A lightweight network state estimation mechanism in ARQ-based wireless networks

Abstract

Most multimedia applications implement some kind of packet losses analysis mechanism to trigger self-adaptation actions. However, error recovery mechanisms such as ARQ variants avoid packet losses at IP level. Then, there appears an impact into the delay that may indeed result on application-level losses due to samples arriving later than the scheduled playout time. In most cases such degradations are not detected until they are so severe than even losses at IP level appear. We propose a lightweight method that achieves a finer grain estimation. We prove mathematically the capability of modified versions of statistics of delay to predict the error ratio of the wireless link and the load of the wired backhaul. After deducing also simplified heuristics for proposed method, we analyze their estimation capabilities and provide guidance for selecting appropriate parameters. Finally, we test and adjust the algorithm to an specific scenario including mobile video streaming and VoIP calls.

Appendix: Calculus of truncated e2e delay

The following calculus procedure is applied to the simplified CN scenario in Sect. 4 under the M/M/1 assumption. However, a totally equivalent one can be applied to other delay distributions either analytically or by applying inverse Laplace numerical methods. We have included such calculus here in order to obtain a closed analytical expression by using the Laplace transform, a common method for computing waiting time in queuing systems (see [16]). This way, there would be no need to compute the inverse Laplace transform of D _CN . Instead, we should compute the convolution directly.

Then, the objective of the calculus is to compute the PDF of the ideal truncated distribution (probability of truncated first retransmission section) $D'_{\mathit{AN}}=D'_{\mathit{AN}_{1}}$. Considering that $\sqcap_{\mathit{TTI}}{ (t-\frac{\mathit{TTI}}{2} )}=u(t)-u(t-\mathit{TTI})$ where u(t) is the Heaviside step function, we can obtain the Laplace transform:

Similarly for D _CN (t) (taking advantage of methods in [16]).

The convolution in time is the product in Laplace ‘s’-domain:

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$$\begin{aligned} &\quad{}+K\cdot\frac{\mathit{AN}_1[0]\cdot(1-\rho) }{\mathit{TTI}}\cdot \biggl[\frac{1}{s}- \frac {e^{-\mathit{TTI}\cdot s}}{s} \biggr] \end{aligned}$$

(22)

Considering that expression can be calculated as:

Similarly for :

Finally we obtain the whole expression for $D'_{e2e}$ considering the effect of D _TTI , considering that K, the constant to make the area of the PDF equal to 1, has the following expression:

$$K=\frac{1}{\mathit{AN}_1[0] \cdot(1-\rho\cdot e^{-A\cdot \mathit{RTT} })} $$

Then:

$$\begin{aligned} D'_{e2e}(t)&=\frac{1}{\mathit{TTI}\cdot (1-\rho\cdot e^{-A \cdot \mathit{RTT} } ) }\cdot\sqcap_{\mathit{RTT} }{ \biggl(t-\frac{\mathit{TTI}}{2} \biggr)} \\ &\quad{}- \frac{\rho}{\mathit{TTI}\cdot (1-\rho\cdot e^{-A \cdot \mathit{RTT} } ) } \\ &\quad{}\cdot \biggl[e^{-A\cdot t}\cdot \sqcap_{\mathit{RTT} }{ \biggl(t-\frac {\mathit{TTI}}{2} \biggr)} \biggr] \\ &\quad{}- \frac{\rho}{\mathit{TTI}\cdot (1-\rho\cdot e^{-A \cdot \mathit{RTT} } ) } \\ &\quad{}\cdot \bigl[ \bigl(1-e^{A\cdot \mathit{TTI}} \bigr)\cdot e^{-A\cdot t}\cdot u(t-\mathit{TTI}) \bigr] \end{aligned}$$

Therefore:

$$D'_{e2e}(t) = \left\{ \begin{array}{l} \frac{1}{\mathit{TTI}\cdot (1-\rho\cdot e^{-A \cdot \mathit{RTT} } ) }\cdot [1 - \rho\cdot e^{-A\cdot t} ] \\ \quad\forall0<t<\mathit{TTI} \\ - \frac{\rho}{\mathit{TTI}\cdot (1-\rho\cdot e^{-A \cdot \mathit{RTT} } )}\cdot [ (1-e^{A\cdot \mathit{TTI}} )\cdot e^{-A\cdot t} ] \\ \quad t \geq \mathit{TTI} \end{array} \right. $$