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{\bf Directions:} Work on this sheet (both sides, if needed) only;
{\bf do not turn in any supplementary sheets of paper}.  
In order to get full credit, {\bf SHOW YOUR WORK.}

{\bf 1.}  (10) In the worked-out CRC example in our handout, suppose the
message had been 1001100 instead of 1001101.  Then what would
have the resulting CRC field been?
  
{\bf 2.}  (10) One particular divisor for CRC coding has been found to be
so effective that it has its own name.  What is that name?

{\bf 3.}  (10) Say we are in some setting which uses CRC, such as HDLC or
Ethernet.  Suppose in some hapless instance a frame happens to be
received with every bit in the CRC field in error, but no errors in the
rest of the frame.  Then which of the following necessarily holds? (i)
The receiver will definitely decide that there was an error; (ii) the
receiver will definitely decide that there was no error; (iii) neither
(i) nor (ii) is necessarily true.

{\bf 4.}  In the description of GBN in Section 6.5.1 of Walrand,
the main variables are W, T and $\tau$.  

\begin{itemize}

\item [(a)] (5) To which variable (or numeric constant), if any,
does W correspond to in our handout material on GBN?  Answer using
the $\leftrightarrow$ symbol.  For example, if W corresponds to a
variable named $\gamma$, then answer $W \leftrightarrow \gamma$.  If W
corresponds to no variable (or numeric constant) in our handout, answer
$W \leftrightarrow none$.

\item [(b)]  (5) Do the same for T.

\item [(c)]  (5) Do the same for $\tau$.

\end{itemize}

{\bf 5.}  (10) {\bf Cite a specific page in our course materials}
(handouts and Walrand) which gives advice as to the proper layer (in the 
the 7-layer model) to do error-checking.

{\bf 6.}  (10) Suppose in our handout's analysis of Stop and Wait, the return
link is a separate line, with a latency of $\beta$.  What would equation
(2) change to?

{\bf 7.}  (5) Consider the context of our handout's analysis of error-free
sliding-window protocols which leads to equation (6).  Suppose N = 32
and $\alpha$ is 5.  What is the narrowest that the sender's window ever
becomes?

{\bf 8.} (10) Suppose a certain T1 line is being used to transmit data.
There seems to be a contradiction here:  Every parameter in T1 is
fixed---fixed-length frames (193 bits), fixed number of channels (24),
and so on---yet HDLC allows variable-length frames.  Resolve this
seeming contradiction.  {\bf Be very specific and clear in your
answer.}

{\bf 9.}  Consider the ``state space'' analysis of Go Back N in our
handout.

\begin{itemize}

\item [(a)]  (10) Suppose all frames are k bytes long, and let $\mu$ denote
the long-run average buffer space needed at the sender's end.  Express
$\mu$ as a function of the state probabilities $\pi_i$.

\item [(b)]  (10) Let $\eta$ denote the long-run average time between
ACKs.  Express $\eta$ in terms of p and $\alpha$.  (Hint:  This does
\underline{not} require an elaborate derivation.)


\end{itemize}

{\bf Solutions:}

{\bf 1.}  110

{\bf 2.}  CRC-CCITT

{\bf 3.}  (i)

{\bf 4a.}   N

{\bf 4b.}   none

{\bf 4c.}   1

{\bf 5.}  p.157

{\bf 6.} $\frac{1}{\alpha + \beta + 1}$

{\bf 7.}  11 (accepted 21, due to typo in notes)

{\bf 8.}  One HDLC frame will be spread across a multiple (and
variable) number of T1 frames.

{\bf 9a.} $\mu = k \sum_i \pi_i i$

{\bf 9b.}  $\eta = 1/u = \frac{1+2\alpha p}{1-p}$


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