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Unified approach to crosslayer scheduling and resource allocation in OFDMA wireless networks
EURASIP Journal on Wireless Communications and Networking volume 2012, Article number: 145 (2012)
Abstract
Orthogonal frequency division multiple access (OFDMA) has been selected as the core physical layer access scheme for stateoftheart and nextgeneration wireless communications standards. In these systems, scheduling and resource allocation algorithms, jointly assigning transmission data rates, bandwidth and power, become crucial to optimize the resource utilization while providing support to multimedia applications with heterogeneous quality of service (QoS) requirements. In this article, a unified framework for channeland queueaware QoSguaranteed crosslayer scheduling and resource allocation algorithms for heterogeneous multiservice OFDMA wireless networks is presented. The framework encompasses different types of traffic, uniform and continuous power allocation, discrete and continuous rate allocation, and protocols with different amounts of channel and queueawareness. System parameters and QoS requirements are projected into utility functions and the optimization problem is then formulated as a constrained utility maximization problem. Optimal solutions for this problem are obtained for the uniform power allocation schemes, and novel quasioptimal algorithms are proposed for the adaptive power allocation strategies. Remarkably, these techniques exhibit complexities that are linear in the number of resource units and users. Simulation results demonstrate the validity and merits of the proposed crosslayer unified approach.
1 Introduction
Due to its high spectral efficiency, inherent robustness against frequencyselective fading and flexibility in resource allocation, orthogonal frequency division multiple access (OFDMA), combined with multipleinput multipleoutput (MIMO) strategies, has been chosen as the multiple access technique for stateoftheart and nextgeneration wireless communications standards such as IEEE 802.16e/mbased WiMAX systems [1] and Third Generation Partnership Project (3GPP) technologies based on the longterm evolution (LTE) and LTEadvanced (LTEA)^{a}[2]. These systems have been designed with different quality of service (QoS) frameworks and strategies to allow the delivery of the wide range of emerging Internet multimedia applications with diverse QoS requirements [3]. In this context, scheduling and resource allocation algorithms jointly assigning transmission data rates (AMCadaptive modulation and coding), subcarriers, time slots and power become crucial for maximizing the resource utilization while providing satisfactory service delivery to end users.
The instantaneous characteristics of the transmission channel used by wireless MIMOOFDMA networks are inherently varying in time and frequency due to multipath propagation, changing positions of mobile stations (MS) relative to the base station (BS), and nonstationary environment. Consequently, the result is that different users sharing a BS experience different channel conditions at the same time and frequency. This phenomenon, referred to as multiuser diversity, constitutes the basis of opportunistic or channelaware scheduling algorithms. The goal of these strategies is to jointly allocate resources (i.e., power, subcarriers and/or time slots) in order to either minimize the weighted sum of powers under a prescribed minimum rate budget [4–6] or maximize the weighted sumrate under a prescribed power budget [7–9]. Nevertheless, greedyopportunistic schedulers serving only the users with favorable channel quality conditions raise the issue of fairness, as those users experiencing bad channel quality conditions may suffer from starvation. Therefore, besides channel state information (CSI), fairness is also an important issue that has been taken into account when designing scheduling algorithms for OFDMAbased multiservice networks [10–16]. Fairness, however, may lead to low spectral efficiency, and this may become an issue when facing realtime services with stringent QoS requirements in terms of delay and error tolerance. Thus, beyond channel quality conditions and fairness, another important issue that should be considered to maximize users' satisfaction is the one raised by the wide range of QoS requirements of heterogeneous applications supported by emerging OFDMAbased wireless networks.
In order to tackle all previously mentioned issues, the data link layer (DLC) bursty packet arrivals and queueing behavior should be jointly taken into consideration, in a crosslayer fashion, with the physical layer (PHY) channel conditions when designing scheduling and resource allocation algorithms. In this context, publications such as [17–23], reporting optimal and suboptimal crosslayer algorithms for very specific wireless multiuser OFDMA network configurations, lack a complete overview of the full problem, making it difficult to extract general conclusions. Song and Li [17, 18] present a framework for crosslayer optimization of downlink multiuser singlecell OFDMA systems, where the interactions between the physical and DLC layers are modeled using a utility function that trades fairness for throughput efficiency. This work assumes, however, that the system has an infinite number of subcarriers and proposes suboptimal allocation algorithms for practical realization. A crosslayer scheduling scheme for OFDMA wireless systems with heterogeneous delay requirements taking into account both queueing theory and information theory in modeling the system dynamics is presented in [19]. The objective of maximizing system throughput with constraints on the delay and the maximum transmitted power is formulated as a mixed convex and combinatorial optimization problem. Mohanram and Bhashyan [20] propose a suboptimal joint subcarrier and power allocation for channel and queueaware schedulers aiming at the maximization of the global average long term throughput. This scheduler, however, seems to be only applicable for traffic types without any constraint on delays. In [21] the authors propose a QoSaware proportional fairness (QPF) scheduling policy based on a crosslayer design where the scheduler is aware of both the channel and the queue state information. The proposed approach, however, apart from using suboptimal modified greedy multicarrier proportional fairness algorithms, only considers Shannon's capacitybased data rate allocation schemes and uniform power allocation (UPA) in the frequency domain. In [22], Song et al. propose a joint channeland queueaware scheduler, which is called the maxdelayutility (MDU) scheduling, designed to efficiently support delaysensitive applications. However, this scheduler is only effective for traffic types without explicit constraints on the minimum achievable average data rate and/or the maximum allowable absolute delay. Furthermore, only suboptimal sortingsearch algorithms for the subcarrier (subband) allocation problem and greedy algorithms for the power allocation problem are proposed. Finally, Zhou et al. [23] propose a packetdependent adaptive crosslayer design for downlink multiuser OFDMA systems, designed to maximize the weighted sum capacity of users with multiple heterogeneous traffic queues and based on the suboptimal algorithms proposed in [19].
Scheduling and resource allocation based on crosslayer principles can be regarded as a multiobjective optimization problem taking into account not only the system throughput but also the transmitted power, the QoS constraints on traffic delay and minimum and maximum data rates, the priority levels of different traffic classes and amount of backlogged data in the queues. In general, there is not a single optimal solution to a multiobjective optimization problem, however, using tools from information theory, queueing theory, convex optimization, and stochastic approximation [24], a unified framework for channel and queueaware QoS guaranteed scheduling and resource allocation for heterogeneous multiservice OFDMA wireless networks is proposed in this article. To this end, this study introduces a framework able to account for different types of traffic (e.g., best effort, nonrealtime and realtime), different allocation strategies (e.g., continuous and discrete rate allocation (DRA), uniform and adaptive power allocation (APA)), protocols with different amounts of channel and queueawareness, and different utility functions measuring user's satisfaction in terms of, for instance, throughput, queue length and/or service time (waiting time in the queues). Channel state, physicallayer characteristics, queueing delay and/or QoS requirements are projected into utility functions and the multiobjective optimization problem is then formulated as a constrained utility maximization problem, where the objective function is the maximization of the user services' utility functions. The constraints are related to the specifications of the network and offered services under consideration, namely, power limitations, perservice rate limits, and exclusive chunk (frequency/time resource unit) assignment. The unified algorithmic framework presented in this article generalizes results presented in, for instance, [19, 21, 22, 25, 26]. The proposed approach is based on dual decomposition optimization [27] and stochastic approximation techniques [24] exhibiting complexities that are linear in the number of resource units and users, and that achieve negligible duality gaps in numerical simulations based on current standardslike scenarios. Algorithms presented in this article optimize nonstatic utility functions based on the temporal evolution of throughput and/or waiting time of packets in the queues. Stochastic approximation techniques are used that allow these strategies to be implemented in real time.
This article is organized as follows. Section 2 presents a brief description of the system model under consideration alongside with the key assumptions made in the formulation of the optimization problem. A thorough description of the singlecell scenario, transmitter and receiver architectures, as well as of the channel model employed is also provided. As part of the crosslayer unified framework, the variables involved in the optimization problem are described in Section 3. Next, Section 4 presents a unified framework for constrained channeland queueaware QoS guaranteed scheduling and resource allocation for heterogeneous multiservice OFDMA wireless networks. Both continuous (Shannoncapacitybased) and discrete (AMCbased) strategies are considered, and solutions based on dualoptimization techniques are provided. In Section 6, numerical results illustrating the different performance/complexity tradeoffs of the proposed unified optimization framework are presented. Special emphasis is paid to efficiency, fairness and the fulfillment of QoS requirements. Finally, Section 7 summarizes the contributions of this article, and outlines the most interesting avenues for further research.
This introduction ends with a notational remark. Vectors and matrices are denoted by lower and uppercase bold letters, respectively. The Kdimensional identity matrix is represented by I_{ K }. The symbols ${\mathbb{R}}_{+}$and serve to denote the set of nonnegative real numbers and the set of complex numbers, respectively. Superscripts (·)^{T}and (·)^{†} are used to denote the transpose and the conjugate transpose (hermitian) of a matrix. Finally, and for the sake of clarity, a list of the most important symbols (in order of appearance) is also provided in Table 1.
2 System model and assumptions
Let us consider the downlink of a timeslotted MIMOOFDMA wireless packet access network as the one depicted in Figure 1. In this setup, a BS with a total transmit power P_{ T } and equipped with N_{ T } transmit antennas provides service to N_{ m } active MS, each equipped, without loss of generality, with an equal number of receive antennas, denoted by N_{ R } .
Transmission between the BS and active MSs is organized in time slots of a fixed duration T_{ s } , assumed to be less than the channel coherence time. Thus, the channel fading can be considered constant over the whole slot and it only varies from slot to slot, i.e., a slotbased block fading channel is assumed. Each of these slots consists of a fixed number N_{ o } of OFDM symbols of duration T_{ o } + T_{CP} = T_{ s } / N_{ o } , where T_{CP} is the cyclic prefix duration. Slotted transmissions take place over a bandwidth B, which is divided into N_{ b } orthogonal subbands, each consisting of N_{sc} adjacent subcarriers and with a bandwidth B_{ b } = B/N_{ b } small enough to assume that all subcarriers in a subband experience frequency flat fading. One subband in the frequency axis over one slot in the time axis forms a basic resource allocation unit. Active MS and frequency subbands in a given slot are indexed by the sets ${\mathcal{N}}_{m}=\left\{1,\dots ,{N}_{m}\right\}$ and ${\mathcal{N}}_{b}=\left\{1,\dots ,{N}_{b}\right\}$, respectively.
Without loss of generality, and in order to simplify the mathematical notation of the problem, only one service data flow (also known as connection or session) per active MS will be assumed. Depending on the traffic type, three classes of service and the associated QoS requirements and priorities must be accounted for in wireless communications [24]:

Best effort (BE) low priority services with a prescribed maximum allowable error rate but without specific requirements on rate or delay guarantees. Examples of besteffort services include applications such as email or HTTP web browsing.

Nonrealtime (nRT) services entail applications such as file transfers (FTP). They do not impose any constraint on delays but, in addition to a maximum allowable error rate, they require sustained throughput guarantees.

Realtime (RT) high priority services are used for applications such as video conferencing and streaming entailing QoS guarantees on maximum allowable error rate, minimum throughput, and maximum delay.
Traffic flows arriving from higher layers are buffered into the corresponding N_{ m } firstin firstout (FIFO) queues at the DLC layer. At the beginning of each scheduling time interval, based on the available joint channel and queuestate information (CSI/QSI), the crosslayer scheduling and resource allocation algorithms select some packets in the queues for transmission, which are then forwarded to the OFDM transmitter, at a rate R_{ m }(t) for all $m\in {\mathcal{N}}_{m}$, where they are adaptively modulated and channel encoded (AMC), and are allocated power and subbands, just before MIMO processing.
2.1 PHY layer modeling
2.1.1 Transmitter
Multipleinput multipleoutput technology provides a great variety of techniques to exploit the multiple propagation paths between the N_{ T } transmit antennas and the N_{ R } receive antennas. Notably when CSI is available at the transmitter and receiver sides, and multiplexing in the spatial domain is not used, the joint use of maximum ratio transmission (MRT) [28] at the transmitter and maximal ratio combining (MRC) at the receiver is known to provide optimum performance in the sense of maximizing the received signaltonoise ratio (SNR).
Let us assume that subband b has been allocated to MS m and that the BS uses an MRT scheme to exploit the spatial diversity provided by the MIMO channel. In this case, bits from the queue of MS m are channel encoded and mapped onto a sequence of symbols drawn from the allocated normalized unit energy complex constellation (e.g., BPSK, QPSK, 16QAM, 64QAM). Furthermore, before the usual OFDM modulation steps on each transmit antenna (IFFT, cyclic prefix appending and upconversion), the symbols are allocated power and are processed in accordance with the MRT transmission scheme. Denoting by ${d}_{m,b}^{\left(c,o\right)}\left(t\right)$ the symbol to be sent to MS m over subcarrier c ∈ {1, ..., N_{sc}} of subband b and OFDM symbol o ∈ {1, ..., N_{ o } } during time slot t, then the corresponding N_{ T } × 1 transmitted vector can be expressed as
where p_{ m, b } (t) is the power allocated to MS m on subband b during the time slot t (in a given subband, power is uniformly allocated to subcarriers), and ${\upsilon}_{m,b}\left(t\right)\in {\u2102}^{{N}_{T}\times 1}$ denotes the unit energy linear transmit filter used by the MRT transmission system.
2.1.2 Channel model
The propagation channel between the BS and MS m is characterized by a power delay profile [29], common to all pairs of transmit and receive antennas, that can be expressed as
where L_{ p } denotes the number of independent propagation paths, and ${\sigma}_{m,l}^{2}$ and τ_{ l } are, respectively, the power and delay of the l th propagation path. Hence, assuming that the channel coherence time is greater than T_{ s } , the channel impulse response between transmit antenna n_{ T } and the receive antenna n_{ R } of MS m, over the whole frame period t, can be written as
where $E\left\{{\left{h}_{m,l}^{{n}_{R},{n}_{T}}\left(t\right)\right}^{2}\right\}={\sigma}_{m,l}^{2}$. The corresponding frequency response, when evaluated over subband b (with center frequency f_{ b } ), can be safely approximated by
Accordingly, the MIMO channel between the BS and MS m, for subband b and over the whole time slot period t, will be characterized by the complex valued N_{ R } × N_{ T } matrix
2.1.3 Receiver
At the receiver side, as usual, ideal synchronization and sampling processes, and an OFDM cyclic prefix duration greater than the maximum delay spread of the channel impulse response are assumed. In this case, the received samples at the output of the N_{ R } FFT processing stages of MS m over subcarrier c of subband b and OFDM symbol o during time slot t are given by the N_{ R } × 1 complex valued vector
where ${\mathbf{\nu}}_{m,b}^{\left(c,o\right)}\left(t\right)\in {\u2102}^{{N}_{R}\times 1}$ is a noise vector with elements modeled as independent identically distributed (i.i.d.) zeromean complex circularsymmetric Gaussian random variables, with covariance matrix $E\left\{{\u2225{\mathbf{\nu}}_{m,b}^{\left(c,o\right)}\left(t\right)\u2225}^{2}\right\}={\sigma}_{\nu}^{2}{\mathit{I}}_{{N}_{R}}$.
According to the MRT strategy [28], the transmission filter υ _{ m, b } (t) that maximizes the SNR at the receiver side is the right singular vector of matrix H_{ m, b }(t) associated with its largest singular value, denoted as σ_{max}( H _{ m, b } (t)). In this case, the instantaneous SNR experienced by all subcarriers in subband b at the output of the maximal ratio combiner (MRC) used at the receiver side can be expressed as
where ${\delta}_{m,b}\left(t\right)={\sigma}_{\text{max}}^{2}\left({\mathit{H}}_{m,b}\left(t\right)\right)$, which coincides with the largest eigenvalue of the N_{ R } × N_{ R } positive semidefinite Hermitian matrix ${\mathit{H}}_{m,b}\left(t\right){\mathit{H}}_{m,b}^{\u2020}\left(t\right)$.
2.2 DLC layer modeling
In order to characterize the queueing behavior at the DLC layer, a slightly modified version of the model proposed by Kong et al. [[21], Section IV.A] is assumed. At the beginning of time slot t, MS m is assumed to have Q_{ m } (t) bits in the queue. If there are A_{ m } (t) bits arriving during time slot t, the queue length at the end of this time slot, assuming queues of infinite capacity, can then be expressed as
where
with r_{ m } (t) denoting the data rate allocated to user m during time slot t. A crosslayer resource allocation strategy that, in order to avoid the waste of resources, selects a transmission rate
is said to fulfill the frugality constraint (FC) [22].
As will be shown in Section 4.1, most of the schedulers and resource allocation schemes that have been proposed in the literature can be interpreted as decision making algorithms that, at the beginning of time slot t estimate or predict the future behavior of QoS quantitative performance measures such as the throughput, average delay, queue length and/or headofline delay, and decide which users will be granted a transmission opportunity and the amount of resources that they will be allocated.
2.2.1 Predicting the queue length
As A_{ m } (t) is unknown at the beginning of time slot t, and assuming that the DLC layer only knows the average arrival data rate λ_{ m } , then a prediction of the queue length at the end of this time slot can be obtained from (8) as
where E_{ x } {⋅} denotes the statistical expectation operator with respect to the random variable x.
2.2.2 Predicting the average waiting time
Using standard stochastic approximation recursions, a recursive estimate of the slotbyslot queue length sample average can be obtained as [24]
where the stepsize β_{ t } ∈ (0, 1) implements a forgetting factor in the averaging and can be selected to be either constant (i.e., β_{ t } = β) or asymptotically vanishing (e.g., β_{ t } = 1/t). Little's law [30] asserts that with stable queues the average delay at the end of time slot t can be obtained as
Using (11) and (12), this in turn leads to a recursive prediction of the slotbyslot average delay via
2.2.3 Estimating the average throughput
Stochastic approximation tools can also be used to obtain a recursive estimate of the framebyframe throughput sample average as
2.2.4 Predicting the headofline delay
The HOL delay of user m at the beginning of time slot t (or equivalently, the end of time slot (t  1)) can be written as ${W}_{\text{H}\text{O}\text{L},m}\left(t\right)=t{T}_{s}{\tau}_{m}^{\left(A\right)}\left(t\right)$, where ${\tau}_{m}^{\left(A\right)}\left(t\right)$ denotes the arrival time of the HOL packet in the queue of user m. Hence, a prediction of the HOL delay at the end of time slot t can be readily obtained as
3 Optimization variables
3.1 Power allocation
Let p_{ b }(t) = [p_{1,b}(t) ... p_{ Nm, b } (t)]^{T}denote the vector of power allocation values for subband b and time slot t. For a given set of constraints, the scheduling and resource allocation algorithm will be in charge of determining the power allocation vector
optimizing a prescribed objective function. In addition to determining the power allocation values, the resource allocation algorithms should also allocate subbands and transmission rates. Nevertheless, as it will be shown next, the power allocation vector p(t) can also be used to represent the allocation of all these resources, thus simplifying the formulation of the optimization problem [9].
3.2 Subband allocation
As usual, it is assumed that subband allocation is exclusive, that is, only one MS is allowed to transmit on a given subband. Hence, the subband allocation constraints can be captured by constraining the power allocation vectors as
where
with ${\mathbb{R}}_{+}$ denoting the set of all nonnegative real numbers. Hence, the power allocation vector satisfies
where × denotes the Cartesian product (or product set).
3.3 Rate allocation
In the downlink of multirate systems based on AMC, a channel estimate is obtained at the receiver of each MS and it is then fed back to the BS so that the transmission scheme, comprising a modulation format and a channel code, can be adapted in accordance with the channel characteristics.
If MS m is allocated subband b over time slot t, then the BS selects a modulation and coding scheme (MCS) that can be characterized by a transmission rate ρ_{ m, b } (t) (measured in bits per second). As each subband contains N_{sc} subcarriers, the aggregated data rate allocated to MS m over time t will be given by
Transmission rate ρ_{ m, b }(t) can be related to bit error rate (BER) observed by MS m, denoted as ɛ_{ m }, and instantaneous SNR γ_{ m, b } (t) as [[31], Chapter 9] (see also [24])
where κ_{1} and κ_{2} are modulationand codespecific constants that can be accurately approximated by exponential curve fitting. This expression is general enough to obtain the BER performance of any transmission system for which the joint effects of transmission filters, channel coefficients and reception filters can be represented through an instantaneous SNR γ_{ m, b } (t). For the special case of MRT/MRC scheme, γ_{ m, b } (t) is defined by (7).
3.3.1 Discreterate AMC
Realistic AMC strategies can only use a discrete set ${\mathcal{N}}_{k}=\left\{0,1,\dots ,{N}_{k}\right\}$ of MCSs that can differ for different MSs. Each MCS is characterized by a particular transmission rate ${\varrho}_{m}^{\left(k\right)}$, with ${\varrho}_{m}^{\left(1\right)}<\cdots <{\varrho}_{m}^{{N}_{k}}$, and ${\varrho}_{m}^{\left(0\right)}=0$ denoting the case where MS m does not transmit.
Given p_{ m, b } (t), δ_{ m, b } (t) and the noise variance ${\sigma}_{\nu}^{2}$, we can use (7) to find γ_{ m, b } (t) and then, considering the maximum allowable BER ${\stackrel{\u02c7}{\epsilon}}_{m}$, employ (21) to select the most adequate MCS scheme as the one with transmission rate
In fact, the transmission rate ρ_{ m, b } (t) can be expressed using the staircase function
where ${\left\{{\Gamma}_{m}^{\left(k\right)}\right\}}_{k=1}^{{N}_{k}1}$, with ${\Gamma}_{m}^{\left(k\right)}\le {\Gamma}_{m}^{\left(k+1\right)}$, are the instantaneous SNR boundaries defining the MCS intervals, which can be obtained from (21) as
3.3.2 Continuousrate AMC
A useful abstraction when exploring rate limits is to assume that each user's set of MCSs is infinite. In this case, the maximum allowable transmission rate fulfilling the prescribed BER constraint with equality can be obtained from (21) as
where ${\Lambda}_{m}={\kappa}_{2}^{1}\text{ln}\left({\kappa}_{1}/{\stackrel{\u02c7}{\epsilon}}_{m}\right)\ge 1$ represents the coding gap due to the utilization of a practical (rather than ideal) coding scheme. With Λ_{ m } = 1 this expression results in the Shannon's capacity limit and allows the comparison of practical AMCbased schemes against fundamental capacityachieving benchmarks.
4 Problem formulation
The main objective of crosslayer scheduling and resource allocation algorithms over a wireless network is the establishment of effective policies able to optimize metrics related to spectral/energy efficiency and fairness, while satisfying prescribed QoS constraints. The issues of efficient and fair allocation of resources have been intensively investigated in the context of economics, where utility functions have been used to quantify the benefit obtained from the usage of a pool of resources. In a similar way, utility theory can be used in wireless communication networks to evaluate the degree up to which a given network configuration can satisfy users' QoS requirements [17, 18].
Utility functions are used to map the resources (e.g., bandwidth, power, ...), performance criteria (e.g., throughput, delay,...) and QoS requirements (e.g., maximum tolerable error rate, maximum absolute delay, maximum allowable outage delay probability, ...) into the corresponding user's satisfaction. Different applications can be characterized by different utility functions and/or even different performance quantitative measures and QoS requirements. For instance, utility functions for BE applications are typically characterized in terms of throughput, whereas those for nRT or RT delaysensitive applications are characterized in terms of queuing delay with QoS requirements on the sustainable throughput, and/or the average or absolute delay. Thus, in general, the satisfaction of MS m at time t can be expressed by a utility function ${U}_{m}\left({\mathit{\theta}}_{m}\left(t\right),{\stackrel{\u02c7}{\mathit{\Omega}}}_{m}\right)$, where ${\mathit{\theta}}_{m}\left(t\right)=\left\{{\theta}_{m}^{1}\left(t\right),\dots ,{\theta}_{m}^{{N}_{z}^{\left(m\right)}}\left(t\right)\right\}$ is the set of quantitative QoS measures used to characterize the satisfaction of MS m (e.g., throughput ${\overline{T}}_{m}\left(t\right)$, average delay ${\overline{W}}_{m}\left(t\right)$, queue length Q_{ m } (t) or HOL delay W_{HOL,m}(t)) and ${\stackrel{\u02c7}{\mathit{\Omega}}}_{m}=\left\{{\stackrel{\u02c7}{\Omega}}_{m}^{1},\dots ,{\stackrel{\u02c7}{\Omega}}_{m}^{{N}_{y}^{\left(m\right)}}\right\}$ is the set of QoS requirements for user m (e.g., maximum tolerable error rate ${\stackrel{\u02c7}{\epsilon}}_{m}$, maximum tolerable absolute delay Ď_{ m }, maximum outage delay probability ${\stackrel{\u2323}{\xi}}_{m}$). Hence, assuming the availability of perfect CSI/QSI, the utilitybased crosslayer scheduling and resource allocation scheme can be formulated as the following optimization problem,
4.1 Gradientbased scheduling and resource allocation
The first order Taylor's expansion of ${U}_{m}\left(\mathit{\theta},{\stackrel{\u02c7}{\mathit{\Omega}}}_{m}\right)$ in a neighborhood of θ= θ_{ m }(t) can be written as
where ∇_{ θ }denotes the vector differential operator or gradient function with respect to θ. Thus, using this approximation, the variation of utility for MS m during time slot t is given by
Using this result, the crosslayer longterm optimization problem in (26) can be rewritten, as shown in [17, 18, 32], as the instantaneous gradientbased optimization problem
Although utility functions based on QoS quantitative performance measures other than the throughput, the average delay, the queue length and/or the HOL delay could be devised, most practical utility functions used in stateoftheart wireless communications are based on either one of these performance measures or a combination of them. Therefore, let us assume hereafter that ${\mathit{\theta}}_{m}\left(t\right)=\left\{{\theta}_{m}^{1}\left(t\right),{\theta}_{m}^{2}\left(t\right),{\theta}_{m}^{3}\left(t\right),{\theta}_{m}^{4}\left(t\right)\right\}=\left\{{\overline{T}}_{m}\left(t\right),{\overline{W}}_{m}\left(t\right),{Q}_{m}\left(t\right),{W}_{HOL,m}\left(t\right)\right\}$. In this case, using (11)(2.2.4), then
and
Finally, using these expressions in (29) and eliminating constants not affecting the optimization process yields
where the weighing (prioritization) coefficient for MS m during the time slot t is given by
4.2 Marginal utility functions
4.2.1 Maxsumrate (MSR) rule
The MSR scheduler [33] is based on a channelaware scheduling rule that, using
maximizes the slotbyslot aggregated transmission rate
However, as stated by Song et al. [22], although it maximizes the spectral efficiency, it can lead to unfairness and queue instability, especially for nonuniform traffic patterns and MSs operating in uneven channel conditions. Furthermore, since MSs with unfavorable channel conditions can experience long deep fading periods, long delays are expected and consequently, the MSR rule is not able to support delaysensitive applications.
4.2.2 Proportional fair (PF) rule
The PF scheduler [34] is based also on a channelaware scheduling rule aiming at maximizing the logarithmicsumthroughput of the system, that is
Thus, the gradientbased PF scheduling algorithm is effected by using
Nevertheless, although PF rule can trade off spectral efficiency and fairness among users belonging to the same QoS class, it cannot cope with MSs with disparate QoS requirements, especially those supporting delaysensitive applications. Particularly, long deep fading starvation periods are not solved by this rule.
It is worth pointing out that for incoming lowrate data flows it is quite common that for some users ${\overline{T}}_{m}\left(t\right)={\lambda}_{m}$ no matter how good their average channel condition is; as a result, for those users, ${\overline{T}}_{m}\left(t\right)$ is not a good measure of the actual amount of resources allocated to them and so, it is better to use [35],
4.2.3 Modified largest weighted delay first (MLWDF) rule
The MLWDF scheduler was proposed by Andrews et al. [36] for singlecarrier CDMA networks with a shared downlink channel and was proved to be throughput optimal.^{b} It is based on a channel and queueaware scheduling rule that considers the waiting time in the queues, the instantaneous potential transmission rates and the maximum tolerable delay requirements. At each time slot t, the MLWDF scheduler aims at choosing the best combination of queueing delay and potential transmission rate, serving the users that maximize the sum of marginal utility functions given by [36]
That is,
where χ_{ m } (t) are arbitrary positive constants that can be used to set different priority levels between traffic flows. The MLWDF scheduling rule remains throughput optimal if for all or some queues, the headofline delay W_{HOL,m}(t) is replaced by the queue length Q_{ m } (t). Thus, the scheduler can be easily implemented by time stamping arriving data packets of all MSs, and/or keeping track of the corresponding queue lengths.
In order to guarantee that users with absolute delay requirement Ď_{ m }and maximum outage delay probability requirement ${\stackrel{\u2323}{\xi}}_{m}$ will be satisfied, the authors of [36] propose to properly set the values of χ_{ m } (t) as
providing in this way QoS differentiation among user's flows.
As stated by Andrews et al. [37], services with QoS constraints on the minimum sustainable throughput Ť_{ m }can also be supported by the MLWDF scheduling rule, provided that the scheduler is used in conjunction with a token bucket control. In this case, each queue m is associated to a virtual token bucket, with tokens arriving at a constant rate Ť_{ m }. At each time slot, queues are served according to the MLWDF rule, with W_{HOL,m}(t) denoting the delay of the headofline token in bucket m instead of the headofline packet delay for queue m. After serving a given queue m, the number of tokens in the corresponding bucket must be reduced by the actual amount of data served.
4.2.4 Exponential (EXP) rule
The EXP scheduler, proposed by Shakkottai and Stolyar [38], is also based on a channeland queueaware throughput optimal scheduling rule that considers the waiting time in the queues, the instantaneous potential transmission rates and the maximum tolerable delay requirements. It was proposed for singlecarrier CDMA networks with a shared downlink channel but, similarly to MLWDF, it can easily be extended to multichannel scenarios. In this case, at each time slot t, the EXP scheduler serves the users maximizing the sum of marginal utility functions given by [35]
That is,
for all m, with
As in the MLWDF scheduler, the headofline delay W_{HOL,m}(t) can be replaced, for all or some queues, by the queue length Q_{ m } (t) without affecting the throughput optimality of this strategy. Furthermore, if providing a minimum throughput Ť_{ m }to a given flow is a goal, the EXP rule can also be modified by introducing a virtual token queue, where tokens arrive at a constant rate Ť_{ m }, and serving the queue according to the EXP rule, with W_{HOL,m}(t) denoting the delay of the headofline token in bucket m.
4.2.5 MDU rule
To efficiently support delaysensitive applications, Song et al. [22] proposed another joint channeland queueaware scheduling approach, known as MDU rule, which maximizes the total utility with respect to average delays or average waiting times in the queues. Generalizing the marginal utility functions proposed by [39], the MDU scheduling rule can be treated in the unified optimization framework defined in (34) by setting
where φ_{ m } = {φ_{ m } ,_{1}, φ_{ m } ,_{2}} is a set of constants used to differentiate between heterogeneous services and ${\stackrel{\u0303}{W}}_{m}$ is a delay threshold related to the maximum tolerable delay Ť_{ m }. In [39], based on the corresponding required QoS, these parameters were set to φ_{ m } = {1, 1.5} and ${\stackrel{\u0303}{W}}_{m}=25\phantom{\rule{0.3em}{0ex}}\text{ms}$ for packetswitched voice with endtoend delay required to be less than 100 ms, φ_{ m } = {0.6, 1} and ${\stackrel{\u0303}{W}}_{m}=100\phantom{\rule{0.3em}{0ex}}\text{ms}$ for goodquality streaming transmission requiring endtoend delays between 150400 ms and, finally, φ_{ m } = {0.5, 0} and ${\stackrel{\u0303}{W}}_{m}=100\phantom{\rule{0.3em}{0ex}}\text{ms}$ ms for BE traffic. Actually, using these settings the MDU scheduling for the besteffort traffic becomes the PF scheduling.
4.2.6 Other scheduling rules
Although not treated in this article, the unified crosslayer optimization approach defined in (34) can also be extended to scheduling rules such as those proposed in [16, 23, 40–42]. Notably, AlManthari et al. [41] propose the use of utility functions that are based on both the throughput and the average delay.
5 Unified optimization framework
The optimization problem formulated in (34) is general enough to account for different power and rate allocation strategies, either with or without FC. For clarity of presentation, the following list of acronyms will be used: UPA, APA, continuous rate allocation (CRA), DRA and FC. Furthermore, since optimization is performed on a slotbyslot basis, from this point onwards the time dependence (i.e., (t)) of all the variables will be dropped.
5.1 UPA without FC
Let us assume a system where the scheduling and rate allocation schemes do not consider the FC. In this case, problem (34) can be rewritten as
Let us also assume that the BS transmit power P_{ T } is uniformly allocated to all subbands. In this case, if subband b is allocated to user ${m}_{b}^{*}$, then the subband exclusive allocation constraint (i.e., $\mathit{p}\in \mathcal{P}$) forces that
for all b. Thus, using (20) in (44) it is straightforward to show that subband b must be allocated to MS ${m}_{b}^{*}$ satisfying
with ρ_{ m, b } obtained as in either (23), for the DRA case, or (25), for the CRA case.
5.2 APA without FC
The objective function in (44) is concave, but is a highly nonconvex discrete constraint space. Fortunately, problem (44) is separable across the subbands and, as stated in [9, 27], it can be approached by using Lagrange duality principles. With μ denoting the Lagrange multiplier associated with the power constraint, the Lagrangian of (44) can be expressed as
and the dual problem can then be written as [43]
Now, using the subband exclusive allocation constraint and the separability of power variables across subbands, the dual problem can be simplified as [26]
The solution to the simplified dual problem is given by optimizing (49) over all (p, μ) ≽ 0. This optimization can be done iteratively and coordinatewise, starting with the p variables and continuing with μ.
5.2.1 Optimizing the dual function over p
CRA: In case of using ρ_{ m, b } as defined in (25), and for a given value of μ, the innermost maximization in (49) provides a multilevel waterfilling closedform expression for the optimal power allocation given by
where [x]^{+} ≜ max{0, x}. Now, using (50) in (49) yields
with
Hence, for a fixed dual variable μ, the subband b will be allocated to MS ${m}_{b}^{*}$ satisfying
DRA: In this case ρ_{ m, b } is a nonderivable discontinuous function. However, the approach proposed in [[9] Chapter 3] can be applied to arrive at the optimal solution. Using (23) the set of nonnegative real numbers (i.e., ${\mathbb{R}}^{+}$) can be subdivided, for each MS m and subband b, into N_{ k } segments
Furthermore, given that μ and p_{ m, b } belong to ${\mathbb{R}}^{+}$, if a power allocation p_{ m, b } is used such that ${\Gamma}_{m}^{\left(k\right)}\le {\gamma}_{m,p}<{\Gamma}_{m}^{\left(k+1\right)}$ then
As a consequence, there only exist N_{ k } candidate power allocations
from which the one maximizing ${w}_{m}{N}_{\text{sc}}{\varrho}_{m}^{\left(k\right)}\mu \frac{{N}_{\text{sc}}{\sigma}_{\nu}^{2}{\Gamma}_{m}^{\left(k\right)}}{{\delta}_{m,b}}$ must be selected, that is,
where
Furthermore, as in the CRA case, given μ and ${p}_{m,b}^{*}$, the subband b must be allocated to MS ${m}_{b}^{*}$ satisfying (53).
5.2.2 Optimizing the dual function over μ
Once known the optimal vector p* for a given μ, the dual optimization problem (49) reduces to
Using standard properties of dual optimization problems [9, 27], it can be shown that this problem is convex with respect to μ, and thus, derivativefree line search methods like, for example, Goldensection or Fibonacci, can be used to determine μ*. Once μ* has been found, it can be used to obtain optimal power, subband and rate allocation for each of the data flows in the system.
Algorithm 1 Resource allocation for UPA/APA with FC

1:
i = 1; {Initialize iteration counter}

2:
${\mathcal{N}}_{b}^{\left(1\right)}={\mathcal{N}}_{b}$ {Initialize set of non allocated subbands}

3:
${Q}_{m}^{\left(1\right)}\left(t\right)={Q}_{m}\left(t\right)\phantom{\rule{2.77695pt}{0ex}}\forall m$ {Initialize queue lengths}

4:
${P}_{T}^{\left(1\right)}\left(t\right)={P}_{T}$ {Initialize available power (APA)}

5:
while${\mathcal{N}}_{b}^{\left(i\right)}\ne \mathrm{0\u0338}$ and ${\sum}_{m=1}^{{N}_{m}}{Q}_{m}^{\left(i\right)}\ne 0$ do

6:
{Allocate resources using (61)(63) or (65)(69)}

7:
${\mathcal{N}}_{b}^{\left(i+1\right)}=\left\{{\mathcal{N}}_{b}^{\left(i\right)}\backslash b\right\}$ {Update non allocated subbands}

8:
${Q}_{{m}_{b}^{*}}^{\left(i+1\right)}={Q}_{{m}_{b}^{*}}^{\left(i\right)}{N}_{\text{sc}}{\rho}_{{m}_{b}^{*},b}{N}_{o}{T}_{o}$ {Update queue}

9:
${P}_{T}^{\left(i+1\right)}={P}_{T}^{\left(i\right)}{p}_{{m}_{b}^{*}}$ {Update available power (APA)}

10:
i = i + 1; {Update iteration counter}

11:
end while
5.3 UPA and APA with FC
When considering the socalled FC, the unified optimization problem in (34) can be rewritten as
This problem belongs to the class of nonlinear integer optimization programs, which have no general global optimal solution. In an attempt to provide a fast and efficient suboptimal solution to the joint scheduling and resource allocation problem, an iterative searching algorithm providing quasioptimal solutions is proposed in Algorithm 1. Our approach is based on a modified version of the optimal solutions presented in Sections 5.1 and 5.2. If necessary, the proposed algorithm allocates a subband $b\in {\mathcal{N}}_{b}$ per iteration i, assuming that queue length and available transmit power (APA cases only) are updated, in each iteration, by taking into account the data rate and power (APA cases only) allocated to subband b.
5.3.1 UP A with FC
When implementing UPA strategies, if subband b is allocated to user ${m}_{b}^{*}$ in iteration i, then the subband exclusive allocation constraint (i.e., $\mathit{p}\in \mathcal{P}$) forces that
for all b. Thus, in iteration i, subband b is allocated to MS ${m}_{b}^{*}$ satisfying
where ${Q}_{m}^{\left(i\right)}$ is the updated queue length of user m at the beginning of this iteration, with ${Q}_{m}^{\left(1\right)}={Q}_{m}$ and
The persubband data rate ρ_{ m, b } is obtained as in either (23), for the DRA case, or (25), for the CRA case.
5.3.2 APA with FC
When implementing APA strategies, assuming that vector p*^{(i)}for a given μ^{(i)}fulfils the FC, then the corresponding dual optimization problem, for iteration i, reduces to
which, as previously stated, can be solved by using derivativefree line search methods. Once μ*^{(i)}has been found, it can be used to obtain power, subband and rate allocation.
In the APA/CRA scheme the optimal power allocation in (50) must be redefined in order to fulfil the FC. Thus,
where
which has been obtained from (25), is the minimum power required to fulfill the FC in iteration i. Subband b will be allocated to MS ${m}_{b}^{*}$ satisfying
In the APA/DRA scheme, the power allocation in iteration i is obtained as
where ${k}_{m,b}^{*\left(i\right)}$ is obtained by redefining (58) as
Furthermore, as in the APA/CRA case, given μ*^{(i)}and ${p}_{m,b}^{*\left(i\right)}$, the subband b must be allocated to MS ${m}_{b}^{*}$ satisfying (67).
6 Numerical results
Based on the unified crosslayer framework previously described, this section is devoted to the performance comparison of different scheduling and resource allocation algorithms in the downlink of a MIMOOFDMA network. The following performance metrics will be discussed:

Average system throughput: average number of transmitted bits per second by the BS. It is obtained as the average sumrate achieved by the whole set of users connected to the BS.

Average delay: average amount of time the bits spent in the queue at the BS in addition to transmission time. Notice that delay can be interpreted as an indirect throughput measure. If a given MS is not allocated enough resources, the achievable throughput is below the traffic arrival rate, the corresponding queue gets unstable and delay grows toward infinity. On the other hand, if the MS is overprovisioned, the traffic arrival rate is below the maximum achievable throughput, the queue is stable, and the mean delay remains bounded.

Fairness: Jain's fairness index [44] will be used to calculate fairness among users of the same class of service. With Ω_{ m }denoting the performance metric for user m (i.e., throughput or average delay), then Jain's fairness index is calculated as
$$\text{JF}{\text{I}}_{\mathcal{C}}=\frac{{\left({\sum}_{m\in \mathcal{C}}{\Omega}_{m}\right)}^{2}}{\left\mathcal{C}\right{\sum}_{m\in \mathcal{C}}{\Omega}_{m}^{2}},\phantom{\rule{1em}{0ex}}{\Omega}_{m}\ge 0\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}\forall m,$$(70)where is the set of users belonging to a given class of service and $\left\mathcal{C}\right$ denotes the cardinality of this set. The Jain's fairness index is constrained to the set of values $1/\left\mathcal{C}\right\le \text{JF}{\text{I}}_{\mathcal{C}}\le 1$. If all the users in get the same Ω_{ m } , then $\text{JF}{\text{I}}_{\mathcal{C}}=1$ and maximum fairness is achieved. Lower Jain's fairness index values indicate a higher variance in their achieved QoS, revealing unfairness in scheduling and resource allocation.

Service coverage: percentage of users who achieve their QoS requirements in terms of minimum throughput or maximum allowable average or absolute delay.
6.1 Simulation configuration
Let us consider a singlecell downlink scenario where the BS, transmitting with a power of P_{ T } = 37 dBm over a carrier frequency f_{0} = 2 GHz, is assumed to be located at the center of a circular coverage area with a radius R = 500 m. This BS serves a set of N_{ u } MSs that are uniformly distributed over the whole coverage area. Unless otherwise specified, a default 2 × 2 MIMO configuration will be assumed. The entire system bandwidth is B = 5.6 MHz, and is divided into N_{ b } = 64 orthogonal subbands, each with a bandwidth B_{ b } = 87.5 kHz and consisting of N_{sc} = 8 adjacent subcarriers. Transmission between the BS and active MSs is organized in time slots of duration T_{ s } = 2.0571 ms, and each of these slots consists of N_{ o } = 20 OFDM symbols of duration (without considering the cyclic prefix) T_{ o } = 91.4286 μ s. Thus, the basic resource allocation unit is formed by 8 adjacent subcarriers and 20 OFDM symbols. We would like to point out that, without loss of generality, most of the chosen parameters are very much aligned with those considered in the Mobile WIMAX standard (see, for instance, [[45] Table 2.3]).
When using DRA strategies, the set of achievable transmission rates in bits/symbol has been fixed to {0, 0.5, 1, 1.5, 2, 3, 4, 4.5}, the coding gap has been set to Λ_{ m } = 3, and the switching thresholds between transmission modes have been obtained as ${\Gamma}_{m}^{\left(k\right)}={\Lambda}_{m}\left({2}^{{\varrho}^{\left(k\right)}}1\right)$. In contrast, a coding gap of Λ_{ m } = 1 (Shannon's capacity limit) has been set when using CRA strategies, whose performance serves as a benchmark against which practical DRA strategies can be measured.
The channel model describing the pathlosses, shadowing effects and frequency, timeand spaceselective fading experienced by the transmitted signal on its way from the BS to the MSs, has been implemented by using Stanford University Interim (SUI) channel model 4 [46] with a shadow fading standard deviation of 6 dB. The power delay profile of this model is characterized by L_{ p } = 3 Rayleigh distributed paths with power gains ${\sigma}_{m,0}^{2}=0\phantom{\rule{0.3em}{0ex}}\text{dB,}{\sigma}_{m,1}^{2}=4\phantom{\rule{0.3em}{0ex}}\text{dB}$ and ${\sigma}_{m,2}^{2}=8\phantom{\rule{0.3em}{0ex}}\text{dB}$, and corresponding delays τ_{0} = 0 μs, τ_{1} = 1.5 μ s and τ_{2} = 4 μ s. Moreover, a per subcarrier AWGN power of ${\sigma}_{\nu}^{2}=163.6\phantom{\rule{0.3em}{0ex}}\text{dBW}$ has been assumed at the receiver frontend.
To demonstrate the ability of our proposed unified framework to schedule and allocate resources to service flows with different QoS requirements, three traffic classes are considered, i.e., real time (RT), non real time (nRT) and BE. As in [22], traffic arrivals have been modeled as Poisson random variables, with a mean that depends on the average arrival rate per flow (measured in bits/s). Without loss of generality, the maximum tolerable delays (Ď_{ m }) for each traffic class have been set to 100 ms (RT), 2 s (nRT) and 20 s (BE), and the outage delay probabilities (ξ_{ m } ) to 0.01 (RT), 0.01 (nRT) and 0.1 (BE).
All the numerical results presented in this article have been obtained by averaging the outcomes of a dynamic discrete event simulation performed over 60 scenarios, each with a particular random distribution of MSs over the coverage area, and transmitting 15,000 slots per scenario. To guarantee that the presented results correspond to the steadystate of the system, initial transitory periods of 1,000 slots per scenario, which are not accounted for in the performance evaluation process, have been used.
6.2 Comparing scheduling rules
The performance metrics versus traffic load for MSR, PF, EXP and MLWDF scheduling rules are compared in Figure 2 for an adaptive MIMOOFDMA system serving N_{ u } = 20 RT users with the same average arrival rate. The use of uniform power and CRA strategies over a 2 × 2 MIMO system has been considered. Furthermore, since the FC can be implemented with all scheduling rules, performance results have been obtained for each scheduling rule either with or without FC.
Without FC, the EXP and MLWDF rules provide the best joint results in terms of throughput, delay, Jain's fairness indexes, and service coverage, with MLWDF achieving a slightly higher throughput, lower delay and better service coverage that EXP, at the cost of lower throughput and delay fairness indexes.^{c} The PF scheduler, although achieves a quite good result in terms of average throughput per flow, fails in providing QoS requirements. In fact, the PF rule can only guarantee a 99% service coverage for average arrival rates per flow less than 0.3 Mbps compared to the 0.8 and 1 Mbps that can be guaranteed by EXP and MLWDF rules, respectively. The MSR scheduling rule, which only considers CSI as a quality indicator, allocates all the resources to the users with favorable channel quality conditions, and those users experiencing bad channel quality conditions suffer from starvation. Hence, as it wastes resources, MSR rule is not capable of achieving queue stability and presents a very low average throughput and an infinite^{d} average delay per flow, irrespective of the average traffic arrival rate.
Except for a slight increase in delay Jain's fairness index, which is only perceptible for light or moderate traffic loads, the effect of implementing FC on the performance of EXP and MLDF scheduling rules is very small. This can be explained by the fact that, when calculating the weighting coefficients w_{ m } (t), the EXP and MLDF schedulers use QSI and thus, the performance gains provided by the introduction of the FC are just incremental. On the contrary, the performance improvement induced by the implementation of FC is considerable for the PF rule, and specially important for the MSR scheduler, which do not use QSI when calculating w_{ m } (t). In fact, even though the PF and MSR rules provide poorer Jain's fairness indexes than those delivered by the EXP and MLWDF rules, they can guarantee a 99% service coverage for average arrival rates per flow less than approximately 0.8 Mbps, which is almost the same that can be guaranteed when using the EXP scheduler.
6.3 Comparing allocation strategies
Figure 3 shows the performance metrics versus traffic load for a MIMOOFDMA system using MLWDF scheduling rule and different combinations of UPA, APA, DRA and CRA strategies, with and without FC. A set of N_{ u } = 20 RT users with the same average arrival rate has been assumed. As it can be observed, APAbased strategies improve the performance of UPAbased ones. Nevertheless, this performance improvement, although noticeable for discrete ratebased systems, becomes almost negligible when using continuous ratebased schemes. This result suggests that using AMC schemes with a large set of modulation formats combined with powerful channel codes with adaptive coding rates can make unnecessary the use of power allocation strategies.
The effect of implementing FC on the system performance metrics is practically identical irrespective of the power and rate allocation strategies implemented at the crosslayer resource allocation unit. The average throughput per flow, delay, throughput JFI and service coverage are basically unaffected, and only an improvement in delay JFI is obtained with light and moderate traffic loads. Although not shown in the graphs, when implementing APA strategies, the use of FC also introduces a decrease in power consumption. This is due to the fact that resources (power and subbands) are only allocated when necessary, that is, when there is enough information in the queues ready to be transmitted.
6.4 Comparing MIMO configurations
The effects of using different N_{ T } × N_{ R } MRT/MRC MIMO configurations on the average throughput per flow and service coverage are depicted in Figure 4. Results have been obtained for an adaptive MIMOOFDMA system using the MLWDF scheduler, uniform power and CRA strategies, without FC, to serve N_{u} = 20 RT users with the same average arrival rate. As it can be observed, increasing the number of transmit and/or receive antennas at the PHY can significantly improve the system capacity. In fact, the increase of N_{T} and/or N_{R} translates into a widening of the stability region, which proves the convenience of employing MIMO spatial diversity at the PHY to support statistical QoS for upper layer protocols. For instance, Figure 4b shows that, using this particular configuration, a single transmit/receive antenna system can only guarantee a 99% service coverage for average arrival rates per flow less than 0.45 Mbps, compared to the 0.95 or 1.5 Mbps that can be guaranteed by using 2 × 2 or 4 × 4 MIMO configurations, respectively.
6.5 Performance results for heterogeneous traffic scenarios
Figure 5 shows the performance metrics versus traffic load for a MIMOOFDMA system serving a set of heterogeneous traffic flows. The behavior of three scheduling rules, namely, MLWDF, EXP and MDU, are compared for a system implementing UPA and CRA, without FC. Simulations have been performed assuming that ${N}_{m}^{\left(\text{RT}\right)}={N}_{m}^{\left(n\text{RT}\right)}={N}_{m}^{\left(\text{BE}\right)}=10$ users are always active in the system. Furthermore, based on the required QoS of the different traffic flows, the parameters of the MDU scheduler have been set to φ_{ m } = {1, 1.5} and ${\stackrel{\u0303}{W}}_{m}=25\phantom{\rule{0.3em}{0ex}}\text{ms}$ ms for RT users, φ_{ m } = {0.6, 1} and ${\stackrel{\u0303}{W}}_{m}=500$ ms for nRT users and, finally, φ_{ m } = {0.5, 0} and ${\stackrel{\u0303}{W}}_{m}=500$ ms for BE users. As it can be observed, crosslayer scheduling and resource allocation strategies are able to fairly allocate resources among traffic classes, according to the assigned priorities χ_{ m } (t), obtained from the QoS requirements. Obviously, RT users, which exhibit stringent absolute delay requirements, tend to be allocated more resources than nRT and BE users as the arrival data rates increase. For the same reasons, nRT users are allocated more resources than BE users. The result is that, although for light traffic arrivals the three classes of service can achieve good performance figures, for moderate traffic arrivals, RT and nRT users can only maintain acceptable performance at the cost of a decrease in the performance of BE users. Furthermore, for heavy traffic arrivals, the performance of RT users can only be maintained by sacrificing that of nRT and BE users.
In this particular scenario, except for very heavy traffic arrivals, the MLWDF scheduling rule provides the best performance results in terms of average throughput per flow and service coverage at the cost of a worse behavior of the delay Jain's fairness index. The MDU scheduler provides the best results in terms of both throughput and delay Jain's fairness indexes for RT and nRT traffic classes, but such a fair behavior is obtained at the cost of service coverage. The EXP scheduler sacrifices the average throughput and delay per flow of BE users to obtain a good tradeoff between service coverage and delay Jain's fairness index.
7 Conclusions
The emergence of stateoftheart and nextgeneration wireless communications networks based on adaptive MIMOOFDMA PHY access schemes, will enable the support of a wide range of multimedia applications with heterogeneous QoS requirements. In order to optimize the resource utilization while maintaining the QoS provided to as many users as possible, these systems require of adaptive scheduling and resource allocation algorithms able to grant a proper trade off between efficiency and fairness. In this context, using tools from information and queueing theories, mathematical convex programming, and stochastic approximation, a unified framework for channel and queueaware QoSguaranteed crosslayer scheduling and resource allocation algorithms has been developed in this article. The proposed unified framework generalizes previous work on this topic by encompassing different types of traffic, different utility functions measuring user's satisfaction, uniform and adaptive power allocation, continuous and DRA, and protocols with different amounts of channel and queueawareness. System parameters and QoS requirements have been projected into utility functions, which have then been used to formulate a unified constrained utility maximization problem, whose main aim is to balance the efficiency and fairness of resource allocation. Optimal solutions for this problem have been obtained for the UPA schemes, and novel quasioptimal algorithms have been proposed for the APA strategies, exhibiting complexities that are linear in the number of resource units and users.
The proposed unified optimization framework allows for a fair performance comparison of different scheduling rules, different allocation strategies, different MRT/MRCbased MIMO configurations, and different traffic scenarios. Simulation results presented in this article have shown that:

Without FC, the EXP and MLWDF rules provide the best joint performance results, with MLWDF achieving a slightly higher throughput, lower delay and better service coverage that EXP, at the cost of lower throughput and delay fairness indexes. The PF and MSR scheduling rules, which only consider CSI as a quality indicator, fail in providing QoS. However, although implementing FC has a negligible impact on the performance of EXP and MLDF scheduling rules, the performance improvement induced by FC is remarkable for the PF and MSR schedulers.

APAbased strategies improve the performance of UPAbased ones. Nevertheless, this performance improvement, although noticeable for DRA systems, becomes almost negligible when using CRA schemes. Thus, using AMC schemes with a large set of modulation and coding formats can make unnecessary the use of power allocation strategies.

Increasing the number of transmit and/or receive antennas at the PHY translates into a widening of the stability region, proving in this way the convenience of employing MIMO spatial diversity to support statistical QoS provision to upper layer protocols.

Channel and queueaware crosslayer scheduling and resource allocation strategies can fairly allocate resources among heterogenous traffic classes, with different scheduling policies (e.g., EXP, MDU and MLWDF) providing different tradeoffs between efficiency, delay, fairness and service coverage.
Simulation results have demonstrated the validity and merits of the proposed crosslayer unified approach. However, the optimization problem treated in this article is only applicable to single cell scenarios using MRT/MRCbased MIMO techniques. Therefore, to widen its application scope, current work focusses on extending the crosslayer unified approach to distributed scheduling and resource allocation in generalized MIMOOFDMA multicellular wireless heterogeneous networks, possibly including more sophisticated MIMO techniques, one and twoway relays, shared relays, femtocells and/or clusters of coordinated BSs.
Endnotes
^{a}LTE was introduced in 3GPP Releases 8 and 9 as a major step forward for UMTSbased networks, and LTEAdvanced is the fourth generation (4G) LTE standard in 3GPP Release 10.
^{b}A scheduling algorithm is said to be throughput optimal if it can keep all the queues stable if this is at all feasible to do.
^{c}Typically, the delay Jain's fairness index is high for light traffic arrival rates because, in this case, all the flows can be served after very low average waiting times in the queues. For moderate traffic arrival rates, the variance of per flow waiting times in the queues increases and, consequently, the delay Jain's fairness index decreases. Heavy traffic arrivals tend to cause queue instability, with almost all the flows experiencing large average delays, thus producing again an increase of the delay fairness index.
^{d}The average delay per flow would be infinite if simulations were performed over an infinite period of time.
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Acknowledgements
This was supported in part by the MEC and FEDER under projects COSMOS (TEC200802422) and AM3DIO (TEC201125446), and in part by the Govern de les Illes Balears through a Ph.D. scholarship.
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Femenias, G., Dañobeitia, B. & RieraPalou, F. Unified approach to crosslayer scheduling and resource allocation in OFDMA wireless networks. J Wireless Com Network 2012, 145 (2012). https://doi.org/10.1186/168714992012145
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Keywords
 Power Allocation
 Mobile Station
 Queue Length
 Orthogonal Frequency Division Multiple Access
 Best Effort