Measuring the impact of adversarial errors on packet scheduling strategies

Abstract

In this paper, we explore the problem of achieving efficient packet transmission over unreliable links with worst-case occurrence of errors. In such a setup, even an omniscient offline scheduling strategy cannot achieve stability of the packet queue, nor is it able to use up all the available bandwidth. Hence, an important first step is to identify an appropriate metric to measure the efficiency of scheduling strategies in such a setting. To this end, we propose an asymptotic throughput metric which corresponds to the long-term competitive ratio of the algorithm with respect to the optimal. We then explore the impact of the error detection mechanism and feedback delay on our measure. We compare instantaneous with deferred error feedback, which requires a faulty packet to be fully received in order to detect the error. We propose algorithms for worst-case adversarial and stochastic packet arrival models, and formally analyze their performance. The asymptotic throughput achieved by these algorithms is shown to be close to optimal by deriving lower bounds on the metric and almost matching upper bounds for any algorithm in the considered settings. Our collection of results demonstrate the potential of using instantaneous feedback to improve the performance of communication systems in adverse environments.

Appendix

Lemma 4

When $\eta p{\ell _{\min }} \le \frac{\overline{\gamma }}{\gamma }$ it holds that $\frac{(t_{i^*}\eta p - \overline{\gamma }){\ell _{\min }} + \frac{t_j-t_{i^*}}{{\ell _{\max }}}\overline{\gamma }{\ell _{\min }}}{t_j} \ge \frac{(t_j\eta p - \overline{\gamma }){\ell _{\min }}}{t_j}$.

Proof

Let us assume first the case of $i^*<j$. This means that:

$$\begin{aligned}&\eta p{\ell _{\min }} \le \frac{\overline{\gamma }}{\gamma } = \frac{(j-i^*)\overline{\gamma }}{(j-i^*)\gamma } =\frac{(j-i^*)\overline{\gamma }{\ell _{\min }}}{(j-i^*){\ell _{\max }}}\\&\quad \Rightarrow \eta p{\ell _{\min }} (i^*-j){\ell _{\max }} + (j-i^*)\overline{\gamma }{\ell _{\min }} \ge 0\\&\quad \Rightarrow i^*{\ell _{\max }} \eta p{\ell _{\min }} + (j - i^*)\overline{\gamma }{\ell _{\min }} \ge j{\ell _{\max }} \eta p{\ell _{\min }} \\&\quad \Rightarrow t_{i^*}\eta p {\ell _{\min }} + \frac{t_j-t_{i^*}}{{\ell _{\max }}}\overline{\gamma }{\ell _{\min }} \ge t_j\eta p{\ell _{\min }} \\&\quad \Rightarrow (t_{i^*}\eta p - \overline{\gamma }){\ell _{\min }} + \frac{t_j-t_{i^*}}{{\ell _{\max }}}\overline{\gamma }{\ell _{\min }} \ge (t_j\eta p - \overline{\gamma }){\ell _{\min }}. \end{aligned}$$

What is more, for the case when $i^* = j$, we have that:

$$\begin{aligned}&(t_{i^*}\eta p - \overline{\gamma }){\ell _{\min }} = (t_j\eta p - \overline{\gamma }){\ell _{\min }} \\&\quad \Rightarrow (t_{i^*}\eta p - \overline{\gamma }){\ell _{\min }} + \frac{t_j-t_{i^*}}{{\ell _{\max }}}\overline{\gamma }{\ell _{\min }} \ge (t_j\eta p - \overline{\gamma }){\ell _{\min }}. \end{aligned}$$

Both cases conclude to the same, which proves the lemma. $\square$

Lemma 5

When $\eta p{\ell _{\min }} > \frac{\overline{\gamma }}{\gamma }$ it holds that $\frac{(t_{i^*}\eta p - \overline{\gamma }){\ell _{\min }} + \frac{t_j-t_{i^*}}{{\ell _{\max }}}\overline{\gamma }{\ell _{\min }}}{t_j} \ge \frac{\frac{(t_j - t_{\sqrt{j}})}{{\ell _{\max }}}\overline{\gamma }{\ell _{\min }}}{t_j}$.

Proof

When $\eta p{\ell _{\min }} > \frac{\overline{\gamma }}{\gamma }$, the following is also true:

$$\begin{aligned} \eta p{\ell _{\min }} \ge \frac{\overline{\gamma }}{\gamma } + \frac{(1 - \sqrt{j})\overline{\gamma }}{i^*\gamma }. \end{aligned}$$

This means that:

$$\begin{aligned}&\eta p{\ell _{\min }} \ge \frac{(1 + i^* - \sqrt{j})\overline{\gamma }{\ell _{\min }}}{i^*{\ell _{\max }}}\\&\quad \Rightarrow \eta p{\ell _{\min }} i^*{\ell _{\max }} + \overline{\gamma }{\ell _{\min }} (j - i^* - 1 - j + \sqrt{j}) \ge 0\\&\quad \Rightarrow i^*{\ell _{\max }} \eta p{\ell _{\min }}- \overline{\gamma }{\ell _{\min }} + (j - i^*)\overline{\gamma }{\ell _{\min }} \ge (j - \sqrt{j})\overline{\gamma }{\ell _{\min }} \\&\quad \Rightarrow (t_{i^*}\eta p - \overline{\gamma }){\ell _{\min }} + \frac{t_j-t_{i^*}}{{\ell _{\max }}}\overline{\gamma }{\ell _{\min }} \ge \frac{(t_j - t_{\sqrt{j}})}{{\ell _{\max }}}\overline{\gamma }{\ell _{\min }}. \end{aligned}$$

$\square$