Analysis and optimization of a cross-layer adaptation mechanism for real-time applications in wireless networks

Abstract

The theoretical analysis of a cross-layer mechanism for improving the quality of service of real-time applications in wireless networks is presented. The mechanism coordinates adaptations of the modulation order at the Physical layer and the media encoding mode at the Application layer, to improve packet loss rate, throughput and mean delay. With the use of Continuous Flow Modeling, the system is considered as a “fluid” queue with inflow and outflow rates representing its traffic generation and service rates, respectively. Each data source is modeled as a Markov chain, from the steady-state of which the optimal adaptation thresholds of the cross-layer mechanism are derived. Extensive performance evaluation results show that the optimized operation of the mechanism attains a significant performance improvement compared to both the sub-optimal case, and a legacy system, which adjusts the modulation order and encoding mode separately and independently of each other.

Appendix

Let the Markov chain that models the operation of the system employing CLEMA be in state $i=\left( d_{i}, m_{i}, x_{i}\right)$, $i \in E$. For this state, the BER thresholds based on which the probabilities of the possible transitions are derived are the following:

• 1. $th_{\varepsilon_{max}}=1-\left( \frac{1-\varepsilon_{max}} {1-\frac{L_{i}}{\alpha_{i}\cdot T_{f}}}\right) ^{\frac{1} {b_{m_{i}}}}$, so that $R_{loss}>\varepsilon_{max}$ if $BER_{i}>th_{\varepsilon_{max}}$

• 2. $th_{\varepsilon_{min}}=1-\left( \frac{1-\varepsilon_{min}} {1-\frac{L_{i}}{\alpha_{i}\cdot T_{f}}}\right) ^{\frac{1} {b_{m_{i}}}}$, so that $R_{loss}<\varepsilon_{min}$ if $BER_{i}<th_{\varepsilon_{min}}$

• 3. $th_{\delta_{low}}=1-\left( 1-\frac{\frac{L_{i}}{\alpha-{i}\cdot T_{f}}\cdot \delta_{low}}{1-\delta_{low}\cdot\left( 1-\frac{L_{i}}{\alpha_{i}\cdot T_{f}}\right) } \right)^{\frac{1}{b_{m_{i}}}}$, so that δ > δ _low if BER _i  > $th_{\delta_{low}}$

• 4. $th_{\delta_{med}}=1-\left( 1-\frac{\frac{L_{i}}{\alpha-{i}\cdot T_{f}}\cdot \delta_{med}}{1-\delta_{med}\cdot\left( 1-\frac{L_{i}}{\alpha_{i}\cdot T_{f}}\right) } \right)^{\frac{1}{b_{m_{i}}}}$, so that δ > δ _med if BER _i  > $th_{\delta_{med}}$

The state $i=\left( d_{i}, m_{i}, x_{i}\right)$, $i \in E$ is absorbing $\left( P\left( i,i\right)=1 \right)$ if the values of the thresholds δ _low , δ _med and β are such that satisfy the following conditions:

Let the state i is such that 1 < m _i  < M and 1 < d _i  < D. This state is absorbing if all the following conditions are met:

• 1. The probability of transition to a state with lower modulation order is zero: $P\left( \left( d_{i},m_{i},x_{i}\right) , \left( d_{i},m_{i}-1,x_{j}\right)\right) =0$, thus if:

  $$ P\left( BER_{i}>max\left\lbrace th_{\varepsilon_{max}},th_{\delta_{med}}\right\rbrace \right) =0. $$

• 2. The probability of transition to a state with higher modulation order is zero: $P\left( \left( d_{i},m_{i},x_{i}\right) , \left( d_{i},m_{i}+1,x_{j}\right)\right) =0$, thus if:

  • a. If the buffer load x _i is such that the percentage of the mean delay with respect to the maximum tolerable delay is higher than $\beta \, \left( x_{i}:\, \frac{\bar{S}}{S_{max}}>\beta \right)$:

    $$ P\left( BER_{i}>th_{\varepsilon_{max}},\, BER_{i}<th_{\delta_{low}}\right) =0\, \hbox{and}\, P\left( BER_{i}<th_{\varepsilon_{min}}\right) =0. $$

  • b. If the buffer load x _i is such that the percentage of the mean delay with respect to the maximum tolerable delay is lower than $\beta \, \left( x_{i}:\, \frac{\bar{S}}{S_{max}}<\beta \right)$:

    $$ P\left( BER_{i}>th_{\varepsilon_{max}},\, BER_{i}<th_{\delta_{low}}\right) =0. $$

• 3. The probability of transition to a state with lower encoding mode is zero: $P\left( \left( d_{i},m_{i},x_{i}\right) , \left( d_{i}-1,m_{i},x_{j}\right)\right) =0$, thus if

  $$ P\left( BER_{i}>th_{\varepsilon_{max}},\, BER_{i}>th_{\delta_{low}},\, BER_{i}<th_{\delta_{med}}\right) =0. $$

• 4. The probability of transition to a state with higher encoding mode is zero: $P\left( \left( d_{i},m_{i},x_{i}\right) , \left( d_{i}+1,m_{i},x_{j}\right)\right) =0$, thus if

  $$ P\left( BER_{i}<th_{\varepsilon_{min}}\right) =0,\, x_{i}:\, \frac{\bar{S}}{S_{max}}<\beta. $$

• 5. The buffer load after the time interval of T _f seconds equals x _i (see Eq. 1).

In any other case, where either m _i  = 1 or m _i  = M or d _i  = 1 or d _i  = D, the state i is absorbing if all the above conditions are met except the ones that refer to impossible transitions (for example the condition of zero probability of a transition to a state with lower modulation order in case m _i  = 1).

Due to the structure of the Markov chain and the calculation of its transition matrix, some of its states are transient. For example, although the state $i:\, \left( d_{i}, m_{i}, 0\right)$ with $\frac{\mu_{d_{i}}}{b_{m_{i}}}>\varphi$ can be considered as the initial state, the Markov chain cannot return to it due to the fact that the queue cannot empty when the inflow rate is higher that the outflow rate. Thus, in order to calculate the mean loss rate of the system modeled by the Markov chain, the problem is limited to its minimum closed set that is irreducible, as it has a finite number of states.