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A Framework for Cross-layer Design ofEnergy-efficient Communication with QoSProvisioning in Multi-hop Wireless NetworksUlag C. Kozat. Iordanis Koutsopoulos. and Leandros Tassiulas

Department of Electrical and C omputer Engineeringand Institute for Systems Research

University of Maryland. College Park. M D 20742, US AEmail: {kozat,jordan.leandros} @isr.umd.edu

Abstract-Efticient use of energy while providing an ad eq uat elevel of connection to individual sessions is of paraniount im-portance in multi-hop wireless networks. E nergy efficiency a n dconnection quality depend on mechanisms that span severalcommunication layers due to the existing crrchannel interfer-ence among competing Rows that must reuse the limited radiospectrum. Although independent consideration of these layerssimplifies the system dedgn, it is nften insufficient for wirelessnetworks when the overall system performance is examinedcarefully. The multi-hop wireless extensions and the need formuting users' sesenion5 from source to the destination onlyintensify this point of view. In th b wnrk. we present a frameworkfor cross-layer design towards energy-efficient communication.O ur apprnach is characterized hy a synergy between the physicaland the medium access control (MAC) layers with a view to wardsinclusion of higher l aye rs a s well. M ore specifically, we addressthe joint prohlem of power mntrol and scheduling with theohjective of minimizing the total tran5mit power subject to theend-to-end quality of service ( @S ) uarantees for sessions interms of their bandwidth and hit error rate guarantees. Bearingto the NP-hardness of this eomhinatorial optimizstion prohlem,we proposeou r heuristic solutions that follow greedy approaches.

I . I NT RODUCT I ONThe issue of energy-eficient information transmission in

wireless ad hoc networks has received significant attention inrecent years [ l ] .The autonomous nature of such networks ren-ders their lifetime highly dependable on energy consumption.Nevertheless. the primary goal of a communication networkis to deliver an acceptable level of communication to users.Energy efficient multi-hop wireless network are not exemptfrom providing such a quality of service for their own userseither. Then. the first ma.jor i ssu e beco mes th e formu lation ofa meaningful Qo S measure for both the multi-hop wirelessnetworks and the applications running over them.

One can have different interpretations of QoS at differentcommunication layers. At the lowest level. i.e. physical layer,QoS is synonymous to an acceptable hit error rule ( B E R )or sipiul to interference and noise rario ( S I NR) . whereas atthe MAC layer or higher layers. QoS is usually expressed interms of minimum rate or maximum delay guarantees. For themulti-hop communications. network layer QoS pertains to end-

to-end provisioning of the guuaranteed QoS for each session.In accordance with these different interpretations at differentlayers. it is natural to us e a Qo S policy that is explicitly basedon both minimum short-term fate requirements and maximumtolerable BERs of the sessions. Such a QoS policy alsohelps classifying the applications .as high bandwidth or lowbandwidth and as error prone or error resilient. For instance.consider the case of a wireless ad hoc network for a battle-field operation. Users establish audio-visual communicationwith the command center while situation awareness data areexchanged among users. Though all of these applicationsexhibit quite different bandwidth. delay and error tolerancecharacteristics. they can be easily expressed in terms o fminimum short-term rate and target BER requirements persession along each session's path.Having defined the QoS policies for each session. th enext issue is to satisfy each of these policies at a minimdenerg y expenditure. W ireless transmissions mainly suffer fromchannel impairments and other user interference operating inth e same frequency band. Multi-hop wireless operation merelyexacerbates the existing conditions. Unless a coordinationspanning to multiple layers and multiple hops exists- eitherthe session QoS requirements are not satisfied or they areprobably satisfied a t a significantly higher energy consumptionthan the necessary. Once the set of sessions with their source -destination pairs and QoS requirements are given. three layerstogether impact the contention for network resources: physicallayer, medium access control (MAC) layer. and routing layer.For a cross-layer design that satisfactorily enhances the net-work performance. it is essential to highlight the interactionsamong these layers.Physical layer with its key parameters- such as transmitpower. modulation. coding rate- antenna beam coeliicients-has a direct impact on multiple access of rides in wire-less channels through affecting the interference at receiversand susceptibility to it. Local adaptation of these parametersto achieve a target BER restraints both routing and MACdecisions by altering the directed topology graph, feasibletransmission schedules. and payload transmission rates. Phys-

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ical layer features -such as transceiver complexity. powerrequired to drive the R F modules, and the transmit power-accumulatively govern the energy expenditure of transmitters.receivers. and idle nodes.MAC laxer is responsible for scheduling the transmissions

and allocating the wireless channels. While the concurrenttransmissions create mutual interference, the time evolutionof the scheduled transm issions ultimately determines the band-width allocated to each transmitter and the packet delays. Theinterference impo sed by simultaneou s transmissions naturallyaffects the performance of the physical layer in terms ofsuccessfully separating the desired signals from the rest. Onthe other hand. as a result of transmission schedules, highpacket delays andlor low bandwidth can occur. forcing therouting layer to change its rou te decisions. MA C layer affectsthe energy expenditure in two ways: (i) It mainly controls theinterference level at any time instance that may lead to transmitpower adaptation in the physical layer. (ii) Depending on theuansmission schedules. nodes may switch to a power-savingmode. turning off all or som e of their RF components.

Roirting layer selects the wireless links that will eventuallycarry the data packets. Different routing decisions alter these t of links to be scheduled. and thereby inHuence the per-formance of MAC layer. For instance. if the routing protocolchooses How paths that are closer to each other among lhe al-ternatives. the subseq uently higher interference and contentionlevels in the network make it harder for MAC to resolvethe transmission conHicts. Similarly. higher interference levelsforce the adaptation of physical layer parameters to achieve thetarget BER. However. as the number of independent sessionswith distinct source-destination pairs increases. the routingcriterion is expected to play a less important role in contentionresolution as compared to the physical layer adaptations andMAC decisions. When Qo S requiremen ts are ignored and linkcosts that accurately quantify the energy consumption canbe assigned. routing layer becomes the sole determinant ofenergy consum ption. These l ink costs_however. depend on thetransmit power, which is a hnction of decisions in all threelayers. Therefore. the layer interactions necessitate iterativeapproaches to find the most energy efficient communicationscenario.In this paper. as a first step towards solving the problem. weassume that the session paths a re already given and we addressonly the joint power control and scheduling problem with theobjective of minimizing the total average transmission powerwhile providing quality of service for individual sessions interms of payload rate and BER guarantees.

The rest of the paper is organized as follows. Section-I1presents an overview of the works that are closely related toour problem. In section-111. we lay out the detailed systemmodel. Section-IV states the formal problem description withthe objective function and constraint sets. We explain our so-lution framework in s ection-V and then provid e our simulationresults in Section-VI. We conclude the paper with a synopsisand a final discussion t o emp hasiz e the future work in section-VU.

11. RELATEDWORKSPower control has been the focus of single-hop multi-

user wireless networks for more than a decade [31-[131. Thepopularity of the topic stems tiom the facts that i t can beexploited in suppressing multi-user interference. increasingsystem user or throughput capacity. and reducing the trans-mission power hence extending the battery life of the wirelessdevices. Later on. power control has also been adopted asan ef icient protocol design technique for ad hoc wirelessnetworks in different layers a oint or isolated problems [Z],[14]-[20]. Among these highly diverse works. i t is essentialto dwell upon two recent studies. 121 and [141, in order toelucidate our own contribution with this paper.

In [?I. Elbatt and Ephremides investigate the problem ofscheduling m aximum num ber of links in the same time slo t. Inother words. authors try to maxim ize the per hop throughput ofthe network. They adapt the transmit powers to their minimumrequired levels such that all transmissions achieve a PargetSINR threshold. l l e y show that this particular system model isactually equivalent to uplink power control in cellular networksand the iterative algorithms developed for cellular networkscan be employed i n ad hoc wireless networks. In the casewhere the set of links that have buffered packets cannotbe scheduled in the same time slot. these solutions do notconverge and authors suggest to remove one link at a timeuntil a feasible set of links is achieved. However. the criterionfor removing th e link is not precisely addressed: especiallyin the case of varying target SINR thresholds for each link.Also; the system model does not cover a multi-hop wirelessenvironment.

A closer approach to our own is followed by Cmz andSanthanam in [14]. where authors provide long term end-to-end rate guarantees to a set of sessions at the minimumpossihle long term average of the total transmit powers. Theirmain assumption is that the system operates at significantlylow SINR values and that the link rates can be approximatedas linearly dependent on SINR. Hence, the transmit poweris not used for giving a quality of service guarantee in biterror rate (BER) hut rather directly used as a throughputguarantee consu aint. lnste ad o f solving the relatively difficultproblem of minimizing the long term average transmit powersum with the constraints on the power vector and on thelong term session rates. they define and solve a dual problemthat does no t have a drra1il.v gap with the primary problem[ ? I ] . Their results reveal that all the links scheduled in aparticular time slot must transmit at the maximum allowedpower P,,,a,c ather than in more number of slots at a lowerpower level. The solution method to determine the set of linksthat must he activated simultaneously as well as the existenceof schedules to achieve the rate requirements are established inthe paper. Under certain continuity conditions on the optimumdual objective function. authors also extend cross-layering 10the routing layer, where each small increment in session ratesis routed dynamically abiding by the path costs as determinedby the rate of change in dual objective function. Hence. the

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optimal joint routing. scheduling, and power control policy isobtained.

Our system model differs from Cruz and Santhanam inseveral respects. First of all. we want to satisfy the raterequirements of the sessions not only in the long term buta lso i n the short term within a well-defined frame duration.This prevents the sessions with low jitter or bounded delayrequirement suffering from the ambiguity of the long termguarantees. Secondly, the end-to-end rate constraint used in[141 is actually the end-to-end throughput constraint. i.e. thenumber of bits that are successfully reached to the destination.We instead decouple the end-to-end throughput constraints intothe transmission rate and the BER constraints. which betterdifferentiate the applications and which are more amenableto actual system implementation. In this way. we also avoidthe artificial assumptions such as approximating the rate as alinear function of SINR values.

In the following sections. we describe our system modeland solution framework in detail.' .

111. S Y S T E M MODELWe consider a wireless network of 11 nodes. Each node is

capable of transmitting at a power value less than or equal toP,,,,,. A rlirected link exisw: between nodes i and j if the sipnalto noise ratio ( SN R) at receiver j . when i transmits at thismaximum power. is above a threshold T;~. i.e. GljP,,,nz/cr,'27 i j . where G ij represents the path gain from i to j and'.,'is the ambient noise at receiver j . Furthermore. we have Ssessions and each session i is characterized by:

A {source.destination} pair:A set of directed links that constitute the session path.A minimum short-term end-to-end rate requirement inb i t s k c .. aximum RER requirement for each directed link alongthe session path.T h e end-to-end rate requirement for a session dictates that ihe

designated session rate must be supported across all links thatconstitute the session path. BER requirements are derived as alink budget estimation using the information on the total errnrtolerance of the session and its path length. In the rest of thissection. we delineate the specific details of how the sessionrequirements are satisfied.A. Channel iMode1

The data packets are transmitted over the same wirelesschannel. which refers to the same frequency band in this paper.To prevent s rlf-in terfer ence . half-duplex operation is enforced.i.e. a node cannot transmit and receive at the same time. Weal so limit ourselves on ly to point-to-point transmissions andno node is p ermitted to send multiple packets (for the samereceiver or not) at the same time. The payload rate K of linkI over the data channel is given by

A,1. . TdOl ,Tlmm =%m,=Lslobliram ._ ..& 7

Fig. I . Sampld topology an d scheduling fw concurrznl multi-hopsessions.

where b",,,, is the number of hits per symbol. R6 is the codingrate, and T&,n is the symbol duration for the transmissionsover 1. l i m e domain is divided into slots of length ' T 3 l O t andtime slots are further grouped into frames of L slots. We d onot have control over the physical layer parameters b f y n l l RL.and T&m.hut we assume that they can be altered only beforethe start of each frame and that they are kept fixed throughoutthe frame. Hen ce_ for link [: each slot has a constant payloadrate, i.e.

bLvm x R:L x T:ym)'I =

The scheduling is performed per frame basis and each link isassigned to a numb er of slots in a given frame . More precisely.the short-term rate requirement 'ri of each session i . whichtraverses directed link [. necessitates allocating

time slots for link 1 . Here. r.1 stands for the ceiling operation.Note that. in reality. we assign the time slots to the transmitterof a link and different links may have the same transmitter.As i t will be clear later on, transmitters can utilize the sametime slot assigned to them for different sessions only if th esessions have the same BER constraint and they traverse thesame directed link. Therefore. the actual num ber of t ime slotsLf assigned to a directed link I can be bounded as:

where 'Piepresents the flow path of session i an d I ( - ) s theindicator function that is equal to one if it s argument is trueand zero otherwise. k L atisfies the left hand side of the aboveexpression when all sessions traversing link [ have the sameBER requirement and are able to be m ultiplexed together ont othe same slot. In other cases. the upper-bo und on the right han dside becomes valid. Obviously, the lower and upper boundsbecome the same when re's are integer multiples of ri . Hereon. without loss of generality. we restrict our attention to th esession rates that are integer multiples of q.

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Let us examine OUT system model as described so far onFig.1. In the figure, bidirectional arrows show the existence ofdirected links between node pairs they connect. T he .framelength is se t to S slots. There are three sessions initiatedat nodes I , 2. and 5 with flow paths depicted by dasheddirectional mows. Session 1 has a bandwidth requirement of2 slots per frame. whereas sessions 2 and 3 both require 1Slot per frame. Thus. total end to end bandwidth requirementbecomes 8 slots per frame. Since the total figure is above theframe lenglh. different links have t o he activated at the sametime. Also_ the BER requirements at each receiver must b esatisfied in all time slots. A sample link scheduling is given inthe figure. Due to the bandwidth requirements or overlappingRows. the same link can be activated more than once duringa frame period. For instance, ordered vertex pairs (15) an d(5.7) must both he scheduled twice in the sample scenario.Next. we elaborate on how the BER constraints of the con-current transmissions can be satisfied using proper schedulesand transmission powers.B. SINR tlirestrold and Feasibilib of Concrrrrenl Transinis-sions

In this subsection. we dwell upon the relation among themodulation level. coding rate. DER. and SINR. We willassume that DER is a one-to-one monotonically decreasingfunction of SINR . Thereinre. a maximum tolerable BER canhe mapped onto a minimum SINR threshold fo r a successfulreception. In general. uansceiver pairs may support multiplemodulation levels (e.g. M-QAM with AI E { l>?>. . . ,M O } )and code rates (e.g. K, = 314: 18:1). In the presenceof time-varying link quality, the objective of modulation andcoding rate adaptations is to increase transmission rate andto maintain an acceptable BER at the receivers. Lower mod-ulation levels and coding rate can sustain more interferenceor equivalently assist in lowering average transmitted signalpower at the same interference level.

For instance. when W Q A M modulation is used for thetransmissions over link 1. e .g . b& , = log, i1.I; the BER isapproximated as BER O.',zs p[-l.S( SIN R)/i\l - 11 [??I.For a maximum acceptable BER of E _ the SINR should satisfy

Thus. we map each modulation level b,,,, and maximumacceptable BER to the SINR threshold h. hich is equal tolhe right-hand side of (l).' Clearly. decreasing b,,, or R,also reduces the SINR threshold. On the other hand. the left-hand side of ( 1 ) is determined by channel gains. noise power,and the transmit powers of the links assigned to the sametime slot. We allow the adaptation of transmit powers betweenconsecutive time slots. Since we have assumed that the cod ingrate and modulation level are kept fixed throughout the frame.

'In 0cnsral. thz SINR thresholds for each transmission iwen over the samel ink) differ from each other: bsrauss sithar different modulation or codinsschemes arcuscd ior differentlinks of sessionsor each szssion is charactenzedhy it.< ow " BER requirement.

transmit powers and slot assignmenv; are the only controls wehave to satisfy the BER constraints.Suppose that C( i ) denotes the set of links that are assignedto slot 17 : T ( / ) nd R(1) are the transmitter and receiver endpoints of directed link 1 ; PI is the transmission power at node

T(1); nd transmissions in slot n over link / are dedicated topackets of session s i . " , Then. for each link 1 E C ( n ) . at thegiven modulation level and coding rate. BER requirements ofsi.,,'s are mapped onto the following set of constraints:

Constraints in (2) can b e p u t into matrix form by definingIC(n)l by IC(ii)I matrix G and the column vector /9 withentries:

'il 4(,LGT(WJ) . B. - ( 3 )L -1+ 13 G T ( j ) R ( j ) 1+ ii G T ( , ) R ( , )3 -Then. we obtain:

P 2 G P + P . (4 )Here. P is simply the transmit power vector for the linksassigned to slot n. C ( n ) is afeasihle assignment for slot nif (4) is satisfied for a non-negative and finite P.Matrix G is nonlnegative and irreducible. From Perron-Frohenius theorem. G has exactly one ositive real eigenvaluep with p = i i i ax{~X~~}~&,here {Xi},=, are the eigenvaluesof G. p is called the Perron-Frobenius eigenvalue of G. Itis well-established that (4)s satisfied for a non-negative andfinite P i f and only if p < 1 [ l l] . Hence_ the feasibility _ofC( n,) is solely determined by the maximum eigenvalue of G.which is contingent upon the channel gains and the sessions'BER requirements.

It is important to note that. in our model. link gains ofdifferent links remain constant within a time frame. Thus.our approach applies primarily to quasi-stationary or fullystationary wireless networks. when the link gain G, j of eachlink ( i : j ) aptures mainly path loss and shadowing effects. Amore general approach would he to consider a model for linkgain variation from slot to slot. where link gains are availablefor th e entire network at the heginnine of a slot.Next. we present the notion of virtlral links to simplify oursystem model.C. Nolion of Virlitul Links

At this point. i t is useful to introduce the notion of virrrrallink to avoid dealing with the handwidth and BER require-ments of the ses sion s explicitly. Let's denote the index set ofactive links' with A" = {1: 2 E } ,As the same link can bescheduled more than once ( i n different slots). we index eachinstance of such links separately and denote them as virtuallinks. hecause they physically constitute the same link. Thus.we have a populated index set A" = { I , 2 ; . . !\,I} for virtual

%

'This is the se t of links which carry payload traffic as a result of routingdecisions.

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Slinks where !lT = h i ( r i l r l ) an d h ; is the number ofhops that ith session traverses. We continue to use T ( i ) ndR( ) notation to denote the actual transmitting and receivingend points of the virtual link i . We formally define our problemin the next section over these virtual links, but we first needto elaborate on one more subtle point.

Our channel model restricts us to half d uple x operation andpoint-to-point communication with one packet transmission ata time. The former condition is violated if two virtual links iand j that are scheduled in the same slot have the property ofT ( i )= R ( j ) and the latter is violated if T( ) = T ( j ) . heseproperties suggest that the set of l inks scheduled for the sametime slot must be il marching set in the corresponding topologygraph. Nonetheless. we can simply absorb the irlatching setconstraint into the SINR constraints by settin g GT(,)T(L) COand letting the y i ' s to be high enough. In other words. whennode i is scheduledto receive and to transmit at the s ame time.the SINR at node i is driven to zero. violating its positive SI NRrequirement as a receiver. In a similar way, if two virtual linkswith the same transmitter are simultaneously scheduled, theywill he strong interferers for each other. and hence lead tounsatisfied SINR constraintsHaving comprehensively described our system model. weare now read y to Sormally state our pro blem in the next section.

IV . J O I N T POWER ALLOCATI ONAND S C H E D U L EA S S I G \ M t i U l P R O B L E M

A . Fornrul Problern SrcitoiririitWe want to minimize thc tolal transmit power as summed

over all time slots and links whilc satislying the minimumrate and SINR conswdints of the sessions. Since the raterequirements are expressed i n nuinher o f slots per frame.assigning more slots ti l each l ink than the minimum dictatedby the rate requirements and the routing paths unnecessarilyincreases the power consumption. Therefore. assigning oneslot to each virtual link satisties the end-to-end session raterequirements while achieving minimal power consumption.This fact motivates the formulation of the problem in termsof the virtual links. Then. OUT objective becomes:

subject to( 5 )

1-ic ( j ) = c ( < )1c ( i ) ~ ; F = { l : i ). . . L } ; t ' i = l : . . . A { : - (7 )

(8)m,,r 2 P, 2 0 ; v7 = 1 : . . . 1b.l ~where Pi is the transmit power of node T(L) nd e ( ; ) is thetime slot virtual link i is assigned to. Inequalities (6). (7).an d (8 ) are the SINR. frame length. and power requirementsrespectively. Together they define the constraint set

R = { ~ p :2 ~ 2,,, u n d ~ 2Hp+p}. (9)

Here. A : A" - 1:. . .,L } is the time Slot assignment ofvirtual links; is the AT x 1 column vector with ith entryP,; is the ib.l x M diagonal matrix with diagonal entriesTi.; -,,; is the M x l I interference matrix with entriesH;.j = 8; xGT( ; )R (~ ) /GT ( ; )R (~ )or 7 # j an d H,,; = 0 V i ;-is the M x 1 column vector with ith entry -&(i)/Gq;)R(i);and 6; is the assignment function that equals IOi f c ( i ) =c ( j ) .otherwise it is 0. Whenever we have a pair (/LE)E a.we will refer to them as joinflyfeasible allocation. Among allsuch pairs. we search for the ones that minimize ( 5 ) , whichwe will call jointly optirrial allocations.

Given the assignment instance. our problem reduces toclassical power control problem in cellular networks and wecan check if there exists a feasible solution [I 11. Moreover, wecan'find the optimum power allocation at each slot centrallyor iteratively. In hct. the optimum power allocation is ParetooptirtJal. i.e. all the l i n k transmit at their minimum feasiblepower. and the constraint (6) is satisfied with equality [Y].However. finding the jointly optimum transmit power and timeslot allocation is not straight-forward extension to the con-tinuous transmission scheme as in the cellular voice services[13]. Since our constraint set does not satisfy the necessarymonotonicity feature of the standard function [lo]. the existingiterative solutions cannot solve our problem. Besides. theconstraint set is not a convex set in general and we cannotalso apply standard techniques that minimizes a linear iunctionover a convex set. On ly under special conditions -such as whenframe length L is larger than the number of virtual links ,%I-the optimum joint allocation is trivial. e.g. each virtual link isplaced on a distinct time slot and transmission power is setto the value just enough for combating the ambient noise.Then, the main question is whether we can find a jointlyfeasible allocation and an efficient optimal solution undergeneral circumstances. Next. we will prove that the feasibilityproblem is indeed NP -complete [ ?3] $which leads us to devicesuboptimal approximation algorithms.B. Inrractability of the Jointly Feasible Allocation

Let us first define the following problem.Pl(feasihi1ity problem): Given the gain matrix G. framelength L. session rate and SINR constraints. is there a schedule

and power assignment that satisfy both the rate and SINRconstraints'?

To show the NP-completeness of P I _ we provide an alter-native formulation of OU T optimization problem and its corre-sponding feasibility question. In this formulation. we assumethat each session ha s the same B13R (o r SIN R) requirement andthe virtual link notion is put aside. Naturally. the gain matrixG and the SINR constraint define a super-set X of activationvectors S1S?,. . ,S,where each S i have exactly E entriesfrom the binary set {I ll}. The entries with value I correspondto the indices of simultaneously transmitting active links whileeach transmission satisfies the given SINR constraint. Clearly.the vectors majorired by any S; re also the members of X.Suppose that we also h o w the power vectors that achieve theminimum total transmit power P.;,or each Si.hen, our

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objective becomes:

subject to(11)(12)

Here_ pi is the total flow rate through link i as determinedby routing decisions and session rate requirements. in i is thenumber of slots that activation set S,. s used. and L is thenumber of slots in a frame as before. (11) is the short-handrepresentation of the rate requirements. whereas (12) simplystates the total number of slots cannot he larger than theframe length. Objective function ( I O ) and constraints (1 1)-(12) constitute an integer programming problem. Hence. itsfeasibility problem given below is NP-complete [23].PZ(a1ternative feasibility problem): Given the finite se t ofX with the associated minimizing power assignments andSINR threshold. is there a E-tuple of integers such thatconstraint (12) is satisfied for fixed L and the rate constraints

Now. we can easily prove that PI is also NP-complete.Lenirria 1: PI is NP-complete.

T[ X I & . . . .Yx]rJ2 [ p , . . P E ]and to

11.. l]C5 L

in (1 ) hold'?

Procc Given any schedule and power allocation. e.g. aninstance for PI. it takes O(M ' ) time steps to check if thesession rates and SINR constraints are satisfied. Therefore. PIis in NP.Consider the following mapping: ( i ) Since we know theSINR threshold -j of p?. se t yi in PI as y . (ii) Starting fromthe members of X that has the least number of active link,compute the elements of gain matrix using y and minimizingpower vectors. (iii) For the entries of G that cannot hecomputed. enter M . (iv) Create virtual links for the physicallinks that have a rate more than 1 slot/frame. ( v ) Keep th eframe length same.This procedure takes O ( ? ) time steps and an instance ofP2 is mapped onto an instance of PI in polynomial time.Since PI solves exactly the same problem: P2 reduces to PIin polynomial time. However. p? is NP-complete and PI is in

This intractability result demands sup-optimal but eficientalgorithms to perform the joint scheduling and power alloca-tion. Bef ore proceeding w ith our algorithmic proposals. wewill derive some useful upper and lower bounds in the nextsubsection.C. Pe'rrfoniiance Boiinds

The main interest of this section is to derive bounds on thetotal transmit power in a specific s lot 71 in terms of path gainsgiven that the virtual link assignment is feasible. Then. thetransmit power of link i in slot 17 satisfies the inequality:

NE which makes PI NP-complete.

Summing up both sides of inequality (13) over all links inslot n and rearranging the terms in the second summation. weobtain:

* , ~ U 2P; , (14)G r ( i ) ~ ( j )

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Generate anundirsrtrd graphfrom unassigned vimal l i n k

/Find Maximum( Molching

Fig. 2. BIuck Diagram for Algmithrn A

sufficiently small. the upper bound also becomes a tight one.Quite intuitively. both the upper and lower bounds suggest thatwe should minimize a(.). i.e. choose a set of links. in whicheach link has a pood channel gain o r low S INR requirements.These observations are the main ingredients in the design ofour heuristic algorithms which are revealed in the next section.

V. I N T E R F E R E R - B A S E DU B - O P T I M A LEURISTICALGORITHMSWe explore two greedy sub-optimal alzorithms to solve thejoint power allocation and schedule assignment problem that

we refer to as algorithms A and B respectively.A. Algorithm A

The first approach follows a top-down design strategy. Itstans with the feasibility problem P1 and sehrches for theminimum frame length L* to satisfy the rate and SINRrequirements. Clearly. P1 is solvable if and only if L' 5 L.On ce the problem instance is identified to be feasible. the linksfrom congested time slots are shifted to the empty or lesscongested ones to further reduce the transmit powers of thevirtual links. The decision criterion on wlrich link to he shiftedIO which tinre $101 is explained in details below.

Block dkigram in Fig.2 summarizes Algorithm A. We aregiven a set of virtual links. each of which has to be scheduledfor once throughout the frame duration. Initially. all the timeslots are empty. Starting from the first slot. we want topack as many virtual links among the unscheduled links aspossible into a single slot. The unassigned virtual links withtheir transmitters and receivers form a directed graph thatpossibly has multiple directional edges between the Samevertex pair. Because of the point-to-point and half duplexcommunication assumptions. we cannot assign any two of0-7803-8355-9/04/S20.007.004 IEEE. 1452

these directional edges connecting the same vertex pair tothe same slot. Hence, we can replace directional edges withundirectional edges and prune the extra edges connecting samevertex pairs. In this way, we obtain an undirected graph. Thesame assumptions further render only simultaneous schedulingof inarching edges' possible. 'then. putting as many linksas possible in the same slot becomes i i i ax ini i inr matchingproblem. which is solvable in polynomial time [23].

Next step in the algorithm involves (i) one-to-one mappingof the maximum matching back to virtual links and (ii)checking if we have a feasibk power allocation for this se tof virtual links. When an undirected link in the matching setcorresponds to the same directed link. w e pick the one that hasa smaller SINR threshold. which obviously has a better chanceto satisfy the slot feasibility. In the case where the undirectedlink corresponds to the links with opposite polarities. we pickany of them. If the maximum m atching fails to he feasible. weremove the link with maximum interference on the matchingset. This process continues until the matching set is reducedto a feasible one. The matching set is infeasible providedthat: (1 ) Perron-Frobenius eigenvalue p is larger than or equalto one or (2 ) p is smaller than one', hu t any of the linksfails to satisfy maximum power constraint. Removal of themaxim um interferer is beneficial not only in limiting the totaltransmit power of the matching set (see (19)). but also foravoiding the ambiguity in case. where successive removalslead to infeasibility as a result of having p 2 1. The virtuallinks in the resulting matching are pruned from the directedgraph and we continue with the next time slot until all virtuallinks are assigned to B feasible slot.

If we cannot assign all the virtual links for a given framelength L. we declare the problem instance as not j o in t&feasible. In the situations. where all the links are assignedto a number of slots less than L. we run an optimizationstep to shift the l inks to non-utiliredlunder-utilized slots. Agreedy approach would be as follows. For a link reassignmentn that involves reassignment of link i from slot s to slot swe compute the factor " ( a . ) = P ( b e / o r e ) - P(of ter)_where P(before) is the total power consumption before thereassignment and P(after) is the total power consumption afterthe reassignment. The link that is selected for reassignment isthe one that causes the maximal power consumption decreaseAP(a.). he algorilhm terminates when no further link reas-signments can cause power consumption decrease. i.e.. whenA P ( o ) < 0 for all reassignments n of links from slots s toslots si. Evidently. we restrict the re-assignments to the onesthat ensure the joint feasibility.B. Algorirlirrr B

The second strategy on the other hand follows a bottom-up approach. The iteration is performed over the unassignedlinks. Initially. we assign exactly one link to each slot until noempty slot remains. We choose the maximum interfering link'Thw are the edges that do not share B c o r n o n wnrx..'Thsn. w s ca n compute the optimal power allsat ion by mauin invsrsion.


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I O

Fis. 3. Block D i a p m for Alocvithm 8.

among all the unassigned links to place in the next emptyslot. In this respect. we distribute the transmitters that arewithin close proximity of each other onto different time slots.In the second stage. algorithm performs the assignment usinga water-filling argument: We assign a link i to slot s onlyif this assignment is feasible and s has the minimum upper-bound after the assignment as computed by the expression (19)among all feasible assignments. Algorithm terminates whenall the links are exhausted or no feasible assignment can befound for an unassigned link. In the latter case. the probleminstance is declared as not jointly feasible. The block diagramin Fig.(3) summarizes these steps. Evidently. Algorithm Btries to balance the power consumption in each time slot. Analternative scheme would he t o directly rely on the actual totaltransmit power levels rather than on the upper-bounds. Wecould therefore directly assign the cost of an assignment a thatmaps link i onto slot .? by looking at the increase in the totalpower consumption and at each iteration the assignment .withthe minimum cost would be chosen. However. this approachmay potentially fail to distribute the transmitters in closeproximity effec tively leading to more costly slot assignmentslater on.

In the next section. we evaluate Algorithms A and B overa multi-hop enabled cellular network topology under differentrouting decisions. We believe that such topologies are well-suited for actual implementation of our cenualized heuristicsolutions.

V I . SIMULATION RESULTSWe have investigated the performance of our heuristic pro-posals on a 11100n1x 1000,11 quare topolozy. The network is

partitioned into four square cells and four nodes are positionedat the center of each cell. These nodes at the cell centers can beviewed as cluster heads that concentrate traffic in each ad hocdomain to relay to the other domains or access pointshase-stations of an inliastructwe/overlay network. For the conve-nience of expression. we refer to them as base-stations. Theremaining wireless nodes are randomly distributed over thewhole topology. Source nodes are randomly picked and theirdestination Doints are set as the closest base-stations from their

m

Fig. a. Multi-hop cellular topology instance with fixed bass-station locationsa1 thd cenler of squnre calls and randomly disuibuled relryisource oodcs overa lWOmxlOWrn l o po lo ~ y .

own position in th e Euclidean sense. Each session as identifiedby its source node bas a fixed rate requirement of 1 slot/frameas well as an SINR requirement uniformly picked from the set{4: 5,6: T:8). The noise power is assumed to be same at eachreceiver and transmit powe rs are norm alized with respect to thenoise power. The channel gains are computed by only takingthe path loss factor into account with the path loss exponentof two for transceiver pairs close than 100 meters and fourotherwise. We limit the maximum normalized transmissionpower to be 31.25. which corresponds to a transmission rangeof 250 meters at the highest SINR requirement. We havecnnsidered two different shortest path routing schemes foreach given scenario with link costs equal to a unit value (i.e.minimum-hop routing) and transmission power just enough tocombat the noise for the specific session (i.e. minimum-powerrouting). A sample topology instance with the set of links asdetermined by the session source nodes and minimum-powerrouting is depicted in Fig.4.

We have compared our heuristic proposals by (i ) theirsuccess in identifying problem instances as feasible or not and(iij their total transmit powers as averaged over all feasibleproblem instances. As the variable system parameter. we usethe frame length in number of slots while keeping the sessionrequirements fixed. Note that this is essentially equivalentto keeping the frame duration fixed and altering the trafficload in terms of the session rate requirements. For the fewscenario settings where problem instance sizes turn out tobe manageable. we have also computed th e performance ofoptimal solutions.

Figures 5 and 6 show the average performance for thescenarios where we limit the number of sessions to sevenand use the minimum-hop routing. In the plot legends, whenthere is an lipper-boiind label next to the algorithms A andB, i t indicates that the upper-bound in (1 9) is used in theheuristics instead of h e actual total transmission powers of

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the slots. Similarly. thc uarrtul power label corresponds to theutilization of the actual power levels in the greedy heuristics.Quite interestinely. we observe that Algorithm A. which isspecifically designed for. first finding a feasible solution. isactually outperfnrmcd hy Algorithm B. which relies on theupper-hound fnrmulaiion w e derived. In other words. a water-filling argum ent with .a priipcr cost iun ction c an actually b emore su cce ssi d than a tnp-dowii dcsign strategy as A lgorithmA does hy computing maximdl matching sets and then byprnceeding with link pruning. Ilinvcver. an inadequate costfunction that daes not assist ii i distributing the links. whichexhihit high interference to each oihcr. onto different slotsresults with a degraded performance as seen i n Fig.5. Alpo-rithm B also successfully matches the perform ance of optimalsolutioii in finding the feasible solutio ns provided that theyexist. On the nther hand. when the total power consumptionas summed over all the virtual links is observed. Algorithm Aexecutes much better than the other heuristics. Algorithm Balso performs comparable to optimal solution when the actualpower values are computed to decide on the next assignment.However. due to the lower success rate in identifying thefeasible solutions. Algorithm B with the actual power heuristiccan resolve only the less constrained scenarios and the lowerpower consu mption figures should not be mis lead ing. Surely.the optimal solution perlorms ktter at each problem instance.The ovcrall suggestion of the power consumption results isthat greedy approaches, which directly operate on the objec-tive function. have an advantage in minimizing the objectivefunction.Figures 7 and 8 present the same topology and sessionrequirements. but at a different routing strategy. i.e. minimum-power routing. Since the problem size is quite large. we didnot compute the optimal values. Nevertheless. we know thatthe optimum strategy when L gets large enough is to scheduleone link at a time. Therefore. we show this asymptotic resultin total power consumption figures. The relative performances

-

Fig. 6.1 ssssions and mnimum-hop routing.Tail1 transmit power averaged over the jointly feasible scenarios for

Fig. 1. Ratio of jointly fsasxblescenmos for 1 sessions and minimum-powerrouting.

have similar tendencies as in the minimum-hop routing exceptfor the following points: (1) Using pow er as an explicit factorin link costs fo r routing protocols significantly ameliorates theoverall power consump tion. (2 ) Higher number of active linksforces the system to use longer frame lengths to satisfy th esession requirements. Thus. reducing the power consumptionin the routing layer often fails to satisfy the session QoSrequirements even at moderate frame lengths.Figures 9 an d 10 give mnre insights when the networkload is increased by changing the number of sessions from7 o 15. The relative performances remain same with widerperformance gaps and the nominal values of the operatingpoints get worse both in terms of the required frame length tosatisfy the session requirements in ma,jority of the scenariosand the settled down total power consumption.

This co ncludes our main results- 'which make a strong. 0-7803-8355-9/04/S20.00Z W EEX. 1454


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Fig. 8 .7 SCSSIORS and mirumurn-power routing.Total transmit power aviraeed over Ihr jointly feasible scenarios for

Fig. 9.routing. Ratio ul jointly f s s s ihk scenarios for 15 sessions and minimum-hop

connection to our arguments a discussed in detail in theintroduction of th is paper. We now brieHy summa rize and drawthe future direction of onr w ork in the next sec tion.

VII. S t iM l M A R Y A N D F U T U R E D I R E C T I O NWe considered the problem of energy-efficient communi-

cation in wireless multi-hop networks with the objective ofproviding the end-to-end Qo S guarantees to a set of sessions.We have started with first formulating a QoS framework thatis able to capture both the different detinitions of QoS fromnetwork layer to physical layer and the general requirementsof the individual sessions. We stated the close inter-actionbetween these layers and pointed out the fact that independentdecisions on different layers for achieving a local objectivewould deteriorate the performance of other layers leading to ahi lure in achieving the main goal. With an open view towardsincluding more layers and system parameters into the picture,

.. . . . . . . . . . . .. .. . . . .O0 ,'

1" $ 5 NBO Frmalsnghl ("Yrn 01dois)

Fig. IO .fur 15 sessions and minimum-hop rouling.

Total transmit power avzoged over the jointly feasible scenarios

our focus has been on addressing the joint power control andscheduling problem. By introducing the notion of virtual linksand assuming a one-to-one mapping between BER and SINRrequirements for each wireless transmission, we decoupled thisjoint optimization problem from the underlying session basedrequirements. We proved the NP-completeness of the problemand provided the performance bounds for justifying the laterproposed heuristic solutions.

Our comparison of proposed heuristics has revealed ihefollowing observations: ( I ) A top-down design strategy such asfirst solving the feas ibility problem. then m inimizing the powerconsumption performs better in terms oE lhe objective function.This is contrary to the general expectation that i t should alsoperform better to find a feasible solution. (2) Water-fillingargument outperforms the topdown design strategy in findinga feasible solution provided that a proper cost function isdefined. In this respect. the upper-bound expression as foundin (19) serves well as a cost function. ( 3 )Routing layer plays adominant role in reducing power consumption. bui i t happensat the expense of QoS provisioning.

It is worth to mention that we have not distinguished theorder of virtual links within a frame neither in our problemdefinition nor in our proposed solutions. This assumes thatevery link has a packet to transmit in its own turn. However.session pdckeu follow their routing paths and the links closerto the source point must he scheduled before the ones thatare further away to guarantee that there is actually a packetto transmit.' Having noted that. it is not hard to modify ourproblem's constraint set as well as our solutions to reflect thisrequirement. The following rule must be applied to narrowdown th e feasibility constraints in the problem statement andthe search space in the heuristics:

'This assumes that 110 packets are lost due to channel cmors. which is acommon problem for al l reservation h s c d schemes. In such situatiou. thel i n k may use the slots for ah er traffic without violating the BE R requirementsof other transmissions or the node can tamit into the power save mode.

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" A imial link I tho1 is 1aDellrd hv ( s i 1 1 ) -d i e r e s is th esession .sorrr(:e node and I?2 0 is rhe hop distance of T ( / )frorri this soiircp point- can he assigned to slot k i fant l onlythe toral nrriubrr of~drtiral inks rliat ai-e labelled O?. ( s ; - 1)f r o m slor 1 to ( k- 1) is irlore than the total nwnber ofvirtitallinks that or e IaOelled b? ( s i h)."

This work is one of the initial steps for a proper treatment ofcross-layer design in multi-hop wireless networks. The follow-ing research directions are not partially or fully addressed i nthis paper and remain as the major problems to be investigatedin future works:

a) Our algorithms are cenualized in the sense that theyare executed by a central agent that has global networkknowledge. This definitely limits the application towards adhoc networks that have some level of infrasuucture support.An interesting issue would be to devise partially or fullydistributed algorithms based on only local node information.Such algorithms would be executed independently at eachnode. yet the transmission schedules and transmit powersshould converge to an optimal or near-optimal solutions.b) A more general case involves the issue of routing as well.Then. the routes of each session from source to destinationare not given a priori to the central controller. but insteadthey need to be identified. In that case. the problem alsoinvolves an interaction with the network layer. At a firststage. we can assume the existence of several alternativeroutes for each session. which are provided by the routingprotocol. The routing decision then determines the route thateach session will follow. A meaninpful objective would beto consider those routes whose paths are maxim ally disjoint.so that corresponding links are relatively far from each otherand d o not mu ch in terfere with each other. With this rule.scheduling decisions are also facilitated, in the sense that co-chann el link sets will be formed wh ere links are in ZeneraJ farfrom each other. Then, smaller amounts of transmission poweris consumed to serve all links. The m ost general (and difficultto treat) is the case when the routes are completely unspecified.except for the s o w e and the destination of each session.Hence. a cross-layer mechanism that jointly performs routing,scheduling an d power control should be designed. Routes fromeach source to each des unation can be found either proactively(route s ready to use. which is more suitable fo r our centralizedframework) or reactively (routes on demand).

c) Anycasting services. i n which we have a freedom ofselecting the destination point for each source node amonga se t of equivalent nodes. will possibly find a vast classof applications especially in overlaid hd hoc networks. Forinstance. consider the cases where the primary goal of ad hoc. nodes is to reach an infrastructure network that supplies a se tof access poin ts. Even when the routes are fixed. the selectionof access point plays a dominant role to determine the set oflinks to be scheduled. altering the attainable performance.

d ) Finding the performance limits of wireless multi-hopnetworks that constitute a finite field is still an open problem.It is also equally important to find the performance gaps ot

the heuristic solutions with these limits.R E F E R E N C E S

[ I ] Anthony Ephrenides. "Enwpy concerns in wireless networks." IEEEWrrlrss Cornninnicnfions. p.48-59. August 2002.[?I T. Elba11 and A. Ephrrmides. "Joint Scheduling an d Power Control forWireless Ad Hoc Naworks:' in Pruc. l?foco,r~-2OOZ.un c 2002.131 J. Zmdzr. "Distributed co-channel intitiereme control in cellular radiosysten~5,"IEEETnoas.W L r c h r d . vol. 41, no. 3. pp. 305-31 1 August1992.141 1.Zander. "Psrformanccof optimum transmitter power control in cellularradio systcms:' IEEE Tram. Wt . Trclznol.. vol. 11,pp. 5762. F ehm ary1992.

151 S.A. Gmndhi. R. Vijayan. an d D.J. Goodman. "Disfribulcd algorithm forpowar control in cellular radio systems:' in Pm r 30th .bmrml .AllenonCO@Cornm.. Co,%ml C o n p l i n g , Urbmin Illinoir. Septzmhzr 1991.

[6] S.A. Grandhi. R. Vijaya". D.J. Goodmn. an d I. Zander. "Ccnrraliaedpower ~o n l r o l n cellular radio systenx," IEEE Tmns. Iph. Teclmol..vol. 42 . pp. 46M68. No v cm kr 1991.

I71 G.J. Foschini and Z. Miljanic. "A simple distributed autonomous pow ercontrol algorithm and its convergdnce:' IEEE Trms. veb. Teciraol..vol. 42. no. 1.pp. 6 4 4 4 6 . November 1993.[a ] R.D. Wtrs. "A framzwork for uplink power control an d banc stationmsignmcnt," IEEE J. Selecl. .Areas Comnrrm.. vol. 13. pp. 1341-1347.Scptcmkr 1995.

191 N. Bamhos. S . C h r n a nd G . Poltie. "Radio Link Admission Algorithmsfm Wireless Networks with Power Control and Activz Link Qualityhotsflion." in Pmc. Infocorn-1W.i. 1995. pp . 97-104.

[IO1 C. Huanp and R.D. Yates, "Rate of cuowrgcnce for minimum powervssignmant algorithm in cellular radio systems." Wireless Networks.vol. 4, pp. 223-231. 1998.

[ I l l Sznnur Lilukus and R.D. Yatcs. "Stochastic Power Conuol fo r C 4 u l a rRadio System." IEEE Trolrs.on Conirnnn..vol. 46. no. 6. pp. 7Xt79X.Junc 1998.1121 S. H a d ? and D. Tse. '"Power ~001roland capcity of sprzad-spsctrumwireless networks." .A~sornoti~o.ol. 35. pp. 1987-2012. 1999.1131 Frcdrik Bzrggzn. Ssong-Lym Kim. and Jdns Zander. "Joint PowerConuol an d lntraccll Schsduling of DS-CDMA Nonreal Time Data:'IEEE J.Y.4C. vol. 19. no. IO . pp. 1860-1870. Vclokr 2001.1141 R.L. Cm r and A.V. Santhanam. "Optimal Rwting. LinkSchcdulinp andPowsr Control in Multi-hop Wirzless Nztworks," in Pmc. I~f'corn-2003,April 2003.I151 -. "Optimal Link Schcdulins and Power Control in CD~MAMultihopWireless ?letworks." in Proc. Clokcum-1W2. November 2002.1161 R. Wattcnhofcr. Li Li. Paramvir Bahl. an d Yi-Min Wag. "Dist"butedTopology Control for Power Efficient G prar ion in Multihop WirelessAd Hoc Network.,'' in Pmc. Infomrn-2001. 2001.1171 R . R n m a n a h n an d R. Rosales-Hain. "Topology Control of Multi-ho p Wireless Networks using Transmit Power Adjustment." in Pmc.1181 I. Monks. V. Barp_havan and W. wu. "A Powm Controlled MultipleAccsss Protocol for Wireless Packet Networks," in Pmc. In/ocom-2WI.A p d 2001.1191 V. R al op lu an d Tuzsa M m p . "Minimum energy mobile wirclcssnetworks." IEEE LWC. vol. 17. no. 8. pp. 1333-1344. 1999.1201 S. Singh. M. W m nd C.S. Raghaveendm. "Power-awarz r wt in g in mo-bilz ad ho c networks:' in Pmc. of Foirnlr .Annrrol,4CWIEEE Conferenceon Mobile Cmiptrting nnd N ~ t w or k i n g . ctohrr 1998, pp. 181-190.[?I]. P. Bcrtschs . Nonlirlrnr Pmgmmming. Athcnn Scientific. 1995.1221 A.J. Goldsmith an d S.-G. Chua. "Va"able-rate variable-p ower M QA Mfo r fading channels." IEEE Tm m on Cornmut,.. vol. 45. no. IO. pp .1218-1210. October 1997.1231 Michael R. Garcy and Dawd S. ohnson. C o n ~ p u t mnd lnrmctoDiIi~.-lG s d e lo b eWru+ ofNP-Cornplererress. W.H. reeman an d Company.1979.

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