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Lecture No. 41
Vincent P. Heuring & Harry F. Jordan
Computer Systems Design and Architecture
Summary
Numerical Examples related to
·
DRAM
·
Pipelining, Pre-charging and Parallelism
·
Cache
·
Hit Rate and Miss Rate
·
Access Time
Example 1
If a DRAM has 512 rows and its refresh time is 9ms, what should be the frequency of
row refresh operation on the average?
Solution
Refresh time= 9ms
Number of rows=512
Therefore we have to do 512 row refresh operations in a 9 ms interval, in other words
one row refresh operation every (9x10-3)/512 =1.76x10-5seconds.
Example 2
Consider a DRAM with 1024 rows and a refresh time of 10ms.
a. Find the frequency of row refresh operations.
b. What fraction of the DRAM's time is spent on refreshing if each refresh takes 100ns.
Solution
Total number of rows = 1024
Refresh period = 10ms
One row refresh takes place after every
10ms/1024=9.7micro seconds
Each row refresh takes 100ns, so fraction of the DRAM's time taken by row refreshes is,
100ns/9.7 micro sec= 1.03%
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Example 3
Consider a memory system having the following specifications. Find its total cost and
cost per byte of memory.
Memory type
Total bytes
Cost per byte
SRAM
256 KB
30\$ per MB
DRAM
128 MB
1\$ per MB
Disk
1 GB
10\$ per GB
Solution
Total cost of system
256 KB( ¼ MB) of SRAM costs = 30 x ¼ = \$7.5
128 MB of DRAM costs= 1 x 128= \$128
1 GB of disk space costs= 10 x 1=\$10
Total cost of the memory system
= 7.5+128+10=\$145.5
Cost per byte
Total storage= 256 KB + 128 MB + 1 GB
= 256 KB + 128x1024KB + 1x1024x1024KB
=1,179,904 KB
Total cost = \$145.5
Cost per byte=145.5/(1,179,904x1024)
= \$1.2x10-7\$/B
Example 4
Find the average access time of a level of memory hierarchy if the hit rate is 80%. The
memory access takes 12ns on a hit and 100ns on a miss.
Solution
Hit rate =80%
Miss rate=20%
Thit=12 ns
Tmiss=100ns
Average Taccess=(hit rate*Thit)+(miss rate*Tmiss)
=(0.8*12ns)+(0.2*100ns)
= 29.6ns
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Example 5
Consider a memory system with a cache, a main memory and a virtual memory. The
access times and hit rates are as shown in table. Find the average access time for the
hierarchy.
Main memory
cache
virtual memory
Hit rate
99%
80%
100%
Access time
100ns
5ns
8ms
Solution
Average access time for requests that reach the main memory
= (100ns*0.99)+(8ms*0.01)
= 80,099 ns
Average access time for requests that reach the cache
=(5ns*0.8)+(80,099ns*0.2)
=16,023.8ns
Example 6
Given the following memory hierarchy, find the average memory access time of the
complete system
Memory type
Average access time
Hit rate
SRAM
5ns
80 %
DRAM
60ns
80%
Disk
10ms
100%
Solution
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For each level, average access time=( hit rate x access time for that level) + ((1-hit rate) x
average access time for next level)
Average access time for the complete system
= (0.8x5ns) + 0.2 x((0.8x60ns) + (0.2)(1x10ms))
= 4 + 0.2(48+2000000)
=4 + 400009.6
= 400013.6 ns
Example 7
Find the bandwidth of a memory system that has a latency of 25ns, a pre charge time of
5ns and transfers 2 bytes of data per access.
Solution
Time between two memory references
=latency + pre charge time
= 25 ns+ 5ns
= 30ns
Throughput = 1/30ns
=3.33x107 operations/second
Bandwidth = 2x 3.33x107
= 6.66x107 bytes/s
Example 8
Consider a cache with 128 byte cache line or cache block size. How many cycles does it
take to fetch a block from main memory if it takes 20 cycles to transfer two bytes of data?
Solution
The number of cycles required for the complete transfer of the block
=20 x 128/2
= 1280 cycles
Using large cache lines decreases the miss rate but it increases the amount of time a
program takes to execute as obvious from the number of clock cycles required to transfer
a block of data into the cache.
Example 9
Find the number of cycles required to transfer the same 128 byte cache line if page-mode
DRAM with a CAS-data delay of 8 cycles is used for main memory. Assume that the
cache lines always lie within a single row of the DRAM, and each line lies in a different
row than the last line fetched.
Solution
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Memory requests to fetch each cache line=128/2= 64
Only the first fetch require the complete 20 cycles, and the other 63 will take only 8 clock
cycles. Hence the no. of cycles required to fetch a cache line
=20 + 8 x 63
= 524
Example 10
Consider a 64KB direct-mapped cache with a line length of 32 bytes.
a. Determine the number of bits in the address that refer to the byte within a cache
line.
b. Determine the number of bits in the address required to select the cache line.
Solution
n=log2 of number of bytes in line
m=log2 of number of lines in cache
a. For the given cache, the number of bits in the address to determine the byte
within the line= n = log232 = 5
b. There are 64K/32= 2048 lines in the given cache. The number of bits required to
select the required line = m =log22048 = 11
Hence n=5 and m=11 for this example.
Example 11
Consider a 2-way set-associative cache with 64KB capacity and 16 byte lines.
a.  How many sets are there in the cache?
b.  How many bits of address are required to select a set in the cache?
c.  Repeat the above two calculations for a 4-way set-associative cache with
same size.
Solution
a. A 64KB cache with 16 byte lines contains  4096 lines of data. In a 2-way set
associative cache, each set contains 2 lines, so there are 2048 sets in the cache.
b. Log2(2048)=11. Hence 11 bits of the address are required to select the set.
c. The cache with 64KB capacity and 16 byte line has 4096 lines of data. For a 4-
way set associative cache, each set contains 4 lines, so the number of sets in the
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cache would be 1024 and Log 2 (1024) =10. Therefore 10 bits of the address are
required to select a set in the cache.
Example 12
Consider a processor with clock cycle per instruction (CPI) = 1.0 when all memory
accesses hit in the cache. The only data accesses are loads and stores, and these constitute
60% of all the instructions. If the miss penalty is 30 clock cycles and the miss rate is
1.5%, how much faster would the processor be if all instructions were cache hits?
Solution
Without any misses, the computer performance is
CPU execution time = (CPU clock cycles + Memory stall cycles) x Clock cycle
=(IC x CPI+ 0)x Clock cycle = IC x 1.0 x Clock cycle
Now for the computer with the real cache, first we compute the number of memory stall
cycles:
Memory accesses
= IC x Instruction x Miss Rate x Miss Penalty
Memory stall cycles
= IC x (l + 0.6) x 0.015 x 30
= IC x 0.72
where the middle term (1 + 0.6) represents one instruction access and 0.6 data accesses
per instruction. The total performance is thus
CPU execution time cache = (IC x 1.0 + IC x 0.72) x Clock cycle
= 1.72 x IC x Clock cycles
The performance ratio is the inverse of the execution times
CPU execution time cache = 1.72 x IC x clock cycle
CPU execution time
1.0 x IC x clock cycle
The computer with no cache misses is 1.72 times faster
Example 13
Consider the above example but this time assume a miss rate of 20 per 1000 instructions.
What is memory stall time in terms of instruction count?
Solution
Re computing the memory stall cycles:
Memory stall cycles=Number of misses x Miss penalty
=IC * Misses  * Miss penalty
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Instruction
=IC / 1000 * Misses * Miss penalty
Instruction * 1000
=IC / 1000 * 20 * 30
= IC /1000 * 600= IC * 0.6
Example 14
What happens on a write miss?
Solution
The two options to handle a write miss are as follows:
Write Allocate
The block is allocated on a write miss, followed by the write hit actions. This is just like
No-Write Allocate
Here write misses do not affect the cache. The block is modified only in the lower level
memory.
Example 15
Assume a fully associative write-back cache with many cache entries that starts empty.
Below is a sequence of five memory operations (the address is in square brackets):
Write Mem;
Write Mem;
Write Mem;
WriteMem;
What is the number of hits and misses when using no-write allocate versus write allocate?
Solution
For no-write allocate, the address 300 is not in the cache, and there is no allocation on
write, so the first two writes will result in misses. Address 400 is also not in the cache, so
the read is also a miss. The subsequent write to address 400 is a hit. The last write to 300
is still a miss. The result for no-write allocate is four misses and one hit.
For write allocate, the first accesses to 300 and 400 are misses, and the rest are hits since
300 and 400 are both found in the cache. Thus, the result for write allocate is two misses
and three hits.
Example 16
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Which has the lower miss rate?
a 32 KB instruction cache with a 32 KB data cache or a 64 KB unified cache?
Use the following Miss per 1000 instructions.
size
Instruction
Data cache
Unified
cache
cache
32 KB
1.5
40
42.2
64 KB
0.7
38.5
41.2
Assumptions
·
The percentage of instruction references is about 75%.
·
Assume 40% of the instructions are data transfer instructions.
·
Assume a hit takes 1 clock cycle.
·
The miss penalty is 100 clock cycles.
·
A load or store hit takes 1 extra clock cycle on a unified cache if there is only one
cache port to satisfy two simultaneous requests.
· Also the unified cache might lead to a structural hazard.
· Assume write-through caches with a write buffer and ignore stalls due to the write
buffer.
What is the average memory access time in each case?
Solution
First let's convert misses per 1000 instructions into
miss rates.
Misses
Miss rate =
1000 Instructions
Memory accesses
Instruction
Since every instruction access has exactly one memory access to fetch the instruction, the
instruction miss rate is
Miss rate32 KB instruction = 1.5/1000 = 0.0015
1.00
Since 40% of the instructions are data transfers, the data miss rate is
Miss Rate 32 kb data = 40 /1000
= 0.1
0.4
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The unified miss rate needs to account for instruction and data accesses:
Miss Rate 64 kb unified = 42.2 /1000  = 0.031
1.00+ 0.4
As stated above, about 75% of the memory accesses are instruction references. Thus, the
overall miss rate for the split caches is
(75% x 0.0015) + (25% x 0.1) = 0.026125
Thus, a 64 KB unified cache has a slightly lower effective miss rate than two 16 KB
caches. The average memory access time formula can be divided into instruction and data
accesses:
Average memory access time
= % instructions x (Hit time + Instruction miss rate x Miss Penalty) + % data x (Hit time
+ Data miss rate x Miss Penalty)
Therefore, the time for each organization is:
Average memory access time split
= 75%x(l +0.0015x 100) + 25%x(l +0.1x100)
= (75% x 1.15) + (25% x 11)
= 0.8625+2.75= 3.61
Average memory access time unified
= 75% x (1+0.031 x 100) +25% x (1 + 1+0.031 x 100)
= (75% x 4.1) + (25% x 5.1) = 3.075+1.275
= 4.35
Hence split caches have a better average memory access time despite having a worse
effective miss rate. Split cache also avoids the problem of structural hazard present in a
unified cache.
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