Cross-Layer Scheduling and Resource Allocation inOFDMA Wireless Networks

Borja Dañobeitia and Guillem FemeniasMobile Communications Group - Universitat de les Illes Balears (UIB) - SPAIN

e-mail: {borja.danobeitia,guillem.femenias}@uib.es

Abstract—One of the fundamental requirements of state-of-the-art and next-generation OFDMA-based wireless communica-tion systems is the provision of quality of service (QoS) guar-antees. This paper addresses the design of cross-layer channel-and queue-aware scheduling and resource allocation algorithmsjointly assigning transmission data rates, bandwidth and/orpower to active users while providing support to a wide range ofmultimedia applications with heterogeneous QoS requirements.Using tools from information and queueing theories, convex op-timization, and stochastic approximation, a utility-based unifiedanalytical framework is proposed. The merits and performancebehavior of the proposed cross-layer algorithms are confirmedby a comprehensive simulation study.

I. INTRODUCTION

Orthogonal frequency division multiple access (OFDMA)has been selected as the core air interface for state-of-the-art and next-generation wireless communications standards(i.e., IEEE 802.16e/m, 3GPP-LTE and LTE-Advanced). Inthese systems, the cross-layer design of channel- and queue-aware scheduling and resource allocation algorithms becomescrucial to optimize the resource utilization while providingsupport to a wide range of multimedia broadband serviceswith heterogeneous quality of service (QoS) requirements.

Scheduling and resource allocation based on cross-layerprinciples can be regarded as a multi-objective optimizationproblem taking into account not only the system throughputbut also the transmitted power, the QoS constraints, the prioritylevels of different traffic classes and the amount of backloggeddata in the data link control (DLC) layer queues. In general,there is not a single optimal solution to a multi-objectiveoptimization problem, however, using tools from informationtheory, queueing theory, convex optimization, and stochasticapproximation, our main aim in this paper is to propose aunified framework for channel- and queue-aware QoS guar-anteed scheduling and resource allocation for heterogeneousmultiservice OFDMA wireless networks. To this end, thisstudy introduces a framework able to account for differenttypes of traffic (e.g., best effort, non-real- and real-time),different allocation strategies (e.g., continuous and discreterate allocation, uniform and adaptive power allocation), anddifferent utility functions measuring user’s satisfaction in termsof, for instance, throughput, queue length and/or service time.Channel state, physical-layer characteristics, queueing delayand/or QoS requirements are projected into utility functions

and the multi-objective optimization problem is then formu-lated as a constrained utility maximization problem.

The unified algorithmic framework adopted in this papergeneralizes results presented in, for instance, [1]–[4]. Theproposed approach is based on dual decomposition optimiza-tion [5] and stochastic approximation techniques [6] exhibitingcomplexities that are linear in the number of resource units andusers, and that achieve negligible duality gaps in numericalsimulations based on current standards-like scenarios.

II. SYSTEM MODEL AND ASSUMPTIONS

Let us consider the downlink of a time-slotted MIMO-OFDMA wireless packet access network. In this setup, a basestation (BS) with a total transmit power 𝑃𝑇 and equipped with𝑁𝑇 transmit antennas provides service to 𝑁𝑚 active mobilestations (MS), each equipped with 𝑁𝑅 receive antennas.

Transmission between the BS and active MSs is organizedin time slots of a fixed duration 𝑇𝑠, assumed to be less thanthe channel coherence time. Thus, the channel fading can beconsidered constant over the whole slot and it only variesfrom slot to slot, i.e., a slot-based block fading channel isassumed. Each of these slots consists of a fixed number 𝑁𝑜

of OFDM symbols of duration 𝑇𝑜 + 𝑇𝐶𝑃 = 𝑇𝑠/𝑁𝑜, where𝑇𝐶𝑃 is the cyclic prefix duration. Slotted transmissions takeplace over a bandwidth 𝐵, which is divided into 𝑁𝑏 orthogonalsubbands, each consisting of 𝑁𝑠𝑐 adjacent subcarriers andwith a bandwidth 𝐵𝑏 = 𝐵/𝑁𝑏 small enough to assume thatall subcarriers in a subband experience frequency flat fading.One subband in the frequency axis over one slot in the timeaxis forms a basic resource allocation unit. Active MSs andfrequency subbands in a given slot are indexed by the sets𝒩𝑚 = {1, . . . , 𝑁𝑚} and 𝒩𝑏 = {1, . . . , 𝑁𝑏}, respectively.

PHY Layer Modeling: MIMO technology provides a greatvariety of techniques to exploit the multiple propagation pathsbetween transmit and receive antennas. Notably, when channelstate information (CSI) is available at the transmitter side, thejoint use of maximum ratio transmission (MRT) and maximalratio combining (MRC) at the transmitter and receiver sides,respectively, is known to provide optimum performance in thesense of maximizing the received signal-to-noise ratio (SNR)[7]

𝛾𝑚,𝑏(𝑡) =𝑝𝑚,𝑏(𝑡)𝛿𝑚,𝑏(𝑡)

𝑁𝑠𝑐𝜎2𝜈

, (1)

978-1-4577-2028-4/11/$26.00 c⃝ 2011 IEEE

where 𝑝𝑚,𝑏(𝑡) is the power allocated to MS 𝑚 on subband𝑏 during the time slot 𝑡 (in a given subband, power isuniformly allocated to subcarriers), 𝜎2

𝜈 denotes the variance ofthe additive white Gaussian noise (AWGN) samples at each ofthe 𝑁𝑅 receive antennas, and 𝛿𝑚,𝑏(𝑡) is the largest eigenvalueof the 𝑁𝑅 × 𝑁𝑅 positive semi-definite Hermitian matrix𝑯𝑚,𝑏(𝑡)𝑯

†𝑚,𝑏(𝑡), with 𝑯𝑚,𝑏(𝑡) being the frequency-domain

MIMO channel matrix and (⋅)† denoting the transposed andcomplex conjugated matrix.

DLC Layer Modeling: At the beginning of time slot 𝑡, MS𝑚 is assumed to have 𝑄𝑚(𝑡) bits in the queue. If there are𝐴𝑚(𝑡) bits arriving during time slot 𝑡, the queue length at theend of this time slot, assuming queues of infinite capacity, canthen be expressed as

𝑄𝑚(𝑡+ 1) = 𝑄𝑚(𝑡) +𝐴𝑚(𝑡)−𝑅𝑚(𝑡)𝑁𝑜𝑇𝑜, (2)

where𝑅𝑚(𝑡) = min{𝑟𝑚(𝑡), 𝑄𝑚(𝑡)/𝑁𝑜𝑇𝑜}

with 𝑟𝑚(𝑡) denoting the data rate allocated to user 𝑚 duringtime slot 𝑡. A cross-layer resource allocation strategy that, inorder to avoid the waste of resources, selects a transmissionrate

𝑟𝑚(𝑡) ≤ 𝑄𝑚(𝑡)

𝑁𝑜𝑇𝑜,

is said to fulfill the frugality constraint (FC).Using stochastic approximations [6], a recursive estimate of

the slot-by-slot throughput can be obtained as

𝑇𝑚(𝑡+ 1) = (1− 𝛽𝑡)𝑇𝑚(𝑡) + 𝛽𝑡𝑅𝑚(𝑡). (3)

Additionally, the head-of-line (HOL) delay (or service time)can also be approximated as

𝑊HOL,𝑚(𝑡+ 1) =𝑊HOL,𝑚(𝑡) + 𝑇𝑠 −𝑅𝑚(𝑡)𝑁𝑜𝑇𝑜/𝜆𝑚. (4)

III. OPTIMIZATION VARIABLES

A. Power allocation

Let 𝒑𝑏(𝑡) = [𝑝1,𝑏(𝑡) ⋅ ⋅ ⋅ 𝑝𝑁𝑚,𝑏(𝑡)]𝑇 denote the vector of

power allocation values for subband 𝑏 and time slot 𝑡. For agiven set of constraints, the scheduling and resource allocationalgorithm will be in charge of determining the power vector

𝒑(𝑡) =[(𝒑1(𝑡))

𝑇 ⋅ ⋅ ⋅ (𝒑𝑁𝑏(𝑡))𝑇 ]𝑇

, optimizing a prescribedobjective function. In addition to determining the power allo-cation values, the resource allocation algorithms should alsoallocate subbands and transmission rates. Nevertheless, as itwill be shown next, the power allocation vector 𝒑(𝑡) can alsobe used to represent the allocation of all these resources.

B. 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 byconstraining the power allocation vectors as

𝒑𝑏(𝑡) ∈ 퓟𝑏 ≜{𝒑𝑏 ∈ ℝ

𝑁𝑚+ : 𝑝𝑚,𝑏𝑝𝑚′,𝑏 = 0, ∀ 𝑚′ ∕= 𝑚

},

where ℝ+ denotes the set of all non-negative real numbers.Hence, 𝒑(𝑡) ∈ 퓟 = 퓟1 × ⋅ ⋅ ⋅ ×퓟𝑁𝑏

⊂ ℝ𝑁𝑚𝑁𝑏+ .

C. Rate allocation

1) Discrete-rate AMC: Realistic adaptive modulation andcoding (AMC) strategies can only use a discrete set 𝒩𝑘 ={0, 1, . . . , 𝑁𝑘} of modulation and coding schemes (MCS) thatcan differ for different MSs. Each MCS is characterized by aparticular transmission rate 𝜚(𝑘)𝑚 , with 𝜚

(1)𝑚 < . . . < 𝜚𝑁𝑘

𝑚 , and𝜚(0)𝑚 = 0 denoting the case where MS 𝑚 does not transmit.

Given 𝑝𝑚,𝑏(𝑡) and 𝛿𝑚,𝑏(𝑡), (1) can be used to obtain 𝛾𝑚,𝑏(𝑡)and then use the staircase function

𝜌𝑚,𝑏(𝑡) = 𝜚(𝑖)𝑚 , Γ(𝑖)𝑚 ≤ 𝛾𝑚,𝑏(𝑡) < Γ(𝑖+1)

𝑚 ∀ 𝑖 ∈ 𝒩𝑘, (5)

to select the transmission rate, where

Γ(0)𝑚 = 0 < Γ(1)

𝑚 < ⋅ ⋅ ⋅ < Γ(𝑁𝑘−1)𝑚 < Γ(𝑁𝑘)

𝑚 = ∞are the instantaneous SNR boundaries defining the MCSintervals.

2) Continuous-rate AMC: A useful abstraction when ex-ploring rate limits is to assume that each user’s set of MCSsis infinite. In this case,

𝜌𝑚,𝑏(𝑡) =1

𝑇𝑜log2

(1 +

𝛾𝑚,𝑏(𝑡)

Λ𝑚

), (6)

where Λ𝑚 ≥ 1 represents the coding gap due to the utilizationof a practical (rather than ideal) coding scheme.

IV. PROBLEM FORMULATION

The satisfaction of MS 𝑚 at time 𝑡 can be expressedby a utility function 𝑈𝑚(𝜽𝑚(𝑡), Ω̌𝑚) [2], where 𝜽𝑚(𝑡) =

{𝜃1𝑚(𝑡), . . . , 𝜃𝑁(𝑚)

𝑧𝑚 (𝑡)} is the set of quantitative QoS measures

used to characterize the satisfaction of MS 𝑚 (e.g., throughput

𝑇𝑚(𝑡), queue length 𝑄𝑚(𝑡)) and Ω̌𝑚 = {Ω̌1𝑚, . . . , Ω̌

𝑁(𝑚)𝑦

𝑚 }is the set of QoS requirements for user 𝑚 (e.g., maximumtolerable delay �̌�𝑚, maximum tolerable error rate 𝜖𝑚). Thus,the unified utility-based cross-layer scheduling and resourceallocation scheme can be formulated as,

max𝒑(𝑡)∈퓟

𝑁𝑚∑𝑚=1

𝑈𝑚(𝜽𝑚(𝑡), Ω̌𝑚)

subject to𝑁𝑚∑𝑚=1

𝑁𝑏∑𝑏=1

𝑝𝑚,𝑏(𝑡) ≤ 𝑃𝑇 .

(7)

The first order Taylor’s expansion of 𝑈𝑚(𝜽, Ω̌𝑚) in aneighborhood of 𝜽 = 𝜽𝑚(𝑡) can be written as

𝑈𝑚(𝜽, Ω̌𝑚) ≃𝑈𝑚(𝜽𝑚(𝑡), Ω̌𝑚)

+ (𝜽 − 𝜽𝑚(𝑡))𝑇 ∇𝜽 𝑈𝑚(𝜽𝑚(𝑡), Ω̌𝑚),

where ∇𝜽 denotes the vector differential operator or gradientfunction with respect to 𝜽. Thus, using this approximation, thevariation of utility for MS 𝑚 during time slot 𝑡 is given by

𝑈𝑚(𝜽𝑚(𝑡+ 1), Ω̌𝑚)− 𝑈𝑚(𝜽𝑚(𝑡), Ω̌𝑚)

≃𝑁(𝑚)

𝑧∑𝑧=1

∂ 𝑈𝑚(𝜽𝑚(𝑡), Ω̌𝑚)

∂ 𝜃𝑧𝑚(𝑡)[𝜃𝑧𝑚(𝑡+ 1)− 𝜃𝑧𝑚(𝑡)] .

Using this result, the objective function of the cross-layer long-term optimization problem in (7) can be rewritten, as shownin [8]–[10], as the gradient-based optimization problem

max𝒑(𝑡)∈퓟

𝑁𝑚∑𝑚=1

𝑁(𝑚)𝑧∑

𝑧=1

∂ 𝑈𝑚(𝜽𝑚(𝑡), Ω̌𝑚)

∂ 𝜃𝑧𝑚(𝑡)[𝜃𝑧𝑚(𝑡+ 1)− 𝜃𝑧𝑚(𝑡)] .

Although utility functions based on QoS quantitative perfor-mance measures other than the throughput, the HOL delayand/or the queue length could be devised, most practicalutility functions are based on either one of these performancemeasures or a combination of them. In these cases, using (2)-(4), and eliminating constants not affecting the optimizationprocess, it is straightforward to show that the optimizationproblem can be rewritten as a constrained weighted sum-ratemaximization problem, that is,

max𝒑(𝑡)∈퓟

𝑁𝑚∑𝑚=1

𝑤𝑚(𝑡)𝑅𝑚(𝑡)

subject to𝑁𝑚∑𝑚=1

𝑁𝑏∑𝑏=1

𝑝𝑚,𝑏(𝑡) ≤ 𝑃𝑇 .

(8)

Max-sum-rate (MSR) rule [11]: It is a channel-awarescheduling rule that, using 𝑤𝑚(𝑡) = 1, for all 𝑚, maximizesthe system throughput

∑𝑁𝑚

𝑚=1𝑅𝑚(𝑡).Proportional fair (PF) rule [12]: This is also a channel-

aware scheduling rule aiming at maximizing the logarithmic-sum-throughput of the system, that is

∑𝑁𝑚

𝑚=1 ln(𝑇𝑚(𝑡)). Thus,the gradient-based PF scheduling algorithm is effected byusing1 𝑤𝑚(𝑡) = 1/𝑇𝑚(𝑡), for all 𝑚.

Modified largest weighted delay first (M-LWDF) rule [12]:In each time slot 𝑡, the M-LWDF scheduler aims at choosingthe best combination of queueing delay and potential transmis-sion rate, serving the users that maximize the sum of marginalutility functions with weights

𝑤𝑚(𝑡) = 𝜒𝑚(𝑡)𝛼𝑚(𝑡)/𝑟𝑚(𝑡) ∀𝑚, (9)

where 𝜒𝑚(𝑡) are arbitrary positive constants, and 𝛼𝑚(𝑡) can bethe head-of-line packet delay or the queue length for user 𝑚. Inorder to guarantee that users with absolute delay requirement�̂�𝑚 and maximum outage delay probability requirement 𝜉𝑚will be satisfied, [12] proposes to set 𝜒𝑚(𝑡) = − log(𝜉𝑚)/�̂�𝑚,providing in this way QoS differentiation between user’s flows.

Exponential (EXP) rule [13]: The EXP scheduler is alsobased on a channel- and queue-aware scheduling rule that,in each time slot 𝑡, serves the users maximizing the sum ofmarginal utility functions with weights

𝑤𝑚(𝑡) =𝜒𝑚(𝑡)

𝑟𝑚(𝑡)exp

(𝜒𝑚(𝑡)𝛼𝑚(𝑡)− 𝜒𝑊

1 +√𝜒𝑊

), (10)

for all 𝑚, with 𝜒𝑊 = 1𝑁𝑚

∑𝑁𝑚

𝑚=1 𝜒𝑚(𝑡)𝛼𝑚(𝑡).

1For incoming low-rate data flows it is quite common that for some users𝑇𝑚(𝑡) = 𝜆𝑚 no matter how good their average channel condition is; as aresult, for those users, 𝑇𝑚(𝑡) is not a good measure of the actual amount ofresources allocated to them and so, it is better to use 𝑟𝑚(𝑡).

Other scheduling rules: Although not treated in this paper,the unified cross-layer optimization approach defined in (8)can also be extended to scheduling rules such as those pro-posed in [14]–[16].

V. OPTIMIZATION FRAMEWORK

A. Uniform power allocation (UPA) without FC

A power 𝑃𝑇 /𝑁𝑏 is allocated to all subbands and, using thesubband exclusive allocation constraint, it is straightforward toshow that subband 𝑏 must be allocated to MS 𝑚∗

𝑏 satisfying2

𝑚∗𝑏 = arg max

𝑚∈𝒩𝑚

{𝑤𝑚𝑁𝑠𝑐𝜌𝑚,𝑏} , ∀𝑏. (11)

B. Adaptive power allocation (APA) without FC

In this case, let us approach problem (8) by using Lagrangeduality principles [5]. With 𝜇 denoting the Lagrange multiplierassociated with the power constraint, the Lagrangian of (8) canbe expressed as

ℒ (𝒑, 𝜇) =

𝑁𝑚∑𝑚=1

𝑤𝑚𝑟𝑚 + 𝜇

(𝑃𝑇 −

𝑁𝑚∑𝑚=1

𝑁𝑏∑𝑏=1

𝑝𝑚,𝑏

), (12)

Using the subband exclusive allocation constraint (i.e., 𝒑 ∈퓟) and the fact that the power variables are separable acrosssubbands, the dual problem can then be written as [4]

𝑔(𝒑, 𝜇) = min𝜇≥0

{max𝒑∈퓟

ℒ (𝒑, 𝜇)

}

= min𝜇≥0

{𝑁𝑏∑𝑏=1

max𝑚∈𝒩𝑚

{max

𝑝𝑚,𝑏≥0{𝑤𝑚𝑁𝑠𝑐𝜌𝑚,𝑏 − 𝜇𝑝𝑚,𝑏}

}+ 𝜇𝑃𝑇

}.

(13)

1) Optimizing the dual function over 𝒑: Continuous rateallocation (CRA): In case of using 𝜌𝑚,𝑏 as defined in (6), andfor a given value of 𝜇, the innermost maximization in (13)provides a multilevel water-filling closed-form expression forthe optimal power allocation given by

𝑝∗𝑚,𝑏 =

[𝑁𝑠𝑐𝑤𝑚

𝜇𝑇𝑜 ln 2− 𝑁𝑠𝑐Λ𝑚𝜎

2𝜈

𝛿𝑚,𝑏

]+, (14)

where [𝑥]+ ≜ max{0, 𝑥}. Furthermore, the subband 𝑏 will beallocated to MS 𝑚∗

𝑏 satisfying

𝑚∗𝑏 = arg max

𝑚∈𝒩𝑏

{𝑤𝑚𝑁𝑠𝑐𝜌

∗𝑚,𝑏 − 𝜇𝑝∗𝑚,𝑏

}, ∀𝑏. (15)

Discrete rate allocation (DRA): In this case 𝜌𝑚,𝑏 is anon-derivable discontinuous function. However, the approachproposed in [17, Chapter 3] can be applied to arrive at theoptimal solution. That is,

𝑝∗𝑚,𝑏 = 𝑁𝑠𝑐𝜎2𝜈Γ

(𝑘∗𝑚,𝑏)

𝑚 /𝛿𝑚,𝑏, (16)

where

𝑘∗𝑚,𝑏 = arg max𝑘∈𝒩𝑘

{𝑁𝑠𝑐𝑤𝑚𝜚

(𝑘)𝑚 − 𝜇𝑁𝑠𝑐𝜎

2𝜈Γ

(𝑘)𝑚 /𝛿𝑚,𝑏

}. (17)

Furthermore, as in the CRA case, given 𝜇 and 𝑝∗𝑚,𝑏, thesubband 𝑏 will be allocated to MS 𝑚∗

𝑏 satisfying (15).

2Since optimization is performed on a slot-by-slot basis, from this pointonwards the time dependence of all the variables (i.e., (𝑡)) will be dropped.

Algorithm 1 Resource allocation for UPA/APA with FC1: 𝑖 = 1; {Initialize iteration counter}2: 𝒩 (1)

𝑏 = 𝒩𝑏; {Initialize set of non allocated subbands}

3: 𝑄(1)𝑚 (𝑡) = 𝑄𝑚(𝑡) ∀𝑚; {Initialize queue lengths}

4: 𝑃(1)𝑇 (𝑡) = 𝑃𝑇 ; {Initialize available power (APA)}

5: while 𝒩 (𝑖)𝑏 ∕= ∅ and

∑𝑁𝑚𝑚=1 𝑄

(𝑖)𝑚 ∕= 0 do

6: {Allocate resources using (15), (16), (18)-(22)}7: 𝒩 (𝑖+1)

𝑏 ={𝒩 (𝑖)

𝑏 ∖ 𝑏}

; {Update non allocated subbands}

8: 𝑄(𝑖+1)𝑚∗

𝑏= 𝑄

(𝑖)𝑚∗

𝑏−𝑁𝑠𝑐𝜌𝑚∗

𝑏𝑁𝑜𝑇𝑜; {Update queue}

9: 𝑃(𝑖+1)𝑇 = 𝑃

(𝑖)𝑇 − 𝑝𝑚∗

𝑏; {Update available power (APA)}

10: 𝑖 = 𝑖+ 1; {Update iteration counter}11: end while

2) Optimizing the dual function over 𝜇: Once known theoptimal vector 𝒑∗ for a given 𝜇, the dual optimization prob-lem (13) reduces to a one dimensional convex optimizationproblem in 𝜇,

𝑔(𝜇) = min𝜇≥0

{𝑁𝑏∑𝑏=1

(𝑤𝑚𝑁𝑠𝑐𝜌

∗𝑚∗

𝑏 ,𝑏− 𝜇𝑝∗𝑚∗

𝑏 ,𝑏

)+ 𝜇𝑃𝑇

}. (18)

Using standard properties of dual optimization problems [5],[17], it can be shown that the objective function for the dualproblem is convex with respect to 𝜇, and thus, derivative-free line search methods like, for example, Golden-section orFibonacci, can be used to determine 𝜇∗.

C. UPA and APA with FC

When considering the so called frugality constraint, theobjective function in (8) can be expressed as

max𝒑∈퓟

𝑁𝑚∑𝑚=1

𝑤𝑚 min

{𝑁𝑠𝑐

𝑁𝑏∑𝑏=1

𝜌𝑚,𝑏,𝑄𝑚

𝑁𝑜𝑇𝑜

}. (19)

A novel iterative searching algorithm providing quasi-optimalsolutions to this problem is proposed in Algorithm 1. Ourapproach allocates a subband per iteration 𝑖, by assuming thatqueue length 𝑄

(𝑖)𝑚 and available transmit power 𝑃 (𝑖)

𝑇 (APAcases only) are updated by taking into account the data rateand power (APA cases only) allocated to this specific subband.When implementing UPA strategy, the subband 𝑏 must beallocated, in iteration 𝑖, to MS 𝑚∗

𝑏 satisfying

𝑚∗𝑏 = arg max

𝑚∈𝒩𝑚

𝑏∈𝒩 (𝑖)𝑏

{𝑤𝑚 min

{𝑁𝑠𝑐𝜌𝑚,𝑏(𝑡),

𝑄(𝑖)𝑚

𝑁𝑜𝑇𝑜

}}. (20)

When implementing APA strategies, 𝑃𝑇 in (18) must besubstituted by 𝑃 (𝑖)

𝑇 when optimizing over 𝜇. In the APA/CRAscheme the optimal power allocation in (14) must be

𝑝∗𝑚,𝑏 = min

{[𝑁𝑠𝑐𝑤𝑚

𝜇𝑇𝑜 ln 2− 𝑁𝑠𝑐Λ𝑚𝜎

2𝜈

𝛿𝑚,𝑏

]+, 𝜋

(𝑖)𝑚,𝑏

}, (21)

where

𝜋(𝑖)𝑚,𝑏 =

𝑁𝑠𝑐Λ𝑚𝜎2𝜈

𝛿𝑚,𝑏

(2𝑄

(𝑖)𝑚 /𝑁𝑠𝑐𝑁𝑜 − 1

)

Table I: Transmit/receive parameters

Carrier frequency (𝑓0) 2.0 GHzSystem bandwidth (𝐵) 5.6 MHzBS transmit power (𝑃𝑇 ) 37.0 dBmCell radius (𝑅) 500 mMIMO configuration (𝑁𝑇 ×𝑁𝑅) 2× 2Number of subbands (𝑁𝑏) 64Number of subcarriers per subband (𝑁𝑠𝑐) 8OFDM symbol duration, w/o CP (𝑇𝑜) 91.4286 𝜇sNumber of OFDM symbols per slot (𝑁𝑜) 20Slot duration (𝑇𝑠) 2.0571 𝑚sNoise power per subcarrier (𝜎2

𝜈 ) −163.6 dBWCoding gap (Λ𝑚) 1 (CRA), 3 (DRA)

Discrete date rates (𝜚(𝑘)𝑚 ) (bits/symbol) {0, 0.5, 1, 1.5, 2, 3, 4, 4.5}Switching thresholds (DRA) Γ

(𝑘)𝑚 = Λ𝑚(2𝜚

(𝑘) − 1)

is the minimum power required to fulfill the FC in iteration 𝑖.In the APA/DRA scheme (17) must be substituted by

𝑘∗𝑚,𝑏 = arg max

𝑘∈𝒩𝑘

𝑏∈𝒩 (𝑖)𝑏

{𝑤𝑚 min

{𝑁𝑠𝑐𝜚

(𝑘)𝑚 ,

𝑄(𝑖)𝑚

𝑁𝑜𝑇𝑜

}− 𝜇

𝑁𝑠𝑐𝜎2𝜈Γ

(𝑘)𝑚

𝛿𝑚,𝑏

}.

(22)

VI. NUMERICAL RESULTS

A single-cell downlink scenario with a BS serving a set of𝑁𝑚 MSs uniformly distributed over the whole coverage areais considered. The default system parameters are summarizedin Table I. The channel model describing the path-losses,shadowing effects and frequency-, time- and space-selectivefading has been implemented by using Stanford UniversityInterim (SUI) channel model 4 with a shadow fading standarddeviation of 6 dB.

Three traffic classes are considered, i.e., real time (RT), nonreal time (nRT) and best effort (BE), all of them following aPoisson distribution. Without loss of generality, the maximumtolerable delays (�̌�𝑚) for each traffic class have been set to100 ms (RT), 2 s (nRT) and 20 s (BE), and the outage delayprobabilities (𝜉𝑚) are 0.01 (RT), 0.01 (nRT) and 0.1 (BE).Numerical results have been obtained simulating 60 differentscenarios and transmitting 15000 slots per scenario with initialtransitory periods of 1000 slots.

A. Comparing strategies

Fig. 1 compares the throughput and delay Jain’s fairness in-dex (JFI) [18] of an MLWDF-based resource allocation systemserving 𝑁𝑚 = 20 RT users and using different combinationsof UPA, APA, CRA, and DRA, with and without FC. As itcan be observed, strategies jointly allocating rates and power(APA/CRA and APA/DRA) and fulfilling the FC provide thebest performance metrics. The performance improvement pro-vided by the use of APA-based strategies, although noticeablefor DRA algorithms, becomes almost negligible when usingCRA algorithms, thus suggesting that using a large set ofmodulation schemes with powerful channel coding strategiescan make unnecessary the use of power allocation.

0 0.5 1 1.5 2 2.50

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

Average arrival rate per flow (Mbps)

Ave

rage

thro

ughp

ut p

er fl

ow (M

bps)

MLWDF, NT=2, N

R=2, RT traffic

APA, CRA, FC offAPA, CRA, FC onUPA, CRA, FC offUPA, CRA, FC onAPA, DRA, FC offAPA, DRA, FC onUPA, DRA, FC offUPA, DRA, FC on

(a) Average throughput per flow (Mbps).

0 0.5 1 1.5 2 2.50.6

0.65

0.7

0.75

0.8

0.85

0.9

0.95

1

Average arrival rate per flow (Mbps)

Del

ay J

ain’

s fa

irnes

s in

dex

MLWDF, NT=2, N

R=2, RT traffic

APA, CRA, FC offAPA, CRA, FC onUPA, CRA, FC offUPA, CRA, FC onAPA, DRA, FC offAPA, DRA, FC onUPA, DRA, FC offUPA, DRA, FC on

(b) Delay Jain’s fairness index.

Figure 1: Metrics for differents strategies.

B. Comparing schedulers

Fig. 2 compares the throughput and delay JFI achievedwhen using MSR, PF, MLWDF and EXP scheduling rules,with and without FC, in a UPA/CRA-based system serving𝑁𝑚 = 20 RT users. Without FC, the MLWDF and EXPrules provide the best joint results in terms of throughput andQoS, with MLWDF achieving a slightly higher throughputthan EXP, at the cost of a lower delay JFI. The PF scheduler,although achieves a quite good result in terms of throughput,fails in providing QoS requirements. The MSR schedulingrule, which only considers channel state as a quality indicator,is not capable of achieving queue stability. However, whenimplementing FC, MSR and PF rules achieve results in termsof throughput that are even better than those obtained byMLWDF and EXP schedulers. Nevertheless, these schedulingrules are not able the provide similar improvements in termsof fairness.

C. Heterogeneous traffic

Fig. 3 shows the throughput, throughput JFI and servicecoverage (percentage of users fulfilling QoS requirements)obtained by MLWDF and EXP scheduling rules in a systemimplementing UPA and CRA without FC, when the offeredtraffic load of active heterogeneous traffic flows (𝑁 (𝑅𝑇 )

𝑚 =

𝑁(𝑛𝑅𝑇 )𝑚 = 𝑁

(𝐵𝐸)𝑚 = 10) increases. As it can be observed,

0 0.5 1 1.5 2 2.50

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

Average arrival rate per flow (Mbps)

Ave

rage

thro

ughp

ut p

er fl

ow (M

bps)

UPA, CRA, NT=2, N

R=2, RT traffic

MSR, FC offMSR, FC onPF, FC offPF, FC onEXP, FC offEXP, FC onMLWDF, FC offMLWDF, FC on

(a) Average throughput per flow (Mbps).

0 0.5 1 1.5 2 2.50

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Average arrival rate per flow (Mbps)

Del

ay J

ain’

s fa

irnes

s in

dex

UPA, CRA, NT=2, N

R=2, RT traffic

MSR, FC offMSR, FC onPF, FC offPF, FC onEXP, FC offEXP, FC onMLWDF, FC offMLWDF, FC on

(b) Delay Jain’s fairness index.

Figure 2: Metrics for different schedulers.

cross-layer scheduling and resource allocation strategies areable to fairly allocate resources among traffic classes, ac-cording to the assigned priorities 𝜒𝑚(𝑡), obtained from theQoS requirements. Obviously, RT users, which exhibit higherpriorities than nRT and BE users, tend to be allocated moreresources as the arrival data rates increase. However, as servicecoverage results show, more resources do not always meana greater chance of fulfilling the QoS requirements. In fact,in the simulated scenario, BE users achieve the best servicecoverage thanks to its higher tolerance to delay and outagedelay probability. It is interesting to observe that, compared toMLWDF, the EXP rule assigns higher priority to RT services,sacrificing in this way the performance of BE and nRT flows.

VII. CONCLUSIONS

A cross-layer unified framework for channel- and queue-aware QoS guaranteed scheduling and resource allocationin heterogeneous multiservice OFDMA wireless networkshas been proposed. Simulation results considering differenttypes of traffic (RT, NRT and BE), different scheduling rules(MSR, PF, MLWDF and EXP), and different power andrate allocation strategies (UPA, APA, CRA and DRA), haveshown the validity and merits of our proposal in terms ofefficiency, fairness and fulfillment of the QoS requirements.Strategies jointly allocating rates and power (APA/CRA and

0 0.5 1 1.5 2 2.50

0.5

1

1.5

2

2.5

Average arrival rate per flow (Mbps)

Ave

rage

thro

ughp

ut p

er fl

ow (M

bps)

UPA, CRA, FC off, NT=2, N

R=2

MLWDF: RTMLWDF: nRTMLWDF: BEEXP: RTEXP: nRTEXP: BE

(a) Average throughput per flow (Mbps).

0 0.5 1 1.5 2 2.50.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Average arrival rate per flow (Mbps)

Thro

ughp

ut J

ain’

s fa

irnes

s in

dex

UPA, CRA, FC off, NT=2, N

R=2

MLWDF: RTMLWDF: nRTMLWDF: BEEXP: RTEXP: nRTEXP: BE

(b) Throughput Jain’s fairness index.

0 0.5 1 1.5 2 2.50

10

20

30

40

50

60

70

80

90

100

Average arrival rate per flow (Mbps)

Ser

vice

cov

erag

e (%

)

UPA, CRA, FC off, NT=2, N

R=2

MLWDF: RTMLWDF: nRTMLWDF: BEEXP: RTEXP: nRTEXP: BE

(c) Service coverage (%).

Figure 3: Metrics with heterogenous traffic.

APA/DRA) and fulfilling FC have been shown to provide thebest performance metrics at the cost of a higher complexity.Nevertheless, since the application of APA-based strategiesto CRA algorithms only brings along a marginal improve-ment, it can be concluded that in systems where a rich setof modulation-coding combinations is available, UPA canbe deemed as quasi-optimal. The use of FC also providesan important performance improvement when implementingMSR or even PF scheduling rules, but it only yields a slightimprovement in delay JFI when implementing queue-aware

scheduling strategies such as MLDF and EXP, which havebeen shown to provide the best joint performance results interms of system efficiency and fairness.

ACKNOWLEDGMENTS

This work has been supported in part by the MEC and FEDERunder project COSMOS (TEC2008-02422/TEC) and in part by theGovern de les Illes Balears through a PhD scholarship.

REFERENCES

[1] D. Hui, V. Lau, and W. Lam, “Cross-layer design for OFDMA wirelesssystems with heterogeneous delay requirements,” IEEE Transactions onWireless Communications, vol. 6, no. 8, pp. 2872–2880, 2007.

[2] G. Song, Y. Li, and L. Cimini, “Joint channel- and queue-awarescheduling for multiuser diversity in wireless OFDMA networks,” IEEETran. Commun., vol. 57, no. 7, pp. 2109–2121, 2009.

[3] Z. Kong, Y.-K. Kwok, and J. Wang, “A Low-Complexity QoS-AwareProportional Fair Multicarrier Scheduling Algorithm for OFDM Sys-tems,” IEEE Trans. Vehic. Technol, vol. 58, no. 5, pp. 2225–2235, 2009.

[4] B. Dañobeitia and G. Femenias, “Dual Methods for Channel- and QoS-Aware Resource Allocation in MIMO-OFDMA Networks,” in IFIPWireless Days, 2010.

[5] W. Yu and R. Lui, “Dual methods for nonconvex spectrum optimizationof multicarrier systems,” IEEE Tran. Commun., vol. 54, no. 7, pp. 1310–1322, 2006.

[6] X. Wang, G. Giannakis, and A. Marques, “A unified approach toQoS-guaranteed scheduling for channel-adaptive wireless networks,”Proceedings of the IEEE, vol. 95, no. 12, pp. 2410–2431, 2007.

[7] T. Lo, “Maximum ratio transmission,” IEEE Trans. Commun., vol. 47,no. 10, pp. 1458–1461, 1999.

[8] A. Stolyar, “On the asymptotic optimality of the gradient schedulingalgorithm for multiuser throughput allocation,” The Journal of theOperational Research Society, pp. 12–25, 2005.

[9] G. Song and Y. Li, “Cross-layer optimization for OFDM wirelessnetworks-part I: theoretical framework,” IEEE Tran. Wireless Commun.,vol. 4, no. 2, pp. 614–624, 2005.

[10] ——, “Cross-layer optimization for OFDM wireless networks-part II:theoretical framework,” IEEE Tran. Wireless Commun., vol. 4, no. 2,pp. 625–634, 2005.

[11] R. Knopp and P. Humblet, “Information capacity and power control insingle-cell multiuser communications,” in IEEE International Confer-ence on Communications (ICC), vol. 1, 1995, pp. 331–335.

[12] M. Andrews, S. Borst, F. Dominique, P. Jelenkovic, K. Kumaran,K. Ramakrishnan, and P. Whiting, “Dynamic bandwidth allocationalgorithms for high-speed data wireless networks,” Bell Labs TechnicalJournal, vol. 3, no. 3, pp. 30–49, 1998.

[13] S. Shakkottai and A. Stolyar, “Scheduling for multiple flows sharing atime varying channel: the exponential rule,” Bell Labs Technical Report,2000.

[14] K. Sundaresan, X. Wang, and M. Madihian, “Scheduler design forheterogeneous traffic in cellular networks with multiple channels,” inProceedings of the Third International Conference on Wireless Internet(WICON), 2007.

[15] B. Al-Manthari, H. Hassanein, N. Ali, and N. Nasser, “Fair Class-BasedDownlink Scheduling with Revenue Considerations in Next GenerationBroadband Wireless Access Systems,” IEEE Transactions on MobileComputing, pp. 721–734, 2009.

[16] N. Zhou, X. Zhu, Y. Huang, and H. Lin, “Low complexity cross-layerdesign with packet dependent scheduling for heterogeneous traffic inmultiuser OFDM systems,” IEEE Transactions on Wireless Communi-cations, vol. 9, no. 6, pp. 1912–1923, 2010.

[17] I. C. Wong and B. Evans, Resource allocation in multiuser multicarrierwireless systems. Springer, 2008.

[18] R. Jain, D. M. Chiu, and W. Hawe, “A quantitative measure of fairnessand discrimination for resource allocation in shared systems,” DECResearch Report TR-301, 1984.