Part $1$: Performance Evaluation $(20$ points$)$

Three different processors $P1,P2,P3$ executing the same set of instructions. $P1$ has a $3GHz$ clock rate and a CPI of $1\mathrm{.}5.$ P$2$ has $2GHz$ clock rate and $\frac{C}{P}$I of $1\mathrm{.}2.P3$ has $4GHz$ and $\frac{C}{P}$I of $2\mathrm{.}4.$

a$.[5$ pts$]$ Which processor has the best performance for that set of instruction? Which is worst?

b$.[5$ pts$]$ Assume the CPI for these three processors remains the same as specified to run a program of $1$ billion instructions, how much time does each processors take to run this program?

c$.[5$ pts$]$ A new generation of these processors have been announced by the vendors of each processor and due to microarchitecture innovations, each decreases its CPI by $25\%(()new),$ however due to the added complexity, each processor decreases it frequency by a percentage $\%$ according to the inverse of $2x$ of the old Clock Freq new $=$ Clock Freq old $)old$ which processor has the best performance and which is worst now for this new generation of processors?

d$.[5$ pts$]$ TSMC has moved to a new process node $($a new generation of technology$)$ and each vendor moved the same design to this new technology node to implement these processors. As a result, each processor is able to run $5\frac{C}{P}$I $\%$ faster than the old clock maintaining the same CPI as before $($in c$).($ Clock ${?}_{\text{n}\text{e}\text{w}}=$ Clock CPI $($:${?}_{\text{o}\text{l}\text{d}}\}),$ which processor has the best performance, and which is worst now for these processors implemented in the new technology?