scenario update camera ready 1

paper/sec/analysis.tex

@@ -18,29 +18,30 @@ pessimistic~\cite{chen2017probabilistic} or have a high computational
 complexity~\cite{von2018efficiently}. This limits the applicability
 of these techniques in non-trivial cases. Moreover, there are few
 works dealing with joint probabilities of consecutive jobs,
-like~\cite{tanasa2015probabilistic}, but they still suffer from
-limited scalability.
+like~\cite{tanasa2015probabilistic}, but they still \st{suffer from limited} 
+\textcolor{red}{lack of (SE RIFORMULATO COSI' POSSIAMO RISPARMIARE UNA RIGA)}scalability.
 
 To handle the scalability issue, we adopt a simulation-based
 approach, backed up by the \emph{scenario
 theory}~\cite{calafiore2006scenario}, that \emph{empirically}
 performs the uncertainty characterization, and provides
 \emph{formal guarantees} on the robustness of the resulting
-estimation. The scenario theory is capable of exploiting the fact
-that simulating the taskset execution (with statistical significance)
-is less computationally expensive than an analytical approach that
-incurs into the problem of combinatorial explosion of the different
-possible uncertainty realizations. In practice, this means that we:
-(i) randomly extract execution times from the probability
-distributions specified for each task, $f_i^{\mathcal{C}}(c)$, (ii)
-schedule the tasks, checking the resulting set of sequences $\Omega$, 
-and (iii) find the worst-case sequence $\omega_*$ based on the chosen
-cost function. The probabilities of sequences of hits and misses are
+estimation. The scenario theory \textcolor{red}{allows to exploit}
+\st{is capable of exploiting} the fact that simulating the taskset 
+execution (with statistical significance) is less computationally 
+expensive than an analytical approach that incurs into the problem of combinatorial explosion of the different possible uncertainty 
+realizations. In practice, this means that we: (i) \st{randomly 
+extract} \textcolor{red}{sample the} execution times from the 
+probability distributions specified for each 
+task, $f_i^{\mathcal{C}}(c)$, (ii) schedule the tasks, checking the 
+resulting set of sequences $\Omega$, and (iii) find the worst-case 
+sequence $\omega_*$ based on the chosen cost function. 
+The probabilities of sequences of hits and misses are
 then computed based on this sequence, and used in the design of
 the controller to be robust with respect to the sequence. We use the
-scenario theory to quantify the probability $\varepsilon$ of not
-having extracted the \emph{true} worst-case sequence and the 
-confidence in the process $1-\beta$.
+scenario theory to quantify the probability $\varepsilon$ of not having 
+extracted the \emph{true} worst-case sequence and the  confidence in the 
+process $1-\beta$ \textcolor{red}{according to the number of extracted samples}.
 
 \subsection{Scenario Theory}
 \label{sec:analysis:scenario}
@@ -53,8 +54,11 @@ for all the possible uncertainty realization might be achieved
 analytically, but is computationally too heavy or results in
 pessimistic bounds. The Scenario Theory proposes an empirical method
 in which samples are drawn from the possible realizations of
-uncertainty. It provides statistical guarantees with respect to the
-general case, provided that the sources of uncertainty are the same.
+uncertainty. \textcolor{red}{By providing a lower bound on the number of 
+samples to be drawn from the uncetainty space} it provides statistical 
+guarantees \textcolor{red}{on the value of the cost function} with 
+respect to the general case, provided that the sources of uncertainty 
+are the same. 
 
 One of the advantages of this approach is that there is no need to
 enumerate the uncertainty sources, the only requirement being the
@@ -76,9 +80,9 @@ $J_{seq}(\omega)$, that determines when we consider a sequence worse
 than another (from the perspective of the controller execution).
 Denoting with $\mu_{\text{tot}}(\omega)$ the total number of job
 skips and deadline misses that the control task experienced in
-$\omega$, and with $\mu_{\text{seq}}(\omega)$ the total number of
-consecutive deadline misses or skipped jobs in $\omega$, we use as a
-cost function
+$\omega$, and with $\mu_{\text{seq}}(\omega)$ the \st{total} 
+\textcolor{red}{maximum} number of consecutive deadline misses or 
+skipped jobs in $\omega$, we use as a cost function
 \begin{equation}\label{eq:Jseq}
   J_{seq}(\omega) = \mu_{\text{tot}}(\omega)\,\mu_{\text{seq}}(\omega)
 \end{equation}
@@ -88,8 +92,14 @@ of simulated sequences $\Omega = \{ \omega_1, \dots
 \text{arg}\,\max\limits_{\omega \in \Omega}J_{seq}(\omega)$. The
 number of simulations, $n_{\text{sim}}$ is selected based on the
 scenario approach, and provides probabilistic bounds on the
-uncertainty realization, giving us some formal guarantees on the
-design.
+uncertainty realization, giving us \st{some} formal guarantees on the
+design \textcolor{red}{according to the chosen cost function}.
+\textcolor{red}{
+The choice of the cost function is anyhow not-univocal. For instance the 
+number of sub-sequences of a given length with at least a given number of 
+deadline misses or the shortest subsequence with more than a given number 
+deadline misses would be other viable choices.
+}
 
 \subsection{Formal Guarantees}
 \label{sec:analysis:guarantees}
@@ -135,7 +145,10 @@ We simulate the system for a number $n_\text{job}$ of executions of
 the control task. Clearly, we want to select $n_\text{job}$ to cover
 an entire hyperperiod (to achieve complete analysis of the
 interferences between the tasks). In practice, we want to be able to
-detect cascaded effects, so simulations that include several 
+detect cascaded effects \textcolor{red}{that might happen due to the 
+probabilistic nature of the execution times of the tasks. Some samplings
+could in fact make the utilization of instances of the taskset greater 
+than one. For this reason} \st{, so} simulations that include several 
 hyperperiods should be performed. On top of that significancy with 
-respect the controlled of the physical system is required, hence the 
-length of the simulated sequences should cover its dynamics.
+respect the controlled of the physical system is required \textcolor{red}{(since the existance of the hyperperiod is not always guaranteed)}, hence 
+the length of the simulated sequences should cover its dynamics.