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IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 25, NO. 4, MAY 2007657Cross-layer QoS Analysis of OpportunisticOFDM-TDMA and OFDMA NetworksYu-Jung Chang, Feng-Tsun Chien, Member, IEEE, and C.-C. Jay Kuo, Fellow, IEEEAbstract— Performance analysis of multiuser orthogonal frequency division multiplexing (OFDM-TDMA) and orthogonalfrequency division multiple access (OFDMA) networks in supportof multimedia transmission is conducted in this work. We take across-layer approach and analyze several quality-of-service (QoS)measures that include the bit rate and the bit error rate (BER)in the physical layer, and packet average throughput/delay andpacket maximum delay in the link layer. We adopt a cross-layerQoS framework similar to that in IEEE 802.16, where serviceclassification, flow control and opportunistic scheduling with different subcarrier/bit allocation schemes are implemented. In theanalysis, the Rayleigh fading channel in the link layer is modeledby a finite-state Markov chain, and the channel state information(CSI) is assumed to be available at the base station. With theM/G/1 queueing model and flow control results, our analysisprovides important insights into the performance difference ofthese two multiaccess systems. The derived analytical results areverified by extensive computer simulation. It is demonstratedby analysis and simulation that OFDMA outperforms OFDMTDMA in QoS metrics of interest. Thus, OFDMA has higherpotential than OFDM-TDMA in supporting multimedia services.Index Terms— quality of services (QoS), multiple access,OFDM, OFDMA, cross-layer analysis, opportunistic scheduling.I. I NTRODUCTIONMULTIMEDIA delivery is one of the key objectives ofnext-generation wireless networks. Its success relies onhow the underlying network can support different QoS requirements demanded by a variety of multimedia applications. Asignificant challenge is posted since multimedia applicationshave very diverse characteristics in terms of physical measuressuch as bandwidth and delay. Furthermore, it is desirable thatthe underlying network can serve multiple users and meettheir individual QoS requirement. All of these call for a QoSprovisioning broadband network in conjunction with propermultiple access schemes such as Time Division MultipleAccess (TDMA) and Frequency Division Multiple Access(FDMA). Recently, OFDM-based networks in combinationwith TDMA and FDMA have become a popular choice forManuscript received May 15, 2006; revised November 25, 2006. Thisresearch was funded by the Integrated Media Systems Center, a NationalScience Foundation Engineering Research Center, Cooperative agreement No.EEC-9529152. Any opinions, findings and conclusions or recommendationsexpressed in this material are those of the author(s) and do not necessarilyreflect those of the National Science Foundation.Yu-Jung Chang and C.-C. Jay Kuo are with the Integrated Media SystemsCenter and the Department of Electrical Engineering, University of SouthernCalifornia, Los Angeles, CA 90089-2564, USA (e-mails: yujungc@usc.eduand Chien is with the Department of Electronics Engineering, National Chiao Tung University, Hsinchu, Taiwan, R.O.C. ( Object Identifier 10.1109/JSAC.2007.070503.such an endeavor. The IEEE 802.16 standard, for instance, hasadopted OFDM-TDMA and OFDMA (OFDM-FDMA) as twotransmission schemes at the 2–11 GHz band [1]. In addition, aQoS framework in the medium access control (MAC) layer hasalso been integrated with the multiaccess transmission systemsin the IEEE 802.16 standard [2].Multiuser diversity provided by opportunistic scheduling[3] has been incorporated in multiuser OFDM-TDMA andOFDMA networks recently. Its effect on QoS-provisioningis an interesting yet challenging topic. On one hand, byallocating resources to users with better channel quality, theopportunistic scheduling scheme can maximize the overallsystem throughput [3]. On the other hand, it may degradeother QoS metrics such as delay, since users are suspendedfrom transmission when their channels are poor. The impact ofopportunistic scheduling on QoS provisioning is investigatedin the course of comparison of OFDM-TDMA and OFDMAin this work. The bit error rate (BER) performance of OFDMTDMA and OFDMA with multiuser diversity has been studiedin previous work, e.g., [4], [5]. Specifically, uncoded andcoded systems with opportunistic OFDMA were shown tooutperform those with static OFDM-TDMA by 3 dB and7 dB at BER 10 3 in [4] and [5], respectively. TheBER performance of OFDM-TDMA and OFDMA was alsocompared in [6] without considering multiuser diversity.While the BER analysis can be used to characterize thephysical layer performance, it is not sufficient to reflectother QoS metrics such as packet throughput and delay inthe link layer. The fact that QoS requirements should betreated differently in different layers suggests a cross-layerapproach for QoS provisioning and analysis. In fact, the crosslayer approach has been applied to the design and analysisof QoS-featured multiaccess systems by a few researchersrecently. For example, the analysis of queueing delay for802.16 networks was conducted in [7], [8] by combining linklayer queueing with physical-layer transmission. A vacationqueueing model was adopted in [9] to analyze the link-layerqueueing performance of OFDM-TDMA systems with roundrobin scheduling. A queueing model for OFDMA systemswas used in [10] to design a scheduling scheme that balancesmultiuser diversity and queueing delay.Although the packet-level analysis has been conducted forOFDM-TDMA or OFDMA in [7]–[10], there are a few openissues to be addressed. We discuss these issues and pointout our contributions below. First, an analytical frameworkto account for both OFDM-TDMA and OFDMA systems tofacilitate their comparison is missing. By generalizing resultsin [11], [12], we propose a framework to achieve this goalhere. Second, performance evaluation of 802.16 has beenc 2007 IEEE0733-8716/07/ 25.00

IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 25, NO. 4, MAY 2007Link/PHYLink (MAC)PremiumPriorityschedulingBest effort.Multiple accessandbit allocationMobile k.PremiumFlow Control(a)Net/LinkPacket ClassifierMobile KAppApplicationconducted primarily by simulation in the past, e.g., [13],[14]. We conduct an analysis to demonstrate that OFDMAoutperforms OFDM-TDMA in terms of several QoS metrics.This is consistent with the trend of the latest IEEE Standard802.16e-2005 [15], which adopts OFDMA as its principalmultiaccess scheme. Third, although packet average delay andmaximum delay are useful link-layer performance measuresfor non-real-time (e.g., file transfer and web browsing) andreal-time (e.g., voice and video) applications, respectively,most previous work has focused on the packet average delay.The packet maximum delay and the delay violation probabilitywill be examined in this work. Finally, the performance oftwo well-known scheduling strategies, namely, the round-robinscheduling and the opportunistic scheduling, is examined forOFDM-TDMA and OFDMA networks so as to understandtheir pros and cons in the context of QoS provisioning.Our approach to physical and link layer analysis is simplystated as follows. The ideal channel state information (CSI)is assumed to be available at the base station. This is oftenachieved by feeding the estimated channel information fromthe receiver back to the transmitter through a control channel.The Rayleigh fading channel is modeled by a finite-stateMarkov chain [16] to translate the effect of the physical layerto higher layers. Specifically, the channel effect is manifestedin the link layer as a time-varying server with the M/G/1queueing model for the packet throughput-delay analysis. Toanalyze the packet maximum delay, we adopt and implementa well-known flow control scheme [17] as part of the proposed QoS framework. With such a flow control scheme,we can derive delay bounds based on network calculus results for different scheduling and rate adaptation schemes.All the aforementioned performance metrics provide valuablemeasures in differentiating OFDM-TDMA and OFDMA intheir QoS-provision performance. It is finally concluded thatOFDMA has a higher potential in meeting the requirementsof multimedia delivery.The rest of this paper is organized as follows. The systemmodel, which consists of the QoS-aware framework, the flowcontrol regulation scheme, multiple access and resource allocation schemes, is discussed in Sec. II. The physical and linklayer analysis for OFDM-TDMA and OFDMA is presented inSec. III. Simulation results are shown and discussed in Sec. IV.Finally, concluding remarks are given in Sec. V.Best effortMobile 1(b)Flow Control658X k1(t) (r k1 , w k1)PremiumPriorityschedulinguk (t) (u k, v k)X k2(t) (r k2 , w k2)Best effortFig. 1. (a) A cross-layer QoS-support system model and (b) a queueingsystem with flow-control regulated streams and preemptive priority servicing,shown for mobile user k.framework is a simplification of those used in DiffServ [18]and the 802.16 MAC protocol [2].A. Flow Control RegulationA flow control regulator [17] is adopted to process real-timemultimedia data so as to keep the delay bound and arrivalconstraints [19]. As depicted in Fig. 1(b), Xk1 (t) and Xk2 (t)are the flow-control regulated premium and best-effort streamsof user k, respectively. We write Xk1 (t) (rk1 , wk1 ) if, forany t1 t2 , t2Xk1 (t)dt rk1 (t2 t1 ) wk1 ,(1)t1where rk1 is the predefined average rate of the stream and wk1is the allowed burst degree. In other words, the input streamhas to be regulated so that the output stream, Xk1 (t), can meetthe imposed rate and burstiness constraints. Likewise, we haveXk2 (t) (rk2 , wk2 ).Suppose that the time-varying server process, uk (t), inFig. 1(b) conforms to a similar but slightly different constraint.That is, we write uk (t) (ūk , vk ) if, for any t1 t2 , t2uk (t)dt ūk (t2 t1 ) vk ,(2)t1II. S YSTEM M ODELThe proposed cross-layer QoS framework is shown inFig. 1(a), where QoS provision is achieved by packet categorization and service differentiation. Specifically, packets areclassified into the premium and the best-effort two classes andbetter service is granted to the premium class. Particularly, adelay-sensitive service is offered to the premium class by thepreemptive priority scheduling mechanism, where premiumpackets maintain a higher priority for processing in ourframework. The reason to choose delay as a main performance metric for service differentiation is that the delivery ofmany multimedia applications are delay-sensitive. As shownin Fig. 1(b), a flow control scheme is adopted on top ofservice differentiation to regulate the end-to-end packet delayof the premium and the best-effort classes. The proposed QoSwhere vk is the service lag and ūk is the average service ratedefined by 1 tuk (τ )dτ,(3)ūk limt t 0with probability one. We will show in Sec. III-C that the serverprocess uk (t), which is prescribed by actual multiaccess andscheduling schemes, satisfies (2) asymptotically, and parameters ūk and vk can be derived analytically.B. Multiple Access and SchedulingThe multiple access scheme in Fig. 1(a) is accomplishedby OFDMA or OFDM-TDMA along with subcarrier/time slotassignment and bit allocation mechanisms. Several differentschemes to be examined are summarized in Table I. Note that

CHANG et al.: CROSS-LAYER QOS ANALYSIS OF OPPORTUNISTIC OFDM-TDMA AND OFDMA NETWORKSTABLE IM ULTIACCESS OFDM MODES CONSIDERED IN THIS WORKOFDMA ModeOFDMA IOFDMA IIOFDMA IIIOFDM-TDMA ModeOFDM IOFDM IIOFDM IIISubcarrier AssignmentstaticstaticdynamicTime-slot AssignmentstaticstaticdynamicBit Allocationfixedadaptive modulationadaptive modulationBit Allocationfixedadaptive modulationadaptive modulationmodulation for each user. We consider squared M -QAM modulations with M 22r , r 1, · · · , rm , where rm determinesthe highest modulation allowed. Both fixed and (discrete-rate)adaptive modulation (AM) methods are considered, whereonly the AM methods take channel conditions into account inadaptive bit allocation [21]. The AM methods are describedbelow. A tight BER approximation for squared M -QAM isgiven by [21]:Pb 0.2 exp( we use OFDM time slot and OFDM symbol interchangeablyin this paper.OFDMA and OFDM-TDMA perform multiple access ina frequency-sharing and a time-sharing manner, respectively.Specifically, OFDMA performs subcarrier assignment whileOFDM-TDMA performs time-slot assignment, both staticallyor dynamically. The difference between static (or round-robin)and dynamic (or opportunistic) assignments lies in whetherusers’ channel conditions are considered. For OFDMA, staticassignment allocates an equal, fixed and interleaved set ofsubcarriers to users while dynamic assignment allocates eachsubcarrier to the user with the best signal-to-noise ratio (SNR).Likewise, for OFDM-TDMA, static assignment allocates fixedand alternate time slots to users (i.e., round robin) whereasdynamic assignment assigns a time slot to the user with thebest channel condition. Besides, the chosen user is allocatedall subcarriers exclusively in OFDM-TDMA.The subcarrier SNR distribution for each multiaccess modecan be obtained as a basis for analysis in Sec. III. For aRayleigh fading channel, the received SNR, denoted by Γ,is exponentially distributed with the following probabilitydensity function (pdf) [16]:1γexp( ), γ 0,(4)gΓ (γ) γ0γ0where γ0 is the average SNR. Note that Eq. (4) holds forsubcarrier SNR in a multicarrier system as well [20]. Eq.(4) applies to all modes in Table I except for OFDMA IIIup to a difference in mean γ0 which will be verified bysimulation. In OFDMA III, recall that the dynamic assignment scheme assigns a subcarrier to the user with the bestSNR at that subcarrier. Thus, when K homogeneous usersare considered, the post-assignment subcarrier SNR, Γ , isdistributed according to the maximum of K independent andidentically distributed (i.i.d.) random variables Γ1 , · · · , ΓK ,which represent the received SNR of users 1, · · · , K andfollow the pdf in (4). To derive the pdf of Γ , we first obtainthe cumulative distribution function (cdf):P {Γ γ} P {max(Γ1 , · · · , ΓK ) γ}γγ 0. (1 exp( ))K ,γ0Then, by differentiating the cdf, we haveKγγexp( )(1 exp( ))K 1 ,γ 0. (5)gΓ (γ) γ0γ0γ0C. Bit AllocationThe multiaccess scheme apportions resource among userswhile the bit allocation scheme chooses the type and order of6593β),2(M 1)(6)where β is the channel SNR. With continuous-rate adaptation,bit rate Rbc is given by the following capacity expression:Rbc log2 M log2 (1 1.5β). ln 5Pb(7)Note that (7) is obtained directly by the rearrangement of (6).Discrete-rate adaptation confines the bit rate to integer values (more precisely to 2r, r 0, · · · , rm ), which is describedas follows. First, the set of possible received SNR (i.e., thenonnegative real line) is partitioned into rm 1 disjointregions R0 , · · · , Rrm by boundary points b0 , b1 , · · · , brm 1 ,where Rr is the interval [br , br 1 ) for r 0, 1, · · · , rm andb0 b1 · · · brm 1 with b0 and brm 1 set to 0 and ,respectively. Second, the boundary points are determined by2br (ln 5Pb )(22r 1),3r 1, 2, · · · , rm .(8)Last, when the received SNR falls in Rr and the informationis successfully fed back to the transmitter, 2r bits are loaded tothe corresponding subcarrier. Note that the channel is too poorto support any order of modulation when the SNR falls in R0 .This leads to the following bit rate expression of discrete-rateadaptation: 1.52 21 log2 (1 ln5Pb β) , if β brm ,(9)Rbd 2rm ,if β brm ,where x represents the largest integer that is less than orequal to x. Note that for any Pb and β, Rbd Rbc .III. P ERFORMANCE A NALYSIS AND C OMPARISONMultiaccess schemes in Table I are analyzed in this sectionbased on the system model introduced in Sec. II. The physicallayer performance is considered in Sec. III-A while the linklayer performance metrics are examined in Secs. III-B andIII-C.A. Bit Rate and BER AnalysisFor the bit rate and BER analysis, we focus on AM-basedmodes (e.g., OFDMA II–III, OFDM II–III) since non-AMmodes are trivial special cases. The theoretical bit rate upperbounds are derived under the assumption of continuous-rateadaptation. However, such bounds also hold for the discreterate adaptation case.For fixed Pb and β, the bit rate per subcarrier withcontinuous-rate adaptation is given in (7). With the distribution

660IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 25, NO. 4, MAY 2007Serialof β shown in (4) for OFDMA II and OFDM II–III, the bitrate corresponding to these modes is bounded byRb1 EΓ [log2 M (Γ)] log2 EΓ [M (Γ)] log2M (γ)gΓ (γ)dγ0 log2 (1 S/P1.5γ0 ). ln 5PbP/SN1.5 log2 (1 γ0 ).Ts ln 5PbK N1.51 log2 1 γ0 () .Ts ln (11)Similarly, the total bit rate for OFDMA III, Rt2 , is obtainedby replacing Γ by Γ in (10) and using (5). That is, we haveRt2Data from linklayersSerial(10)The total bit rate Rt1 (in the unit of bits/sec) can be obtainedby scaling with the total number of subcarriers N and OFDMsymbol time Ts asRt1Parallel(12)k 1For the BER analysis, note that the AM scheme describedin Sec. II-C will lead to comparable BER performance for allOFDMA and OFDM modes due to the predetermined Pb inplace. A more informative comparison in BER can be achievedby fixing a target bit rate for all modes. Towards this end,we adopt a method based on [22] to add and subtract bitsfrom proper subcarriers successively until the target bit rateis achieved. That is, when the actual bit rate is smaller (orlarger) than the target bit rate, additional bits are added to (orsubtracted from) subcarriers such that the error probability isincreased as small as possible (or decreased as far as possible).In other words, additional bits are successively added to (orsubtracted from) the subcarrier where the difference betweencapacity (Rbc in (7)) and assigned discrete-rate bits (Rbd in (9))is maximal (or minimal).The equivalence between “maximizing the difference between capacity and the assigned bits” and “minimizing thebit error rate” is intuitive and it can be formally proved.The proof is however omitted here due to the space limit.This operation to increase (or decrease) the BER when bitsare added (or subtracted) will be confirmed by simulationin Sec. IV. Furthermore, the proposed bit rate adjustmentmethod may not be the best solution due to its computationalcomplexity. The main purpose for us to adopt this methodis to demonstrate a more meaningful BER comparison byfixing the bit rate. In a realistic system design, if no targetbit rate is established as a requirement, such a manipulationis unnecessary.B. Packet Average Throughput and Delay AnalysisThe packet-level average throughput and delay performanceis analyzed in this subsection. By delay, we refer exclusively toqueueing delay. To complete delay analysis, a proper queueingmodel for both OFDM-TDMA and OFDMA is needed, andthe M/G/1 model [23] is adopted for this purpose. We simplifyFig. 1(a) by including the premium and the best effort in onecomposite queue. This simplification does not compromise ourcomparison. In fact, the M/G/1 results as well as the analysisgiven here are readily extensible to priority queues [23]. ToFig. 2.:Symboldetection:FFT:S/PGIremovalA typical OFDM transmission the M/G/1 model, the Poisson packet arrival process isassumed to replace the regulated streams as the input to thequeue.The average delay W̄ in the M/G/1 model is given by [23]:W̄ λE[S 2 ],2(1 λE[S])(13)where S is the packet service time and λ is the Poisson packetarrival rate. Note that in the case of infinite-sized queues asassumed in this work, the throughput is proportional to theλ value. Therefore, we only need to determine S up to thesecond moment to complete the throughput-delay relation in(13).Since a packet is served by subcarrier allocation inOFDM/OFDMA systems, we consider the service time (andthus delay) being measured in the unit of the number ofsubcarriers. To be more specific, S is defined to be the numberof subcarriers Ns satisfying Ns S Ns : U Ii α ,i 1where U is the number of bits loaded in Ns subcarriers,Ii is the number of bits loaded to subcarrier i, which isidentically (but not necessarily independently) distributed overall i, and α is the fixed packet size in bits. The idea can beexplained by a typical OFDM transmission system shown inFig. 2, where data streams can be grouped in a parallel or aserial manner. The “time unit” in serial and parallel groupingsare samples and subcarriers, respectively. With the serial-toparallel (S/P) and the parallel-to-serial (P/S) convertions, theconventional notion of delay in the serial grouping translatesto the equivalent delay measured in the number of subcarriersin the parallel grouping, and vice versa. This fits both OFDMTDMA and OFDMA since we are concerned with the averagedelay over all users and packets.We evaluate the first two moments of S in the following.For the first moment, we have Ns Ii UE[U U ] Ei 1 Ns E EIi Ns , UUi 1 E Ns · E [Ii U ] U E[Ii U ] · E[Ns U ].Conditioning on U α, we have S Ns by definition. It

CHANG et al.: CROSS-LAYER QOS ANALYSIS OF OPPORTUNISTIC OFDM-TDMA AND OFDMA NETWORKSfollows thatE[S] α.E[Ii U α](14)The second moment can be derived similarly. That is, Ns 2E[U U ] E (Ii )2 U Ns Ii )2 Ns , UE E ( U i 1E Ns · E[Ii2 U ] E[Ii Ij U ] U (.15)i jThe cross-term in (15) represents the cross-correlation amongsubcarriers, which depends on the coherence bandwidth of thechannel. Without further knowledge or assumption about thechannel, we may proceed in deriving a lower bound. By theSchwartz inequality and the fact that Ii and Ij are nonnegative,we have E[Ii Ij ] E[Ii2 ]E[Ij2 ] E[Ii2 ], i j,and consequently, E[Ii Ij U ] Ns (Ns 1)E[Ii2 U ].(16)i jBy substituting (16) into (15), we obtainE[U 2 U ] E[Ii2 U ] · E[Ns2 U ].Again, conditioning on U α, we have S Ns so thatE[S 2 ] α2.E[Ii2 U α](17)To complete the analysis in (14) and (17), we need to obtainthe first two moments of Ii , i.e. the number of bits loaded tosubcarrier i. Apparently, Ii depends on the AM scheme in(9), the channel condition given in (4) or (5) and the selectedmultiaccess mode. To proceed, we model the channel state inthe link layer by a finite-state Markov chain, where each statecorresponds to a certain channel condition (R0 , · · · , Rrm ) anda particular modulation scheme. The state “0” corresponds toR0 where no bits are transmitted. The transition probability,which was derived before (e.g., [16]), is not of our concernhere. Our interest is to calculate the steady-state probabilitiesπr ’s, which can be obtained by integrating pdfs in (4) or (5)over disjoint regions Rr ’s. That is, br 1 br 1πr gΓ (γ)dγ orgΓ (γ)dγ,(18)brbrwhere r 0, 1, · · · , rm . With these πr ’s, the first and secondmoments of Ii can be obtained byE[Ii ] E[Ii2 ] rm r 0rm πr · (2r),πr · (2r)2 .and then (14) and (17) in (13), we obtain the theoreticalthroughput-delay lower bound curves. This result will beverified by computer simulation in Sec. IV.C. Packet Maximum Delay Analysisi 1 661(19)r 0Note that the condition on U α does not change themoments of Ii . Finally, by substituting (19) in (14) and (17),We obtain analytical packet delay bounds for all modeslisted in Table I in this subsection. A result from networkcalculus is described below, which will be used in analysislater. The delay for the premium stream in Fig. 1(b) is boundedby [17]wk1 vk,(20)dpm ūkif queues are stable (i.e., rk1 rk2 ūk ) and the first-infirst-out (FIFO) service strategy is employed for each queue.Note that only the delay performance of the premium streamis of our interest. Also, the delay bound in (20) refers to delayexperienced by the regulated streams. The waiting time insidethe flow control regulator as analyzed in [24] is not of ourconcern.Recall that the definition in Sec. II-A is tied with continuousprocesses. However, the slotted structure of OFDM/OFDMAsuggests that the server process be represented by a discretetime random process uk [n], where n is the OFDM time slotindex. This change turns all integrations into summations inSec. II-A while leaving flow control parameters unchangedbut in different units. That is, wk1 , wk2 and vk are in theunit of bits and rates rk1 , rk2 and ūk are in the unit ofbits per time slot. Then, the delay bound in (20) is in theunit of the number of OFDM time slots. Note that thereis no discrepancy between the units used in this subsectionand those in Sec. III-B, where the number of subcarriers isused, since the number of subcarriers can be translated toan equivalent number of OFDM symbols, and vice versa.The most remarkable difference is that, instead of averagingover all users and packets in presenting the average delayresults in Sec. III-B, we must treat each packet and each userindividually here since the measure of delay bounds is not anaverage.In the following, we obtain deterministic delay bounds forstatic modes such as OFDMA I and OFDM I, and thenderive probabilistic delay bounds for the remaining modesthat use dynamic allocation. Note that a smaller delay boundguarantees better worst-case delay performance.1) OFDMA I and OFDM I: These two modes employstatic multiple access and bit allocation yet in a differentfashion. Recall from Fig. 1 that the server process uk (t)is prescribed by actual multiaccess and bit allocation, thusresulting different server processes of OFDMA I and OFDMI as depicted in Fig. 3 for the continuous-time representationand Fig. 4 for the discrete-time representation. From thesefigures, we observe that the server rate of OFDMA I is aconstant, since subcarriers are divided evenly among users ineach OFDM time slot. In contrast, in OFDM I, user k’s serverrate peaks at time slots when user k is in service and remainszero during periods in which other users are served.To obtain the delay bound in (20), our task is to find thepair (ūk , vk ) that defines the server process (wk1 is controlledand known in the design of a flow-control regulator). Let N

662IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 25, NO. 4, MAY 2007server processuk [n]server processuk (t)(b)(a).0Ts2Tst1t 2'.KTs (K 1)Ts.2KTs.time t1.2KK 1.2K 2K 1 .OFDMsymbol nt2Fig. 3. Illustration of the continuous-time server process for user k with atotal of K users: (a) OFDMA I and (b) OFDM the total number of subcarriers and r the (fixed) number ofbits loaded to each subcarrier. Therefore, in each OFDM timeslot N r bits are served. OFDMA I has a constant serverrate, i.e.,uk [n] ūk N r/K and vk 0.Consequently, we havewk1. (OFDMA I)(21)ūkFor OFDM I, the average server rate ūk is the same asthat in OFDMA I according to (3). To obtain vk , we needto examine (2) carefully. First, we observe that a sufficientlylarge vk in the RHS will make (2) always hold since the LHSof (2) is non-negative. However, an arbitrarily large value ofvk is not useful in deriving the delay bound in (20). Thus, wewant to find a smallest vk for which (2) is satisfied for anychoice of t1 and t2 . As depicted in Fig. 3, our choice of t1and t2 is the pair that spans the widest among any t1 and t2values that enclose a “peak-rate” time slot of OFDM. Sincet2 t1 2K 1 is the largest in this case, the associated vkwill guarantee that (2) holds also for any other choices of t1and t2 . With such t1 and t2 and (2), we havedpm N r ūk (2K 1) vk .By arranging the terms and the fact that N r K ūk , wehave the smallest vk asvk ūk (K 1).Substituting ūk and vk into (20) yieldswk1 (K 1),(OFDM I)dpm ūk(22)(23)where ūk N r/K. By comparing (21) and (23), we seethat OFDMA I has a smaller delay bound than OFDM I bya fixed amount of K 1. The physical interpretation of thisresult is that the K 1 idle time slots in OFDM I account forthe K 1 extra delay bound.2) OFDMA II and OFDMA III: To obtain delay boundsfor OFDMA III and, as a special case, OFDMA II, we maketwo assumptions in the asymptotic analysis. The number ofsubcarriers, N , is large, and the subcarrier channel coefficientsare i.i.d. These may be regarded as an ideal approximation toreal-world situations. However, as will be demonstrated bysimulation in Sec. IV, the theoretically derived delay boundsFig. 4. Illustration of the discrete-time server process for user k with a totalof K users: (o) OFDMA I and (x) OFDM I.coincide well with experimental results in both the i.i.d. andthe real-world channel setups. Thus, our analysis does provideinsights into performance differences among different modes.OFDMA III, unlike OFDMA I, has a time-varying serverprocess due to the fluctuation of user channels. Specifically,the server process uk [n] is a discrete-time random processdefined by uk [n] Ii [n],(24)i Dk [n]where Ii [n] is the number of bits loaded onto subcarrieri, and Dk [n] is the set of subcarriers assigned to user k,both at OFDM time slot n. Ii [n], i Dk [n], are i.i.d. byour aforementioned assumption. Besides, due to opportunisticselection of users, the probability of each user “winning” aparticular subcarrier is 1/K, which leads to the binomiallydistributed Dk [n] , denoted by Dk [n] B(N, 1/K), where x is the cardinality of set x. Given uk [n] as the summation of Dk [n] i.i.d. random variables, along with the assumption ofbig N , we have (as N and consequently Dk [n] )uk [n] E[uk [n]] N (0, 1),V ar(uk [n])(25)where N (0, 1) is the standard Gaussian distribution accordingto the Central Limit Theorem, andNE[Ii [n]],K1NNV ar(Ii [n]) (1 )(E[Ii [n]])2 ,V ar(uk [n]) KKKwhich can be obtained by applying the standard conditionalmean and variance

802.16 networks was conducted in [7], [8] by combining link-layer queueing with physical-layer transmission. A vacation queueing model was adopted in [9] to analyze the link-layer queueing performance of OFDM-TDMA systems with round-robin scheduling. A queueing model for OFDMA systems was used in [10] to design a scheduling scheme that balances

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