third.tex

   1 \emph{A vessel where some chemical reaction takes place can be cooled by
   2 inserting two different rods, each with different cooling capacities. Only one
   3 rod can be inserted at a certain time. The vessel can be in three different
   4 states:\\
   5 \textbf{no rods:} then the temperature T evolves according to the differential
   6 equation $T'=0.1T − 10.0$.\\
   7 \textbf{with rod1}: equation is $T'=0.1T − 11.2$\\
   8 \textbf{with rod2}: equation is $T'=0.1T − 12.0$ (so rod2 cools better than
   9 rod1).\\
  10 A rod will be inserted if the temperature reaches 110 degrees. A rod that is in
  11 the vessel will be removed from the vessel if the temperature is between 102
  12 and 105 degrees. When a rod is removed, it cannot be used for 20 time units.\\
  13 Initial temperature is 102 degrees. The objective is to keep the temperature
  14 between 102 and 110 degrees. If the temperature is going up to 110 degrees but
  15 no rod is yet available, the system goes into a state Overheating.}
  16
  17 \subsection*{3.a}
  18 \emph{Specify the system in \UPPAAL{}. Show that unfortunately it may reach
  19 state \tt{Overheating}. What is the chance this happens within 100 time units?}
  20
  21 Figure~\ref{fig:uppaal3} shows our \UPPAAL{} specification of the vessel and
  22 rods. It is also included in \tt{3.xml}.  Here \tt{t, r1} and \tt{r2} are
  23 simple \tt{clock}s tracking the temperature and idle times for rod1 and rod2.
  24 When querying for \tt{Pr [<=100] (<> v.overheating)} \UPPAAL{} gives a
  25 probability interval of $[0.0486043,0.148433]$ with a confidence of $0.95$. The
  26 chance of overheating in 100 time units thus lies between $\pm0\%$ and
  27 $\pm1\%$.
  28
  29 \begin{figure}[ht]
  30         \centering
  31         \includegraphics[width=.8\linewidth]{3}
  32         \caption{Modeling vessel and rods in \UPPAAL}\label{fig:uppaal3}
  33 \end{figure}
  34
  35 \subsection*{3.b}
  36 \emph{Someone forwards the idea that the fact that the \tt{Overheating}
  37 state can be reached is due to rod2 cooling too well, and proposes to replace
  38 rod2 by a rod that is similar to rod1. What do you think of this idea?}
  39
  40 At first this idea appears to make sense. If rod2 cools too well then rod1 does
  41 not have time to recover when rod2 is done cooling. When we replace rod2 with a
  42 rod which has the same cooling constant ($T'=0.1T − 11.2$) as rod1 in the model
  43 then running the same query returns an interval of $[0,0.0973938]$, so chances
  44 appear to have decreased a bit, but there still is a considerable chance of
  45 overheating.