Please solve, and explain:

$($a$)$ A scalar pipeline has $4$ pipeline stages A$,$ B$,$ C$,$ D$.$ A data item entering this pipeline must be

processed by the four units A$,$ B$,$ C$,$ and D in the order A $>$ B $>$ C $>$ D$.$ Suppose that units A$,$ B$,$ D

take $2$ clock cycle to process a data item, and unit C takes $3$ clock cycles to process a data item. How

many clock cycles does it take for this scalar pipeline to process $1000$ independent operations? What is

the speedup of this scalar pipeline?

$($b$)$ If the $3$rd stage C is increased to $2$ same$-$function units to make the scalar pipeline a superscalar

pipeline, then how many clock cycles does it take for this superscalar pipeline to process $1000$

independent operations? How many cycles does it take to process $1$ operation by this superscalar

pipeline.

$($c$)$ Instead of increasing the number of processing units at stage C to $2,$ we partition stage A into sub$-$

stages A$1$ and A$2,$ partition stage B into B$1$ and B$2,$ partition stage C into C$1,$ C$2,$ and C$3,$ and partition

stage D into D$1$ and D$2,$ with each sub$-$stage having a processing latency of $1$ cycle. Then we can view

this new pipeline as a $9-$stage scalar pipeline. For this $9-$stage scalar pipeline, how many clock cycles

does it take this scalar pipeline to process $1000$ independent operations?

$($d$)$ Suppose that there are $1000$ operations to be processed by the $9-$stage scalar pipeline in $($c$),$ the

$1000$ operations have the data dependence only between the $8-$th and the $10-$th operations in the

following way: the input to the stage B$1$ of the $10-$th operation is the output of the stage C$3$ of the $8-$th

operation. Then how many cycles does it take this scalar pipeline to process these $1000$ operations?

$($e$)$ If the dependency in $($d$)$ is detectable during software development, can you think of the software

solution method to solve eliminate the stalling caused by the dependence described in $($d$)?$

$($f$)$ Instead of further partitioning each stage into sub$-$stages as in $($c$),$ we use two processing units for

stages A$,$ B$,$ D$,$ and use three processing units for stage C$,$ obtaining a superscalar pipeline. Then how

many clock cycles does it take this superscalar pipeline to process $1000$ independent operations?