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Name: \_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_

\textbf{Directions: Work only on this sheet (on both sides, if needed); do not
turn in any supplementary sheets of paper. There is actually plenty of room
for your answers, as long as you organize yourself BEFORE starting writing.
In order to get full credit, SHOW YOUR WORK.}

{\bf 1.} (5) Consider Fig. 16.11, altered to be a 256Mx64 system, but
still using 16Mx16 chips, and still high-order interleaved.  Then
address lines A\_\_\_\_\_\_ through A\_\_\_\_\_\_, inclusive, would be
the inputs to a decoder which would have \_\_\_\_\_\_\_ outputs.

{\bf 2.} (5) Name two pairs of handshaking signals (that means four
signals in all) discussed in class.

{\bf 3.} This question concerns designing a virtual memory system,
specifically the choice of the page size.

\begin{itemize}

\item [(a)] (5) The page size is set by (i) the hardware designer, (ii) the
writer of the OS, (iii) the writer of the compiler, (iv) all of the
above.

\item [(b)] (5) It is natural to set the page size to some multiple of the
size of \_\_\_\_\_\_\_\_\_ on disk.

\end{itemize}

{\bf 4.} (10) The instruction {\bf outb} has two operands.  If we are
using it to send a command to an 8259A chip on a PC, a likely value for
the second operand is \$0x\_\_\_\_\_\_\_\_\_\_\_.  

{\bf 5.} Consider a circuit with inputs $A_i$, i = 0,1,2,3 and output
$X_j$, j = 0,1, in which $X_1X_0$ tells us the largest index i for which
$A_i$ is 1.  For example, if $A_3$ is 0 but $A_2$ is 1, then the X
output codes 2, i.e. $X_1$ is 1 and $X_0$ is 0.  If $A_3$ is 1, then the
X output codes 3, i.e. $X_1X_0$ is 11.  If the only 1 among the $A_i$ is
$A_0$, then $X_1X_0$ is 00.  We are guaranteed that there always will be
at least one $A_i$ which is 1.

\begin{itemize}

\item [(a)] (10) Show the K-maps for $X_1$ and $X_0$.  Each of these
two K-maps MUST be of the form

\begin{tabular}{|c||c|c|c|c|}
\hline
\begin{tabular}{cc}
\  & $A_1$,$A_0$ \\
$A_3$, $A_2$ & \
\end{tabular} & 00 & 01 & 11 & 10 \\ \hline
\hline
00 & - & - & - & - \\ \hline
01 & - & - & - & - \\ \hline
11 & - & - & - & - \\ \hline
10 & - & - & - & - \\ \hline
\end{tabular}

i.e. the order of the variables must be as prescribed here.

\item [(b)] (10) Use the K-maps to find minimal sum-of-product expressions
for $X_1$ and $X_0$. {\it Each SOP must be a sum of only two terms}. 

\end{itemize}

{\bf 6.} (10) Consider what can happen when there is a cache miss on a
Pentium machine, in terms of reads from and writes to memory.  To the
nearest 1,000, in the worst case \_\_\_\_\_\_\_\_ thousand bytes will be
read from memory, and \_\_\_\_\_\_\_\_ thousand bytes will be written to
memory.

{\bf 7.} Consider the PowerPC pseudo-LRU cache line replacement
policy, summarized in Fig. 17.21.

\begin{itemize}

\item [(a)] (10) Suppose the sequence of lines accessed is L1, L0, L4, L0,
L1, L5, L6, and then there is a miss.  Then the line replaced will be
one of \_\_\_\_\_\_\_\_\_. 

\item [(b)] (10) Show the analog of Table 17.7 for the four-way
associative case.

\end{itemize}

{\bf 8.} (10) Consider a machine with address size n, l cache lines, b
bytes per block, with a direct-mapped cache.  Then
\_\_\_\_\_\_\_\_\_\_\_ blocks would map to the same line.

{\bf 9.} (10) This problem concerns the error correction example on
pp.786-7.  The Hamming distance for this code is \_\_\_\_\_\_\_\_.  A
pair of legal codewords which achieves this minimum value are
\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_ and
\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_.  

{\bf Solutions:}

{\bf 1.}  The word size is 64 bits, i.e. 8 bytes, so A0-A2 are still
used to specify byte within word.

Since we are still using chips with 16M words each, we will still have
16M system words per row.  So, we still need 24 bits to specify the word
within row.  These bits will be A26-A3.

There will be 256M/16M = 16 rows, so we need 4 bits to specify row.
These bits will be A30-A27.  They will feed into the decoder, which will
have 16 outputs.

{\bf 2.}  HOLD, HLDA; DREQ, DACK

{\bf 3.a}  i

{\bf 3.b}  sectors (tracks OK too)

{\bf 4.}  20 or 21

{\bf 5.a}

$X_1$:

\begin{tabular}{|c||c|c|c|c|}
\hline
\begin{tabular}{cc}
\  & $A_1$,$A_0$ \\
$A_3$, $A_2$ & \
\end{tabular} & 00 & 01 & 11 & 10 \\ \hline
\hline
00 & d & 0 & 0 & 0 \\ \hline
01 & 1 & 1 & 1 & 1 \\ \hline
11 & 1 & 1 & 1 & 1 \\ \hline
10 & 1 & 1 & 1 & 1 \\ \hline
\end{tabular}

Take d = 0.  The expression simplifies to $A_3+A_2$.

{\bf 5.b}

$X_0$:

\begin{tabular}{|c||c|c|c|c|}
\hline
\begin{tabular}{cc}
\  & $A_1$,$A_0$ \\
$A_3$, $A_2$ & \
\end{tabular} & 00 & 01 & 11 & 10 \\ \hline
\hline
00 & d & 0 & 1 & 1 \\ \hline
01 & 0 & 0 & 0 & 0 \\ \hline
11 & 1 & 1 & 1 & 1 \\ \hline
10 & 1 & 1 & 1 & 1 \\ \hline
\end{tabular}

Again take d = 0.  The expression simplifies to $A_3+\overline{A_2}A_1$.

{\bf 6.}  The desired block is contained in some page.  (It could not
straddle two pages.)  We need to read that page.

The page table table (table pointing to page tables) is still resident,
but the page table for our process might not be resident.  In the latter
case, the page table must be read from disk (a 4K write to memory), and
if the previously-resident page to be replaced is dirty, it must be
written to disk (a 4K read from memory).  

Finally, we read the page table for our process to see where the desired
block is.  The page table might tell us that that page is not resident;
if so, that would again result in a 4K write to memory, and a 4K read
from memory if the replaced page is dirty.

So, as much as 8K would be read from memory, and as much as 8K would be
written to memory.

{\bf 7.a}  L2, L3

{\bf 7.b}

\begin{tabular}{|r|r|r|r|}
\hline
access & B0 & B1 & B2 \\ \hline  
\hline
L0 & 1 & 1 & NC \\ \hline  
L1 & 1 & 0 & NC \\ \hline  
L2 & 0 & NC & 1 \\ \hline  
L3 & 0 & NC & 0 \\ \hline  
\end{tabular}

{\bf 8.}  Of the n bits, $\log_2b$ bits specify the byte within block and
$\log_2l$ bits specify the cache line.  So, $n-\log_2b-\log_2l$ bits form the
tag.  Thus, there are 

$$
2^{n-\log_2b-\log_2l} = \frac{2^n}{bl}
$$ 

different blocks that map to the same line.

{\bf 9.}  Suppose two data strings (that's data, not the whole codeword)
differ only in D0.  Then their codewords will differ also in P4 and P8,
and thus will be a distance 3 from each other.  This is the closest
case.  So, the Hamming distance for the code is 3.

To construct an example in which this distance is attained, use this
same reasoning.  Take the first data string to consist of eight 0s, and
the second to consist of seven 0s and one 1, the 1 being in D0.  Then
the codeword for the first consists of 12 0s, while the second will have
1s in P4, P8 and D0, with 0s elsewhere.

\end{document}