stationary algorithm of Lemma 1.

Scheduling/Routing: For every link such that, transmit the single commodity , where

is chosen randomly among with probability. However, use only a fraction of

the instantaneous link rate, so that

ifotherwise

If a node does not have enough (or any) bits of a certain com-modity to send over its output links, null bits are delivered.

Note that

(18)

and, hence, the processes are rate convergent withtime average rates . In order to analyze delay, we consider


the rates and the input processes averaged overslot intervals. In particular, for any time , we define

From (16), we have that

(19)

It follows that for a suitably large value of , the expectationsof and satisfy a similar inequality, as specifiedby the next lemma.

Lemma 2: Fix , which defines the convergence in-terval according to (11) and (12). Then, for any time andregardless of the past history , we have

(20)

Proof: Note that for any variablessatisfying and for ,then . The proof follows by applying thisfact to inequality (19), and is given in [1].

A. Lyapunov Drift Analysis

Our stability and delay analysis relies on the following lemmathat specifies a sufficient condition for queueing network sta-bility. Define as a Lyapunov functionof unfinished work, representing a scalar measure of networkcongestion. For a given control policy and a given unfin-ished work at time , define the K-step Lypapunov drift

, as follows:

Theorem 3: Lyapunov Stability—If there exists a positive in-teger such that for all time slots , the -step Lyapunov driftsatisfies

(21)

for some positive constants , and iffor , then the network is stable and

(22)

Proof: The proof is given in Appendix B.

The intuition behind Theorem 3 is that drift is negative when-ever backlog is sufficiently large, leading to stability. The factthat Lyapunov drift is compared after slots (rather than aftera single slot) is required for systems with non i.i.d. dynamics.Similar -slot analysis of Lyapunov drift has been used in [27]to address stability, and similar drift statements for i.i.d. systemswhere are found in [35]–[38].

Lemma 3: For any control policy resulting in decisionvariables , the -step Lyapunov drift at any slotsatisfies

(23)

where is defined in (14) and

(24)

Proof: Note that . For , the -stepdynamics of unfinished work satisfies

where the above expression is an inequality instead of anequality because the total bits arriving to node from othernodes of the network may be less thanif these other nodes have little or no data to send, and becausesome arrivals during the slot interval may also depart in thesame interval. Neglecting the time subscripts and using thesimplified notation , we have

[where the variables are defined after (18)]. By squaringboth sides and noting that , we have


The expression for Lyapunov drift is obtained from the aboveinequality by summing over all nodes and commodities andtaking conditional expectations

where we have used the , and bounds bynoting that the sum of squares of positive numbers is less thanor equal to the square of the sum (see details in [1]).

B. Comparing Performance of DRPC and STAT

Using the definitions of and given in(24) together with inequality (20), we have

(25)

Plugging this inequality directly into the Lyapunov drift expres-sion (23) and using the drift theorem (Theorem 3) yields thefollowing bound on network congestion under STAT:

We now consider another algorithm FRAME, which is aframe-based modification of the DRPC policy: Scheduling,power allocation, and routing under FRAME are done everytime slot exactly as in the DRPC algorithm, with the ex-ception that backlog updates are performed only everyslots. Specifically, for any time slot within a slot frame

, power is allocated to maximizesubject to . Thus, current

channel state information but out of date backlog informationis used every slot.

Lemma 4: .Lemma 5: Comparing FRAME and DRPC, we have

Lemma 4 is proven at the end of this section, and Lemma 5 isproven in Appendix C. Using these lemmas with (25), it followsthat:

Using this bound in the Lyapunov drift expression (23) yields

Applying the drift theorem (Theorem 3) proves that DRPC isstable with the congestion bound as given in Theorem 2.

To complete the analysis, we prove Lemma 4.Proof of Lemma 4: We have the following identity:

(26)

Taking conditional expectations above and summing overyields an alternative way to express

For every , the FRAME algorithmis designed to maximizeover all possible algorithms, including algorithm STAT. Tosee this, note that:

, wherethe first inequality follows by definition of , and the secondinequality follows from (2). By definition, the final expressionis maximized under the FRAME strategy.

VI. DISTRIBUTED IMPLEMENTATION

The DRPC algorithm of the previous section involves solvinga constrained optimization problem every time slot, where cur-rent channel state and queue backlogs appear as parameters inthe optimization. Here, we consider decentralized implemen-tations, where users attempt to maximize the weighted sum ofdata rates in (9) by exchanging information with their neighbors.The current neighbors of a node is defined as the set ,representing the nodes to which node can currently transmitand receive. Theoretically, all nodes could be neighbors, as thepower transmitted from one node may be detected everywhere.However, to limit implementation complexity, it is practical torestrict neighbors to a fixed set of nearby nodes with the bestchannel conditions. We assume the neighbor sets are de-fined according to some such rule, and that nodes have knowl-edge of the link conditions between themselves and their neigh-bors and are informed of the queue backlogs of their neighborsvia a low bandwidth control channel.

A. Networks With Independent Channels

Consider a network with independent channels, so that thetransmission rate on any given link depends only on thelocal link parameters: . Assumethat the rate functions are concave in the singlepower variable for every channel state (representingdiminishing returns in data rate for each incremental increase


in power). These assumptions are valid when all links use or-thogonal coding schemes, beamforming, and/or when links arespacially separated such that channel interference is negligible.

In this case, the optimization problem (9) has a simple decou-pling property, where the weighted sum is maximized by sepa-rately maximizing each term. This corresponds to nodes makingindependent power control and routing decisions based only ontheir local information. Indeed, each node max-imizes subject to its power con-straint . This optimization is a standardproblem of concave maximization subject to a simplex con-straint, and can be solved easily in real-time with any degreeof accuracy [41].

B. Distributed Approximation for Networks With Interference

Consider a network with rate-power curves described by thefunction given in (1). This network has dependent,

interfering channels, and the associated optimization problem(9) is nonlinear, nonconvex, and difficult to solve even in a cen-tralized manner. Here, we provide a simple decentralized ap-proximation, where nodes use a portion of each time slot to ex-change control information with neighbors

1) At the beginning of each slot, nodes randomly decide totransmit with probability . All transmitting nodes send acontrol signal of power , where is some globallyknown scaling factor designed to limit power expendedby the control signal.

2) Define as the set of all transmitting nodes. Eachnode measures its total resulting interference

, and sends this scalar quan-tity over a control channel to all neighbors.

3) Using knowledge of the interference, attenuation values,and queue backlogs associated with all neighboringnodes, each transmitting user decides to transmit usingfull power to the single neighbor who maximizes thefunction

Note that this allows nodes to receive from multiple trans-mitters simultaneously, with rates that correspond to the effec-tive SIR of each transmission. The constraint that a transmittingnode cannot simultaneously receive can easily be incorporatedby setting (or, equivalently, ), for all trans-mitting nodes .

By convexity of the function, itcan be shown that the power allocation that maximizes

[for SIR as defined in (1)] has theform where each transmitting user sends to the single neighboras in step 3 above. However, note that there are possiblesubsets of transmitting users. The above distributed algorithmis not optimal over all possible power allocation strategies, asit chooses transmitters randomly. However, we note that thealgorithm is optimal over the class of all algorithms that useeither zero power or full power, and that choose transmittersrandomly with probability . This holds because the

restriction can be viewed as another constraint describing theset of acceptable power allocations, and because the randomtransmitter selection can be viewed as a random channeloutage. Thus, the above distributed algorithm achieves networkcapacity over this modified channel model (see [1] for details,including bounds on capacity and delay).

The idea of randomly choosing users to transmit is similar tothe technique used in the Grossglauser–Tse relay algorithm of[6], designed to achieve throughput for networks with anarbitarily large number of users . However, rather than trans-mitting to the nearest receiver (as in [6]), our algorithm choosesthe receiver with the largest backlog-rate metric. As it is optimalover all random selection algorithms, it supports a setof data rates that contains the set of rates supportable by theGrossglauser–Tse algorithm. In particular, it also achievesthroughput independent of .

C. Simulation of Centralized and Distributed DRPC

Here, we apply the distributed DRPC policy of the previoussubsection to an ad hoc network with mobility and inter-channelinterference. Consider a square network with 10 mobile users,with user locations discretized to a 5 5 grid. The stochasticchannel process is characterized by the following modelof user mobility: Every time slot, users keep their locationswith probability , and with probability they moveone step in either the North, South, West, or East directions(uniformly distributed over all feasible directions). Each useris power constrained to , is restricted to transmitting toonly one other user in a given time slot, and cannot transmitif it is receiving. Power radiates omnidirectionally, and signalattenuation between two nodes and is determined by thefourth power of the distance between them (as in [15]), so that

coefficients are

ifif

where represent user locations within the net-work. Note that the extra “ ” term in the denominator is in-serted to model the reality that attenuation factors are keptbelow 1 (so that signal power at the receiver is never morethan the corresponding power used at the transmitter). Thevalues are set to infinity as a simple way to enforce the constraintthat transmitting nodes cannot receive.

Multiuser interference is modeled similarly to the rate-powercurve given in (1). However, rather than use thefunction, we use a rate curve determined by four differentquadrature amplitude modulation (QAM) schemes designedfor error probabilities less than 10 . The rate function is thus

where is a piecewise constant function of SIR, and is de-fined by the QAM coding schemes given in Fig. 3.

We consider the enhanced DRPC algorithm withfor all , and , and assume the

power/noise coefficient is normalized to ,


Fig. 3. Piecewise constant rate curve for the four modulation schemesdescribed in the table. Scaled power requirements are shown, where �represents the minimum distance between signal points.

where is the minimum distance between signal points inthe QAM modulation scheme. The algorithm is approximatedusing the distributed implementation described in the previoussubsection, where each node transmits using full power withprobability . As the network is small, we simply definethe neighbor set for each user to be the set of all othernodes in the network. A centralized implementation is alsoconsidered, where the optimization problem (9) is implementedusing a steepest ascent search on the piecewise linear relaxationof the curve (see Fig. 3). The resulting data rates arethen “floored” according the threshold levels of the piecewiseconstant curve . Note that the relaxed problem remainsnonlinear and nonconvex [because SIR is nonconvex in thepower variables, see (1)] and, hence, the result of the steepestascent search may be suboptimal.

We simulate the centralized and decentralized implemen-tations of DRPC and compare to the performance offeredby the two-hop relay algorithm presented in [6]. The relayalgorithm restricts routes to two-hop paths, and hence relies onuser mobility for delivering data. Note that the relay algorithmwas developed to demonstrate nonvanishing capacity for largenetworks, and was not designed to maximize throughput orachieve low delay. Thus, it is not completely fair to compareperformance with the DRPC algorithms. However, the compar-ison illustrates that significant capacity and delay improvementsare possible even among the class of algorithms that choosetransmitters randomly.

We set the sender density parameter of the relay algorithmto . The relay algorithm was designed for nodes totransmit data at a fixed rate, attainable whenever the SIR fora given wireless link exceeds a threshold value. However, inorder to make a fair comparison, we allow the relay algorithm totransmit at rates given by the full curve. Following thescenario of [6], we assume user desires communication withonly one other user (namely, user ). Unit lengthpackets arrive according to Poisson processes, where nine of theusers receive data at rate , and the remaining user receivesdata at rate . In Fig. 4, we plot the average network delayfrom simulation of the three algorithms when the ratesare linearly scaled upwards to the values . FromFig. 4, we see that the centralized DRPC algorithm providesstability and bounded delays at more than four times the datarates of the two-hop relay algorithm, and more than twice thedata rate of the decentralized DRPC algorithm. This illustratesthe advantages of exploiting channel state and queue backlog

Fig. 4. Simulation results for the DRPC algorithm and the relay algorithm asrates are increased toward (� ; � ) = (:585;2:925).

information. We further note that the two-hop relay algorithmrelies on full and homogeneous mobility of all users, while theDRPC algorithms have no such requirement and can be used fornetworks with arbitrary mobility.

VII. MULTICOMMODITY FLOWS AND CONVEX DUALITY

The DRPC algorithm stabilizes the network and offers av-erage delay guarantees whenever the input rate matrix is insidethe capacity region of the wireless network. Here, we considera related problem of computing an offline multicommodity flowgiven a known rate matrix . Classical multicommodity flowproblems for wired networks can be reduced to linear programs,and fast approximation algorithms are developed in [33]. A dis-tributed algorithm was first given in [34], and game theory ap-proaches are developed in [29].

Here, we consider the flow problem for wireless networks,and compare the dynamic Lyapunov drift approach to a moretraditional static optimization technique. A convex optimiza-tion problem corresponding to a multicommodity flow in thewireless network is formulated, and it is shown that a classicalsubgradient search method for solving the problem via convexduality theory corresponds exactly to a deterministic networksimulation of the DRPC policy. Notions of duality are also usedin [10], [13], [29], and [30] to consider static network optimiza-tion, where dual variables play the role of prices charged bythe network to multiple users competing for shared network re-sources in order to maximize their own utility. In our context,the dual variables correspond to queue backlogs, rather than net-work prices. This illustrates a relationship between static op-timization and the dynamic DRPC policy and contributes to agrowing theory of dynamic optimization, suggesting that staticalgorithms can be modified and applied in dynamic settingswhile preserving analytical optimality.


We restrict attention to time invariant systems, so that the rate-power curve is only a function of power: . Givena ratematrix , theproblemoffindingamulticommodityflowcorresponds to the following convex optimization problem:

Maximize:

Subject to:

such that

(27)

where is the set of all variables such that

for all

for all

for some (28)

The maximization function “1” is used as an artifice to posethis multicommodity flow problem in the framework of an op-timization problem. Note that the set is convex and compact(it inherits convexity and compactness from the set consistingof all link transmission rate matrices entrywise less thanor equal to some element of , see [1]). Moreover, the objec-tive function “1” and all inequality constraints are linear. Theoptimization problem is therefore convex [40], and has a dualformulation, where the optimal solution of the dual problem ex-actly corresponds to an optimal solution of the original “primal”problem (27). To form the dual problem, we introduce nonnega-tive Lagrange multipliers for each of the inequality con-straints in (27), and define the dual function

(29)

The dual problem to (27) is

Minimize:

Subject to: for all

The dual problem is always convex, and the minimizingsolution can be obtained using classical subgradient searchmethods (where the function is maximized).Consider a fixed stepsize method with stepsize . Thebasic subgradient search routine starts with an initial set ofvalues for the Lagrange multipliers, and upon eachiteration these values are updated by computinga subgradient for one time unit, and, if necessary, projectingthe result back onto the set of nonnegative values [40]

(30)

However, it is shown in [40] that a particular subgradient ofis

(31)

where the variables are solutions to the maximizationin (29) with . Using (31) in (30) for all , wefind

(32)

From (32), it is apparent that the Lagrange multipliersplay the role of unfinished work in a multinode

queueing system with input rates , where representsthe amount of commodity bits in node . In this way, the

values can be viewed as the transmission rates allocatedto commodity traffic on link . Equation (32), thus,states that the unfinished work at time is equal to theunfinished work at time plus the net influx of bits into node. Thus, the operation of projecting the Lagrangian variables

onto the positive orthant acts exactly as an implementation ofthe standard queueing equation.

It is illuminating to calculate the optimal values by per-forming the maximization in (29). To this end, we need to max-imize subject to the con-straints of (28). However, as in the proof of Lemma 4, we canswitch the sum to find

Remarkably, from the right-hand side above, it is apparentthat the optimal values are identical to the resulting linkrates that would be computed if the DRPC algorithmwere used to calculate routing and power allocation decisionsin a network problem with unfinished work levels . Itfollows that the DRPC algorithm can be viewed as a dynamicimplementation of a subgradient search method for computingthe solution to an optimization problem using convex duality.This suggests a deeper relationship between stochastic networkcontrol algorithms and subgradient search methods. It wouldbe interesting to explore how the two interact and build uponeach other. For example, there are several known improve-ments to classical subgradient search routines. Perhaps suchimprovements could reduce the complexity of optimal andsuboptimal dynamic network controllers. Also, note that theoptimization problem (27), which maximizes the function “1,”can be adjusted to maximize some other performance criteria,which may offer additional quality of service guarantees in thecorresponding dynamic network control problem.

VIII. CONCLUSION

We have formulated a general power allocation problem fora multinode wireless network with time-varying channels and


adaptive transmission rates. The problem was formulated at thenetwork layer assuming a given (but arbitrary) set of rate-powerfunctions corresponding to the particular modulation and codingstrategy being used at the physical layer. These rate-power func-tions provide a simple separation between physical layer andnetwork layer concepts while enabling network control algo-rithms to be adapted to the unique channel characteristics ofthe wireless links. The network capacity region was established,and a DRPC algorithm was developed and shown to stabilizethe network whenever the arrival rate matrix is within the ca-pacity region. Such stability holds for arbitrary ergodic arrivaland channel processes, even if these processes are unknown tothe network controller. Delay bounds were derived and shownto grow asymptotically in , where represents the size ofthe network and represents a measure of distance between thearrival rates and the capacity region boundary. Distributed im-plementations of DRPC were considered for ad hoc mobile net-works, and the use of rate and backlog information was shownto offer considerable performance gains.

The dynamic operation of the DRPC policy was shown tobe fundamentally related to a classical iterative technique forsolving a static convex program, where unification of the twoproblems is achieved through the theory of convex duality. Webelieve that such dynamic optimization contributes to bridgingthe gap between theoretical optimization techniques and imple-mentable control algorithms.

APPENDIX ANETWORK CAPACITY REGION

Here, we establish that is a necessary conditionfor stability in a wireless network. The proof uses the followingpreliminary lemma from [1, Ch. 2 ].

Lemma 6: Necessary Condition for Queue Stability—Letrepresent the unfinished work values in a slotted time

queueing network. If the network is stable, then for any ,there exists a finite value for which arbitrarily large times

can be found so that . Inparticular, for the case , there exists a value such thatthe probability that work in all queues simultaneously dropsbelow is greater than 1/2 infinitely often.

Theorem 1a: Necessary Condition for Stability—The condi-tion is necessary for network stability.

Proof: Consider a system with rate convergent inputswith rates , and let process represent the amountof commodity bits that exogenously enter the network at node

during the interval . Suppose the system is stabilizableby some routing and power control policy, perhaps one thatbases decisions upon complete knowledge of future arrivalsand channel states. We show that multicommodity flow vari-ables must exist that satisfy (5)–(7). Let represent theresulting unfinished work function for commodity in nodeunder this stabilizing policy. Further, let represent thetotal number of bits from commodity transmitted over the

link during the first slots. We have for all time

(33)

(34)

(35)

where (34) follows because the unfinished work in any node isequal to the difference between the total bits that have arrivedand departed. Inequality (35) holds because the total bits trans-ferred over any link is less than or equal to the offeredtransmission rate summed up to time .

By the necessary condition for network stability specified inLemma 6, there must exist some finite value such that at arbi-trarily large times , the unfinished work in all queues is simul-taneously less than with probability at least . Letrepresent the slots of during which the channel is instate . Let denote the total number of these slots, andnote that with probability 1 for each . Fix anarbitrarily small value . We seek to find a timesuch that all of the following inequalities are satisfied:

(36)

for all (37)

for all channel states (38)

Define as the event that (36) is satisfied, and note thatarbitrarily large values of exist such that . Define

as the event that both (37) and (38) are satisfied. Becausethere are a finite number of input processes and channel states,we have that as . It follows there is a time

such that and . Hence,

. That is, with nonzero probability, all inequalities (36)–(38)are simultaneously satisfied.

Now, define variables . It is clear from (33)that these flow variables satisfy the constraints (5). Using (36)and (37) in (34), it follows that for all :

(39)

and, hence, the flow conservation constraint (6) is arbitrarilyclose to being satisfied. Applying inequality (35) at time andconsidering entrywise matrix inequalities, we have

(40)


where the matrices in (40) are elements ofConvex Hull . Using (38) in (40), we find

Card (41)

where Card represents the number of channel states , andrepresents the maximum possible transmission rate of thelink. Hence, the right-hand side of inequality (41) is arbi-

trarily close to a point in (compare with (7)).Hence, with nonzero probability, the multicommodity flows

(defined in terms of the processes) satisfy (39) and(41). It follows that there must exist flow values that sat-isfy (39) and (41) (otherwise, the inequalities would occur withprobability 0). As the multicommodity flow constraints (5)–(7)are arbitrarily close to being satisfied, it follows that they canbe satisfied if each nonzero entry of the rate matrix is re-duced by an arbitrarily small amount. This proves that the inputrate matrix is a limit point of the capacity region . In [1]it is shown that the capacity region is compact. Hence, it con-tains its limit points, so that .

APPENDIX BLYAPUNOV DRIFT THEOREM

Here, we prove Theorem 3, establishing a sufficient conditionfor network stability. The proof uses a simple telescoping seriesargument similar to [35], [38], and [39], together with the ma-chinery of the overflow function.

Proof of Theorem 3 : Consider (21) at times, where . Taking expectations of this

inequality over the distribution of and summingover from to creates a telescoping series,yielding

Dividing by and using nonnegativity of the Lyapunov func-tion, we have

The above inequality holds for all . Summing overyields

Dividing by and taking the lim sup of the above inequality asyields the performance bound (22).

To prove stability, note the performance bound implies thatfor any queue

Now, considering the overflow function , we have

where we have used the fact that forany nonnegative random variable . Taking limits asshows that and proves stability.

APPENDIX CCOMPARING DRPC AND FRAME

Proof of Lemma 5: Consider an implementation of theDRPC algorithm, and let represent the unfinished workmatrix at the start of a frame. Let represent the unfinishedwork at some time during the frame . Atany such time , the DRPC algorithm selects transmission rates

that maximize overall other possible control decisions. Hence

where the values represent the control decisionsthat would be made by the FRAME algorithm at time if thebacklog matrix at time were . Using (26) to switch thesummation, we have

Defining and noting that, it follows that:

(42)


where we used the fact that:

Summing (42) over and taking condi-tional expectations yields [using the definition of in(24)]

The expected magnitude of change in unfinished work from timeto time is at most at any

node, which leads to

Changing variables to and using the fact that, we have

This proves the lemma [note definition of in (15)].

REFERENCES

[1] M. J. Neely, “Dynamic power allocation and routing for satellite andwireless networks with time varying channels,” Ph.D. dissertation,LIDS, Mass. Inst. Technol., Cambridge, MA, 2003.

[2] M. J. Neely, E. Modiano, and C. E. Rohrs, “Dynamic power alloca-tion and routing for time varying wireless networks,” in Proc. IEEE IN-FOCOM, Apr. 2003, pp. 745–755.

[3] L. Tassiulas and A. Ephremides, “Stability properties of constrainedqueueing systems and scheduling policies for maximum throughput inmultihop radio networks,” IEEE Trans. Autom. Control, vol. 37, no. 12,Dec. 1992.

[4] P. Gupta and P. R. Kumar, “Critical power for asymptotic connectivityin wireless networks,” in Proc. IEEE Conf. Dec. Control, 1998, pp.745–755.

[5] , “The capacity of wireless networks,” IEEE Trans. Inf. Theory, vol.46, no. 2, pp. 388–404, Mar. 2000.

[6] M. Grossglauser and D. Tse, “Mobility increases the capacity ofad-hoc wireless networks,” IEEE/ACM Trans. Netw., vol. 10, no. 4,Aug. 2002.

[7] D. Tse and S. Hanly, “Multi-access fading channels: Part 1: Polymatroidstructure, optimal resource allocation, and throughput capacities,” IEEETrans. Inf. Theory, vol. 44, no. 7, pp. 2796–2815, Nov. 1998.

[8] L. Li and A. Goldsmith, “Capacity and optimal resource allocation forfading broadcast channels: Part I: Ergodic capacity,” IEEE Trans. Inf.Theory, vol. 47, no. 3, pp. 1083–1102, Mar. 2001.

[9] D. N. Tse, “Optimal power allocation over parallel broadcast channels,”in Proc. Int. Symp. Inf. Theory, Jun. 1997.

[10] L. Xiao, M. Johansson, and S. Boyd, “Simultaneous routing and resourceallocation for wireless networks,” in Proc. 39th Annu. Allerton Conf.Commun., Control, Comput., Oct. 2001.

[11] D. Julian, M. Chiang, D. O’Neill, and S. Boyd, “QoS and fairness con-strained convex optimization of resource allocation for wireless cellularand ad hoc networks,” in Proc. INFOCOM, 2002, pp. 477–486.

[12] J. W. Lee, R. R. Mazumdar, and N. B. Shroff, “Downlink power al-location for multi-class CDMA wireless networks,” in Proc. IEEE IN-FOCOM, 2002, pp. 1480–1489.

[13] P. Marbach and R. Berry, “Downlink resource allocation and pricing forwireless networks,” in Proc. IEEE INFOCOM, 2002, pp. 1470–1479.

[14] R. Cruz and A. Santhanam, “Optimal routing, link scheduling, andpower control in multi-hop wireless networks,” in Proc. IEEE IN-FOCOM, Apr. 2003, pp. 702–711.

[15] T. ElBatt and A. Ephremides, “Joint scheduling and power controlfor wireless ad-hoc networks,” in Proc. IEEE INFOCOM, 2002, pp.976–984.

[16] B. Radunovic and J.-Y. Le Boudec, “Joint scheduling, power control, androuting in symmetric, one-dimensional, multi-hop wireless networks,”WiOpt, 2003.

[17] R. Berry, P. Liu, and M. Honig, “Design and analysis of downlinkutility-based schedulers,” in Proc. 40th Allerton Conf. Commun.,Control, Comput., , Oct. 2002.

[18] M. J. Neely, E. Modiano, and C. E. Rohrs, “Power allocation and routingin multi-beam satellites with time varying channels,” IEEE Trans. Netw.,vol. 11, no. 1, pp. 138–152, Feb. 2003.

[19] L. Tassiulas and A. Ephremides, “Dynamic server allocation to parallelqueues with randomly varying connectivity,” IEEE Trans. Inf. Theory,vol. 39, no. 2, pp. 466–478, Mar. 1993.

[20] M. Andrews, K. Kumaran, K. Ramanan, A. Stolyar, and P. Whiting,“Providing quality of service over a shared wireless link,” IEEECommun. Mag., vol. 39, no. 2, pp. 150–154, Feb. 2001.

[21] E. Uysal-Biyikoglu, B. Prabhakar, and A. El Gamal, “Energy-efficientpacket transmission over a wireless link,” in IEEE/ACM Trans. Netw.,vol. 10, Aug. 2002, pp. 487–499.

[22] M. J. Neely, E. Modiano, and C. E. Rohrs, “Dynamic routing to par-allel time-varying queues with applications to satellite and wireless net-works,” in Proc. Conf. Inf. Sci. Syst., Princeton, NJ, Mar. 2002.

[23] A. Fu, E. Modiano, and J. Tsitsiklis, “Optimal energy allocation fordelay-constrained data transmission over a time-varying channel,” inProc. IEEE INFOCOM, 2003, pp. 1095–1105.

[24] E. M. Yeh and A. S. Cohen, “Throughput and delay optimal resource al-location in multiaccess fading channels,” in Proc. Int. Symp. Inf. Theory,May 2003, p. 245.

[25] S. Shakkottai, R. Srikant, and A. Stolyar, “Pathwise optimality and statespace collapse for the exponential rule,” in Proc. IEEE Int. Symp. Inf.Theory, 2002, p. 379.

[26] N. Kahale and P. E. Wright, “Dynamic global packet routing in wirelessnetworks,” in Proc. IEEE INFOCOM, 1997, pp. 1414–1421.

[27] L. Tassiulas, “Scheduling and performance limits of networks with con-stantly changing topology,” IEEE Trans. Inf. Theory, vol. 43, no. 3, pp.1067–1073, May 1997.

[28] J. A. Van Mieghem, “Dynamic scheduling with convex delay costs: Thegeneralized c� rule,” Ann. Appl. Probability, Aug. 1995.

[29] F. P. Kelly, A. Maulloo, and D. Tan, “Rate control for communicationnetworks: Shadow prices, proportional fairness, and stability,” J. Oper.Res. Soc., vol. 49, pp. 237–252, 1998.

[30] P. Marbach, “Priority service and max-min fairness,” in Proc. IEEE IN-FOCOM, 2002, pp. 266–275.

[31] R. Johari and J. N. Tsitsiklis, “Network resource allocation and a con-gestion game,” Math. Oper. Res., 2003.

[32] T. Roughgarden, “Selfish Routing,” Ph.D. dissertation, Cornell Univ.,Ithaca, NY, May 2002.

[33] T. Leighton, F. Makedon, S. Plotkin, C. Stein, E. Tardos, and S.Tragoudas, “Fast approximation algorithms for multicommodity flowproblems,” J. Comput. Syst. Sci., vol. 50, no. 2, pp. 228–243, Apr.1995.

[34] R. Gallager, “A minimum delay routing algorithm using distributedcomputation,” IEEE Trans. Commun., vol. COM-25, pp. 73–85,1977.


[35] E. Leonardi, M. Melia, F. Neri, and M. Ajmone Marson, “Boundson average delays and queue size averages and variances in input-queued cell-based switches,” in Proc. IEEE INFOCOM, 2001, pp.1095–1103.

[36] N. McKeown, V. Anantharam, and J. Walrand, “Achieving 100%throughput in an input-queued switch,” in Proc. INFOCOM, 1996, pp.296–302.

[37] P. R. Kumar and S. P. Meyn, “Stability of queueing networks and sched-uling policies,” IEEE Trans. Autom. Control, vol. 40, no. 2, pp. 251–260,Feb. 1995.

[38] S. Meyn and R. Tweedie, Markov Chains and Stochastic Sta-bility. New York: Springer-Verlag.

[39] S. Asmussen, Applied Probability and Queues, 2nd ed. New York:Springer-Verlag, 2003.

[40] D. P. Bertsekas, A. Nedic, and A. E. Ozdaglar, Convex Analysis andOptimization. Boston, MA: Athena Scientific, 2003.

[41] D. P. Bertsekas, Nonlinear Programming. Belmont, MA: Athena,1995.

Michael J. Neely received the B.S. degrees in bothelectrical engineering and mathematics from theUniversity of Maryland, College Park, in 1997, andthe M.S. and Ph.D. degrees in electrical engineeringfrom the Massachusetts Institute of Technology,Cambridge, in 1999 and 2003, respectively.

During the Summer of 2002, he worked in theDistributed Sensor Networks Group, Draper Labora-tories, Cambridge. In 2004, he joined the faculty ofthe Electrical Engineering Department, Universityof Southern California, Los Angeles, where he is

currently an Assistant Professor. His research interests are in the areas ofsatellite and wireless networks, ad hoc wireless networks, and queueing theory.

Dr. Neeley received a Department of Defense NDSEG Fellowship for grad-uate study. He is a member of Tau Beta Pi and Phi Beta Kappa.

Eytan Modiano (S’90–M’93–SM’00) received theB.S. degree in electrical engineering and computerscience from the University of Connecticut, Storrs in1986, and the M.S. and Ph.D. degrees, both in elec-trical engineering, from the University of Maryland,College Park, in 1989 and 1992 respectively.

He was a Naval Research Laboratory Fellow from1987 to 1992 and a National Research Council Post-doctoral Fellow from 1992 to 1993, while he wasconducting research on security and performanceissues in distributed network protocols. From 1993

and 1999, he was with the Communications Division, Massachusetts Instituteof Technology (MIT) Lincoln Laboratory, where he designed communicationprotocols for satellite, wireless, and optical networks and was the ProjectLeader for MIT Lincoln Laboratory’s Next-Generation Internet (NGI) Project.He joined the MIT faculty in 1999, where he is presently an Associate Professorin the Department of Aeronautics and Astronautics and the Laboratory forInformation and Decision Systems (LIDS). His research is on communicationnetworks and protocols with emphasis on satellite, wireless, and opticalnetworks.

Charles E. Rohrs received the B.S. degree from the University of Notre Dame,Notre Dame, IN, and the M.S. and Ph.D. degrees from the Massachusetts Insti-tute of Technology (MIT), Cambridge, in 1976, 1978, and 1982, respectively.

He is a Principle Research Scientist at MIT, and is also affiliated with theTellabs Research Center, Mishawaka, IN, which is the research arm of TellabsOperations, Inc., a manufacturer of telecommunications equipment for publicservice network providers. Before becoming a Tellabs Fellow in 1995, he wasDirector of the Tellabs Research Center for ten years. He also served on thefaculty at the University of Notre Dame from 1982 to 1997, and was a VisitingProfessor at MIT from 1997 to 2000, at which time he began his current po-sition. While best known for his early contributions to the study of robustnessin adaptive control, he has also contributed work in adaptive signal processing,communication theory, and communication networks.

air pollutant & control device characteristics effects on emission

Documents