Problem 2: Duplicated Processors (3 of 4 for 24 pts)

***Answer this problem in the exam booklet**

A new multi-processor system is to be constructed using a gross (144) of microprocessor-based machines and lots of ingenuity, some of it yours.

The 144 processors can be used in one of three modes at any given time: (1) all processors in use, (2) 72 processors in use, (3) serial mode (that's one at a time for you CS majors).

Please answer the following questions regarding this potential system. Be sure to include any assumptions necessary to justify your answer.

2.1

Suppose that the original compiler is replaced by one that eliminates some of the longer instructions, thus decreasing the CPI by 40%. However, it increases the instruction count by 60%. Write an expression for $CR_{\rm new}$, the clock rate necessary to achieve the same performance as that prior to the modification.

2.2

A certain process must use serial mode 30% of the time. Give an expression for the effective speedup over serial mode if half the processors are used for 50% of the original execution time, and all processors are used for the remaining 20% of the original execution time.

2.3

Suppose that we wanted to permit an arbitrary number of processors to be used, from 1 (serial) up to n processors. Let $p_i$ denote the fraction of time spent in a mode having $i$ processors in use. Express Amdahl's law as a function of $p_i$, $i$, and $n$.

2.4

Is there some process for which a speedup of 144 feasible? Justify your answer for full credit.

***Answer to problem 2 in the exam booklet***