A block diagram of the system under consideration is shown in Figure1. As it can be observed, the downlink scenario of a wireless system with a base station (BS) serving *N*_{
s
} users is considered. At the BS, there are *N*_{
s
} separate radio link level buffers that are used to queue packet arrivals corresponding to every user connected to the BS. These buffers operate in a first-in-first-out (FIFO) fashion and can store up to\overline{\mathit{Q}}=\{{\overline{Q}}^{1},\dots ,{\overline{Q}}^{{N}_{s}}\} packets, where{\overline{Q}}^{u} is the queue length of user *u*. The scheduler, based on channel state information (CSI) collected from the *N*_{
s
} users and using a time division multiplexing scheme, takes scheduling decisions to allocate transmission opportunities to active users. Adaptive transmission is performed by using an ARQ error control scheme at the DLC layer and an AMC strategy at the PHY layer. The processing unit at the DLC layer is a packet and the processing unit at the PHY layer is a frame. The link is assumed to support QoS-guaranteed traffic characterized by a maximum average packet delay{{D}_{l}}_{\text{max}} and a target link layer packet loss rate (PLR){{P}_{l}}_{\text{max}}.

The AMC scheme is assumed to have a set{\mathcal{M}}_{p}=\{0,\dots ,{M}_{p}-1\} of *M*_{
p
} *possible* transmission modes (TMs), each of which corresponding to a particular combination of modulation and coding strategies. It is assumed that when the system uses TMn\in {\mathcal{M}}_{p}, it transmits *p*_{
n
} = *b* *R*_{
n
} packets per frame, where *R*_{
n
} denotes the number of information bits per symbol used by TM *n* and *b* is a parameter that determines the number of transmitted packets per frame, which is up to the designer’s choice. For convenience, we consider that{p}_{0}<\cdots <{p}_{{M}_{p}-1}, with *p*_{0} = 0 (i.e., TM 0 corresponds to the absence of transmission) and{p}_{{M}_{p}-1}\triangleq {\mathcal{C}}_{p}. As it was shown in[9], depending on the channel conditions and the QoS requirements of the different users, some of these *M*_{
p
} *possible* TMs may be deemed *useless*, and thus, only a set\mathcal{M}=\{0,\dots ,{M}^{u}-1\} of *M*^{u} *useful* TMs will be available to the AMC scheme for user *u*. It will be assumed that when user *u* is allocated *useful* TMn\in {\mathcal{M}}^{\mathit{u}}, the system transmits *c*_{
n
} packets and, for convenience, we also consider that{c}_{0}<\cdots <{c}_{{M}^{u}-1}, with *c*_{0} ≥ 0 and{c}_{{M}^{u}-1}={\mathcal{C}}^{u}\le {\mathcal{C}}_{p}.

A Rayleigh block-fading model[17] is adopted for the propagation channel, according to which the channel is assumed to remain invariant over a time frame interval *T*_{
f
} and is allowed to vary across successive frame intervals^{a}. Perfect CSI is assumed to be available at the receiver side, and thus, an ideal frame-by-frame TM selection process is performed at the AMC controller of the receiver. Furthermore, an error-free and instantaneous ARQ feedback channel is assumed.

As in[5, 9–12], we assume that the packet generation of user *u* adheres to a discrete batch Markovian arrival process (D-BMAP). As stated by Blondia in[18], a D-BMAP can be described by substochastic matrices{\mathit{U}}_{a}^{u}, *a* = 0,1,2,…,*n*, of the order{\mathcal{A}}^{u}\times {\mathcal{A}}^{u}, with elements{u}_{a}^{u}(i,j) denoting the probability of a transition from phase *i* to phase *j* with a batch arrival of size *a* and{\sum}_{a=0}^{\infty}{\sum}_{j=1}^{{\mathcal{A}}^{u}}{u}_{a}^{u}(i,j)=1. The transition probability matrix can be obtained as{\mathit{U}}^{u}={\sum}_{a=0}^{\infty}{\mathit{U}}_{a}^{u}.

Owing to the Markovian property of the arrival process, we have *ω*^{u} = *ω*^{u}*U*^{u} and{\mathit{\omega}}^{u}{1}_{{\mathcal{A}}^{u}}=1, where *ω*^{u} denotes the D-BMAP steady-phase probability vector and{\mathbf{1}}_{{\mathcal{A}}^{u}} is an all-ones column vector of length{\mathcal{A}}^{u}. Then, the average arrival rate *λ*^{u} can be calculated as

{\lambda}^{u}={\mathit{\omega}}^{u}\sum _{a=0}^{{\mathcal{A}}^{u}-1}a\phantom{\rule{1em}{0ex}}{\mathit{U}}_{a}^{u}{\mathbf{1}}_{{\mathcal{A}}^{u}}.

(1)

It will be assumed that the average arrival rate to the DLC layer *λ*^{u} is a system parameter that can be controlled through a traffic shaping and modeling mechanism in order to comply with the QoS requirements of the system.

### 2.1 Adaptive modulation and coding

Let{\gamma}_{\nu}^{u} denote the instantaneous received signal-to-noise ratio (SNR) of user *u* at time instant *t* = *ν* *T*_{
f
}. For the assumed Rayleigh block-fading channel model,{\gamma}_{\nu}^{u} can be modeled as an exponentially distributed random variable with mean{\overline{\gamma}}^{u}=E\{{\gamma}_{\nu}^{u}\}. Given{\gamma}_{\nu}^{u}, the objective of AMC is to select the TM that maximizes the data rate while maintaining an average PER less than or equal to a prescribed value{P}_{0}^{u}. To this end, and according to[3], the entire SNR range is partitioned into a set of nonoverlapping intervals defined by the partition{\mathbf{\Gamma}}^{u,m}=\left\{0,{\gamma}_{1}^{u,m},{\gamma}_{2}^{u,m},\dots ,{\gamma}_{{M}^{u}-1}^{u,m},\infty \right\} and TM *n* will be selected when{\gamma}_{\nu}^{u}\in \left[{\gamma}_{n}^{u,m},{\gamma}_{n+1}^{u,m}\right]. In this paper, the partition **Γ**^{u,m} is obtained by using the threshold searching algorithm described in[10]. This searching algorithm has the capability to discriminate between *useful* and *useless* TMs, while guarantying that the average PER fulfills the prescribed constraint. We also assume, without loss of generality, that convolutionally coded *M*-QAM, adopted from the IEEE 802.11a standard[19], are used in the AMC pool. All possible TMs are listed in ([8], Table one).

### 2.2 Two-dimensional Markov channel modeling

Let us define the rate of change of the fading as{\delta}_{\nu}^{u}={\gamma}_{\nu -1}^{u}-{\gamma}_{\nu}^{u}. Let us also divide the ranges of{\gamma}_{\nu}^{u} and{\delta}_{\nu}^{u} into sets of nonoverlapping 2D cells defined by the partitions{\mathbf{\Gamma}}^{u,c}=\left\{0,{\gamma}_{1}^{u,c},{\gamma}_{2}^{u,c},\dots ,{\gamma}_{K-1}^{u,c},\infty \right\} and **Δ** = {−*∞*,0,*∞*}, respectively. A first-order 2D Markov channel model can now be defined where each state of the channel corresponds to one of such cells. That is, the Markov chain state of the channel at time instant *t* = *ν* *T*_{
f
} can be denoted as{\mathit{\zeta}}_{\nu}^{u}=\left({\chi}_{\nu}^{u},{\mathrm{\Delta}}_{\nu}^{u}\right), *ν* = 0,1,…,*∞*, where{\chi}_{\nu}^{u}=k if and only if{\gamma}_{k}^{u,c}<{\gamma}_{\nu}^{u}\le {\gamma}_{k+1}^{u,c}, and{\mathrm{\Delta}}_{\nu}^{u}=0 (or{\mathrm{\Delta}}_{\nu}^{u}=1) if and only if{\delta}_{\nu}^{u}<0 (or{\delta}_{\nu}^{u}\ge 0).

In our approach the partition **Γ**^{u,c} is designed assuming that the observable dummy output of our improved first-order 2D Markov model at time instant *t* = *ν* *T*_{
f
} belongs to a codebook of nominal values of SNR{\mathbf{\Psi}}^{u,c}=\left\{{\mathrm{\Psi}}_{1}^{u,c},{\mathrm{\Psi}}_{2}^{u,c},\dots ,{\mathrm{\Psi}}_{K}^{u,c}\right\}. The Max-Lloyd algorithm[20, 21], developed for the optimum design of nonuniform quantizers, is then used to determine the partition and codebook minimizing the mean square error between{\gamma}_{\nu}^{u} and the quantizer output.

### 2.3 Physical layer 2D Markov model

Based on the TM selection process used by the AMC scheme (which is defined by the partition **Γ**^{u,m}) and the first-order 2D Markov channel model (which is characterized by the partitions **Γ**^{u,c} and **Δ**), the range of{\gamma}_{\nu}^{u} is partitioned into the set of nonoverlapping intervals defined by{\mathbf{\Gamma}}^{\mathit{\text{u,m,c}}}=\left\{\left[{\gamma}_{0}^{\mathit{\text{u,m,c}}},{\gamma}_{1}^{\mathit{\text{u,m,c}}}\right)\dots \left[{\gamma}_{{N}_{\text{PHY}}^{u}\text{-}1}^{\mathit{\text{u,m,c}}},{\gamma}_{{N}_{\text{PHY}}^{u}}^{\mathit{\text{u,m,c}}}\right)\right\}, where{N}_{\text{PHY}}^{u} denotes the number of PHY states corresponding to user *u*, and\left\{{\gamma}_{1}^{\mathit{\text{u,m,c}}},\dots ,{\gamma}_{{N}_{\text{PHY}}^{u}\text{-}1}^{\mathit{\text{u,m,c}}}\right\}=\text{sort}\left(\left\{{\gamma}_{1}^{\mathit{\text{u,m}}},\dots ,{\gamma}_{{M}^{u}\text{-}1}^{\mathit{\text{u,m}}}\right\}\cup \left\{{\gamma}_{1}^{\mathit{\text{u,c}}},\dots ,{\gamma}_{K\text{-}1}^{\mathit{\text{u,c}}}\right\}\right), with{\gamma}_{0}^{\mathit{\text{u,m,c}}}=0 and{\gamma}_{{N}_{\text{PHY}}^{u}}^{\mathit{\text{u,m,c}}}=\infty. Each partition interval\left(\right)close=")">\n \n \n \n \n \n \gamma \n \n \n k\n \n \n u,m,c\n \n \n ,\n \n \n \gamma \n \n \n k\n +\n 1\n \n \n u,m,c\n \n \n \n \n \n is characterized by a particular combination of TM and channel state. As in Subsection 2.2, the range of{\delta}_{\nu}^{u} is also partitioned into the set of nonoverlapping intervals **Δ** = {−*∞*,0,*∞*}.

Using this 2D partitioning, a first-order 2D Markov model for the PHY layer of user *u* can be defined where each state corresponds to one of such 2D rectangular-shaped cells. Furthermore, the PHY layer Markov chain state at time instant *t* = *ν* *T*_{
f
} is denoted by{\mathit{\varsigma}}_{\nu}^{u}=\left({\phi}_{\nu}^{u},{\mathrm{\Delta}}_{\nu}^{u}\right), *ν* = 0,1,…,*∞*, where{\phi}_{\nu}^{u}\in \{0,\dots ,{N}_{\text{PHY}}^{u}-1\} represents the combination of TM and channel state in this frame interval and{\mathrm{\Delta}}_{\nu}^{u}\in \{0,1\} is used to denote the *up* or *down*^{b} characteristic of the instantaneous SNR over the time frame interval *t* = (*ν* − 1)*T*_{
f
}. At any time instant *t* = *ν* *T*_{
f
}, the PHY layer state can be univocally identified by an integer number{y}_{\nu}^{u}=2{\phi}_{\nu}^{u}+{\mathrm{\Delta}}_{\nu}^{u}, with{y}_{\nu}^{u}\in \{0,\dots ,2{N}_{\text{PHY}}^{u}-1\}, which can be characterized by an steady-state probability{P}_{\text{PHY}}({y}_{\nu}^{u}) and a corresponding conditional average PER{\overline{\text{PER}}}_{\text{PHY}}({y}_{\nu}^{u}). Additionally, the PHY layer FSMC will be characterized by a transition probability matrix{\mathit{H}}_{s}^{u}={\left[{H}_{i,j}^{u}\right]}_{0\le i,j\le 2{N}_{\text{PHY}}^{u}-1}, where{H}_{i,j}^{u}=\text{Pr}\{{y}_{\nu +1}^{u}=j|{y}_{\nu}^{u}=i\}. Throughout this paper, the steady-state probabilities, the conditional average PERs, and the state-transition probabilities have all been computed either numerically or by simulation.

### 2.4 Joint PHY-MAC layer Markov model

Channel-aware-only schedulers can be incorporated to the joint PHY-MAC Markov model by means of a service-vacation process[14]. When a particular user *u* is selected for transmission in a given time slot, it is said that this user PHY layer is in service, otherwise, it is said to be on vacation. The parameter *z*^{u} ∈ {0,1} is used to denote the service (*z*^{u} = 0) or vacation (*z*^{u} = 1) state. The decision wether a user *u* will be in service or vacation during the next time slot will depend on the possible PHY layer states of all users in the next time slot and on previous scheduling decisions. A *D*-step memory in the service-vacation process represents the scheduling dependence on *D* previous decisions and can be used to account for an increased degree of fairness between users.

The joint PHY-MAC layer FSMC state for user *u* at time instant *t* = *ν* *T*_{
f
} is denoted by the vector of random variables{\mathit{\iota}}_{\nu}^{u}=({z}_{\nu}^{u},{z}_{\nu -1}^{u},\dots ,{z}_{\nu -D+1}^{u},{y}_{\nu}^{u}). At any time instant *t* = *ν* *T*_{
f
}, the joint PHY-MAC layer state can be univocally identified by an integer number{n}_{\nu}^{u} with{n}_{\nu}^{u}\in \{0,\dots ,{N}_{\text{PHY-MAC}}^{u}-1\}, where{N}_{\text{PHY-MAC}}^{u}={2}^{D+1}{N}_{\text{PHY}}^{u}. The joint MAC-PHY layer will be in state{n}_{\nu}^{u} with a steady-state probability{P}_{\text{PHY-MAC}}^{u}({n}_{\nu}^{u}). Taking into account that user *u* transmits only when it is in service, the different PHY-MAC states will have a transmission rate (TR), measured in packets per slot, of{\hat{c}}_{{n}_{\nu}^{u}}={c}_{{y}_{\nu}^{u}}(1-{z}_{\nu}^{u}), where{c}_{{y}_{\nu}^{u}} is the TR characterizing PHY layer state{y}_{\nu}^{u}. Furthermore, the PHY-MAC layer FSMC will be described by a transition probability matrix{\mathit{P}}_{s}^{u}={\left[{P}_{i,j}^{u}\right]}_{0\le i,j\le {N}_{\text{PHY-MAC}}^{u}-1}, with state transition probabilities

{P}_{i,j}^{u}=\text{Pr}\left\{{n}_{\nu +1}^{u}=j|{n}_{\nu}^{u}=i\right\}

(2)

that can be analytically calculated for a significant number of scheduling schemes. Without loss of generality, these probabilities are derived in the following sections for the max-rate and proportional fair algorithms, which, in both cases, can be modeled by a service-vacation process with one-step memory (*D* = 1).