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\section{Experimentation}

  \begin{frame}{Experimentation}
    Models and analyses implemented through the Oris tool\footnote{\scriptsize http://www.oris-tool.org/}\footnote{\scriptsize L. Carnevali, L. Ridi, and E. Vicario, "A Framework for Simulation and Symbolic State Space Analysis of Non-Markovian Models", inInternational Conference on Computer Safety, Reliability, and Security (SAFECOMP), Lecture Notes in Computer Science, pp. 409-422,Springer Berlin Heidelberg, 2011.} APIs
    \begin{itemize}
      \item modelled through \textit{PO-sTPN}
      \begin{itemize}
        \item Partially Observable - stochastic Time Petri Nets
        \item workstations are \textit{state machine PNs} and \textit{1-safe}
      \end{itemize}
    \end{itemize}
    
    \begin{center}\scalebox{0.8}{\input{img/serial_workstation_PN}}\end{center}
    
    Experiments performed on a
    \begin{itemize}
      \item single core of an Intel Core i5-6600K processor
      \item with 16GB of RAM
    \end{itemize}
  \end{frame}

  \begin{frame}{Case study assembly lines}
    Sequential, alternative and cyclic workstations
    
    \vspace{0.5em}
    \begin{minipage}{0.3\textwidth}
      \begin{center}\scalebox{0.5}{\input{img/serial_workstation}}\end{center}
    \end{minipage}
    \begin{minipage}{0.325\textwidth}
      \begin{center}\scalebox{0.5}{\input{img/alternative_workstation}}\end{center}
    \end{minipage}
    \begin{minipage}{0.325\textwidth}
      \begin{center}\scalebox{0.5}{\input{img/cyclic_workstation}}\end{center}
    \end{minipage}
    
    \vspace{1em}
    \begin{minipage}{0.5\textwidth}
      \textit{Simple} assembly line
      \begin{itemize}
        \item two sequential workstations
        \item both in phase $\pl{p}_1$ at inspection
      \end{itemize}
    \end{minipage}
    \begin{minipage}{0.45\textwidth}
      \begin{center}\scalebox{0.7}{\input{img/simple_assembly_line}}\end{center}
    \end{minipage}
    
    \vspace{2em}
    \begin{minipage}{0.5\textwidth}
      \textit{Complex} assembly line
      \begin{itemize}
        \item three repetitions
        \item of sequential/alternative/cyclic ws
        \item all observed in \textit{producing}
      \end{itemize}
    \end{minipage}
    \begin{minipage}{0.45\textwidth}
      \begin{center}\scalebox{0.7}{\input{img/complex_assembly_line}}\end{center}
    \end{minipage}
  \end{frame}
  
  \subsection{Simple assembly line}
    \begin{frame}{Simple assembly line}{TTIdle}
      \begin{center}
        $F_{TTI(1,\omega)}$\\
        \colorbox{white}{\includegraphics[scale=0.6]{simple_tti_cdf}}
      \end{center}
      
      \begin{minipage}{0.45\textwidth}
        \textit{TTIdle} computed in
        \begin{itemize}
          \item 45 min for simulation
          \item 0.18 s for bounds
        \end{itemize}
      \end{minipage}
      \begin{minipage}{0.5\textwidth}
        Due to steady-state probabilities
        \begin{itemize}
          \item steady-state probability of $0.208$
          \item $79.2\%$ of runs are discarded
        \end{itemize}
      \end{minipage}
    \end{frame}
    
    \begin{frame}{Simple assembly line}{TTDone, TTIdle, TTStartNext}
      \begin{minipage}{0.3\textwidth}
        \begin{center}
          {\tiny $F_{TTD(1,\omega)}$}\\
          \colorbox{white}{\includegraphics[scale=0.25]{simple_ttd_cdf}}
        \end{center}
      \end{minipage}
      \begin{minipage}{0.3\textwidth}
        \begin{center}
          {\tiny $F_{TTI(1,\omega)}$}\\
          \colorbox{white}{\includegraphics[scale=0.25]{simple_tti_cdf}}
        \end{center}
      \end{minipage}
      \begin{minipage}{0.3\textwidth}
        \begin{center}
          {\tiny $F_{TTSN(2,\omega)}$}\\
          \colorbox{white}{\includegraphics[scale=0.25]{simple_ttsn_cdf}}
        \end{center}
      \end{minipage}
      
      \begin{minipage}{0.5\textwidth}
        \textit{TTD}, \textit{TTI} and \textit{TTSN} computed in
        \begin{itemize}
          \item 41/45/42 min for simulation
          \item 0.15/0.18/0.10 s for bounds
        \end{itemize}
      \end{minipage}
      \begin{minipage}{0.45\textwidth}
        \vspace{2em}
        Very good approximation results
        \begin{itemize}
          \item especially for \textit{independent remaining times}
        \end{itemize}
        Feasible approach
        \begin{itemize}
          \item very fast bounds evaluation
          \item compared to simulation
        \end{itemize}
      \end{minipage}
    \end{frame}
  
  \subsection{Complex assembly line}
    \begin{frame}{Complex assembly line}{TTIdle}
        \begin{center}
          $F_{TTI(5,\omega)}$\\
          \colorbox{white}{\includegraphics[scale=0.6]{complex_tti_cdf}}
        \end{center}
        
        \textit{TTIdle} computed in
        \begin{itemize}
          \item 0.123 s for bounds
        \end{itemize}
        Simulation would be too computationally expensive
    \end{frame}
  
    \begin{frame}{Complex assembly line}{TTDone, TTIdle, TTStartNext}
      \begin{minipage}{0.3\textwidth}
        \begin{center}
          {\tiny $F_{TTD(5,\omega)}$}\\
          \colorbox{white}{\includegraphics[scale=0.25]{complex_ttd_cdf}}
        \end{center}
      \end{minipage}
      \begin{minipage}{0.3\textwidth}
        \begin{center}
          {\tiny $F_{TTI(5,\omega)}$}\\
          \colorbox{white}{\includegraphics[scale=0.25]{complex_tti_cdf}}
        \end{center}
      \end{minipage}
      \begin{minipage}{0.3\textwidth}
        \begin{center}
          {\tiny $F_{TTSN(5,\omega)}$}\\
          \colorbox{white}{\includegraphics[scale=0.25]{complex_ttsn_cdf}}
        \end{center}
      \end{minipage}
      
      \vspace{4em}
      \begin{minipage}{0.5\textwidth}
        \textit{TTD}, \textit{TTI} and \textit{TTSN} computed in
        \begin{itemize}
          \item 0.126/0.123/0.75 s for bounds
        \end{itemize}
      \end{minipage}
      \begin{minipage}{0.45\textwidth}
        Scalable solution
        \begin{itemize}
          \item in a complex scenario
          \item simulation would be infeasible
        \end{itemize}
      \end{minipage}
    \end{frame}