Channel-aware MAC performance of AMC-ARQ-based wireless systems

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$\left.\left[{\gamma }_k^{\mathit{\text{u,m,c}}},{\gamma }_{k+1}^{\mathit{\text{u,m,c}}}\right.\right)$ 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$y_{\nu }^u\in \{0,\dots ,2N_{\text{PHY}}^u-1\}$, 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${\mathit{H}}_s^u={\left[H_{i,j}^u\right]}_{0\le i,j\le 2N_{\text{PHY}}^u-1}$, 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.

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).

3.1 Calculation of Pr{z _ν  = i|y _ν  = k}

where$\mathcal{U}=\left\{2,\dots ,N_s\right\}$ is the set of competitor users and c _k is the TR corresponding to y _ν  = k.

3.2 Calculation of Pr{z_ν+1 = 0,z _ν  = i|y_ν+1 = l,y _ν  = k}

5.1 Queuing Markov model-based approach

Following the work described in[9–12], the queueing process induced by both the ARQ protocol and the AMC scheme can be formulated in discrete time with one time unit equal to one frame interval. Each user’s subsystem states are observed at the beginning of each time unit. Let${\sigma }_{\nu }^u=\left(q_{\nu }^u,a_{\nu }^u,{\mathit{\iota }}_{\nu }^u\right)$ denote the user u subsystem state at time instant t = ν T _f , where$q_{\nu }^u\in \{0,\dots ,{\overline{Q}}^u\}$ denotes the queue length at this time instant,$a_{\nu }^u\in \{0,\dots ,{\mathcal{A}}^u-1\}$ represents the phase of the D-BMAP, and${\mathit{\iota }}_{\nu }^u\in \{0,\dots ,N_{\text{PHY-MAC}}^u-1\}$ represents the combination of PHY layer state and scheduling decision for user u during this frame interval. Focusing on the set of time instants t = ν T _f , ν = 0,1,…,∞, the transitions between states${\sigma }_{\nu }^u$ are Markovian. Therefore, an embedded Markov chain can be used to describe the underlying queueing process for each user u.

In previous work, we developed the embedded Markov chains describing the underlying queueing process for different AMC schemes, such as the ones described in 802.11 and 802.16 proposals, and different ARQ protocols, including infinitely persistent ARQ[9, 10], hybrid ARQ[11], and truncated hybrid ARQ[12] schemes. Using the same technique described in Subsection 2.4, the MAC layer can be incorporated to the models described in these papers and the multiuser case could be also analyzed for those systems. In this paper, as an example and without loss of generality, we have used the model developed in[10] using the transmission mode pool of the IEEE 802.11a system combined with an infinitely persistent selective repeat (SR) ARQ procedure and it has been adapted to the multiuser case. The state space of the user u embedded finite state Markov chain is${\mathcal{S}}^u={\left\{{\mathcal{S}}_{\mu }^u\right\}}_{\mu =1}^{N_s^u}$, where$N_s^u=(\overline{Q^u}+1){\mathcal{A}}^u{N}_{\text{PHY-MAC}}^u-1$ and${\mathcal{S}}_{\mu }^u=\left(q_{\mu }^u,a_{\mu }^u,{\mathit{\iota }}_{\mu }^u\right)\equiv \left(q_{\mu }^u,a_{\mu }^u,n_{\mu }^u\right)$.

for$q\in \{0,\dots ,{\overline{Q}}^u\}$,$h\in \{0,\dots ,min\{q,{\mathcal{C}}^u\}\}$, and$n_{\mu }^u,n_{{\mu }^{\prime }}^u\in \left\{0,\dots ,N_{\text{PHY-MAC}}^u-1\right\}$. The$N_{\text{PHY-MAC}}^u\times N_{\text{PHY-MAC}}^u$ terms in (10) capturing all the cases where h packets are successfully transmitted given that there are q packets in the queue before transmission can be expressed in matrix form as${\mathit{T}}_{h,q}^u=\mathcal{D}\left({\mathit{p}}_{h,q}^u\right){\mathit{P}}_s^u$, for$q\in \{0,\dots ,{\overline{Q}}^u\}$ and$h\in \{0,\dots ,min\{q,{\mathcal{C}}^u\}\}$, where${\mathit{p}}_{h,q}^u=\left(p_{h,min\left\{{\mathcal{C}}_0^u,q\right\}}^{(0)}\cdots p_{h,min\left\{{\mathcal{C}}_{N_{\text{PHY-MAC}}^u-1},q\right\}}^{(N_{\text{PHY-MAC}}^u-1)}\right)$ and$\mathcal{D}(\mathit{x})$ is used to denote a diagonal matrix with the elements of the vector x in its main diagonal. Notice that for$q\ge {\mathcal{C}}_{N_{\text{PHY-MAC}}^u-1}={\mathcal{C}}^u$, the probabilities in these matrices do not depend on q and${\mathit{T}}_{h,q}^u={\mathit{T}}_{h,{\mathcal{C}}^u}^u$.

where${\overline{\mathit{A}}}_{q,i}^u={\sum }_{a=0}^i{\mathit{A}}_{q,a}^u.$ Notice that for$q\ge {\mathcal{C}}^u$, the transition probabilities in these matrix blocks do not depend on q, and therefore, for simplicity this index can be omitted, that is,${\mathit{A}}_{q,l}^u={\mathit{A}}_l^u$ and${\overline{\mathit{A}}}_{q,l}^u={\overline{\mathit{A}}}_l^u$ for all$q\ge {\mathcal{C}}^u$.

To derive the system performance measures, we need to obtain the steady-state probability vectors corresponding to each level of the transition matrix, which can be calculated using the fact that the transition probability matrix P ^u and steady-state probability vector${\mathit{\pi }}^u=\left[{\mathit{\pi }}_0^u{\mathit{\pi }}_1^u\cdots {\mathit{\pi }}_{{\overline{Q}}^u}^u\right]$ satisfy π ^u P ^u  = π ^u along with the normalization condition${\sum }_{i=0}^{{\overline{Q}}^u}{\mathit{\pi }}_i^{u}1=1$, where 1 is a column vector of all ones with the appropriate length. To calculate π ^u , the method described by Le et al. in[5] is used to reduce the complexity in solving the matrix P ^u .

5.2 Performance measures

where$L_q^u$ denotes the average number of packets in queue of user u, which can be obtained as$L_q^u={\sum }_{q=1}^{{\overline{Q}}^u}q{\mathit{\pi }}_q^{u}1.$

5.3 Effective bandwidth/capacity-based approach

The DLC layer can also be modeled by applying the effective bandwidth/capacity-based approach[16]. The effective capacity and effective bandwidth allow the analysis of the so-called PLR bound probability. The analysis is analogous to the one developed in[10]. The effective bandwidth of the D-BMAP arrival process of user u, characterized by a transition matrix U ^u , can be calculated as[25]$E_B^u(\psi )={\psi }^{-1}log({\mathrm{{\rm Y}}}_U^u(\psi )),$ where${\mathrm{{\rm Y}}}_U^u$ is the Perron-Frobenius eigenvalue of the matrix${\mathit{U}}_{\varpi }^u=\mathcal{D}({\mathit{\varpi }}^u){\mathit{U}}^u$, with$\mathit{\varpi }\triangleq \left(e^{{\lambda }_0^u\psi },\dots ,e^{{\lambda }_{{\mathcal{A}}^u-1}^u\psi }\right)$, where${\lambda }_n^u$ denotes the number of packets per frame generated when the source of user u is in state n. The effective capacity of the service process that models the behavior of the MAC and PHY layers for the user of interest, which is characterized by a transition probability matrix${\mathit{P}}_s^u$, can be obtained as$E_C^u(\psi )=-{\psi }^{-1}log({\mathrm{{\rm Y}}}_P^u(-\psi )).$ In this expression,${\mathrm{{\rm Y}}}_P^u$ denotes the Perron-Frobenius eigenvalue of the matrix${\mathit{P}}_{{\upsilon }_{\nu }}^u=\mathcal{D}({\mathit{\upsilon }}_{\nu }^u){\mathit{P}}_s^u$, with${\mathit{\upsilon }}_{\nu }^u\triangleq \left(e^{-{\breve{c}}_0^u\psi },\dots ,e^{-{\breve{c}}_{(4N_{\text{PHY}}^u-1)}^u\psi }\right)$, where${\breve{c}}_n^u$ denotes the number of packets per frame leaving the queue when the PHY-MAC for user u is in state n, which, for an SR-ARQ infinitely persistent scheme, can be calculated as${\breve{c}}_{n^u}={\sum }_{k=0}^{{\hat{c}}_{n^u}}k{p}_{k,{\hat{c}}_{n^u}}^{(n^u)}$ with$p_{k,{\hat{c}}_{n^u}}^{(n^u)}$ defined in (9).