A processor accesses main memory with an average access time

A processor accesses main memory with an average access time of T2. A smaller cache memory is interposed between the processor and main memory. The cache has a significantly faster access time of T1 < T2. The cache holds, at any time, copies of some main memory words and is designed so that the words more likely to be accessed in the near future are in the cache. Assume that the probability that the next word accessed by the processor is in the cache is H, known as the hit ratio.
a. For any single memory access, what is the theoretical speedup of accessing the word in the cache rather than in main memory?
b. Let T be the average access time. Express T as a function of T1, T2, and H. What is the overall speedup as a function of H?
c. In practice, a system may be designed so that the processor must first access the cache to determine if the word is in the cache and, if it is not, then access main memory, so that on a miss (opposite of a hit), memory access time is T1 + T2. Express T as a function of T1, T2, and H. Now calculate the speedup and compare to the result produced in part (b).

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