Problem $2.(35$ points$)$

Consider the SDF graph shown below:

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

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

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

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

$($d$)$ Assume that $w=x=y=0$ and $z$ is whatever value you found in part $($b$).$ Let $b_w,b_x,b_y,$ and $b_z$ be the buffer sizes for connections $w,x,y,$ and $z,$ respectively. What is the minimum for these buffer sizes?