$10.$ Consider the SDF graph shown below:

In this figure, $\backslash ($ A$,$ B $\backslash ),$ and $\backslash ($ C $\backslash )$ are actors. Adjacent to each port is the number of tokens consumed or produced by a firing of the actor on that port, where $\backslash ($ N $\backslash )$ and $\backslash ($ M $\backslash )$ are variables with positive integer values. Assume the variables $\backslash ($ w$,$ x$,$ y $\backslash ),$ and $\backslash ($ z $\backslash )$ represent the number of initial tokens on the connection where these variables appear in the diagram. These variables have non$-$negative integer values.

Lee $\backslash$& Seshia, Introduction to Embedded Systems

EXERCISES

$($a$)$ Derive a simple relationship between $\backslash ($ N $\backslash )$ and $\backslash ($ M $\backslash )$ such that the model is consistent, or show that no positive integer values of $\backslash ($ N $\backslash )$ and $\backslash ($ M $\backslash )$ yield a consistent model.

$($b$)$ Assume that $\backslash ($ w$=$x$=$y$=0\backslash )$ and that the model is consistent and find the minimum value of $\backslash ($ z $\backslash )($as a function $\backslash ($ N $\backslash )$ and $\backslash ($ M $\backslash ))$ such that the model does not deadlock.

$($c$)$ Assume that $\backslash ($ z$=0\backslash )$ and that the model is consistent. Find values for $\backslash ($ w$,$ x $\backslash ),$ and $\backslash ($ y $\backslash )$ such that the model does not deadlock and $\backslash ($ w$+$x$+$y $\backslash )$ is minimized.

$($d$)$ Assume that $\backslash ($ w$=$x$=$y$=0\backslash )$ and $\backslash ($ z $\backslash )$ is whatever value you found in part $($b$).$ Let $\backslash ($ b$\_\{$w$\},$ b$\_\{$x$\},$ b$\_\{$y$\}\backslash ),$ and $\backslash ($ b$\_\{$z$\}\backslash )$ be the buffer sizes for connections $\backslash ($ w$,$ x$,$ y $\backslash ),$ and $\backslash ($ z $\backslash ),$ respectively. What is the minimum for these buffer sizes?