DESCRIPTION

Students:  Please keep in mind the OMSI rules.  Save your files often,
make sure OMSI fills your entire screen at all times, etc.  Remember
that clicking CopyQtoA will copy the entire question box to the answer
box.

In questions involving code which will PARTIALLY be given to you in the
question specs, you may need add new lines.  There may not be
information given as to where the lines should be inserted.

If a question includes test code, make sure to include it in your
submission.

Do not simplify R expressions.  If your answer is wrong but largely
corrected, the full expressions may show that you deserve partial
credit.

Do not answer any question with simulation code unless this is specified. 

MAKE SURE TO RUN THE CODE IN PROBLEMS INVOLVING CODE!  Hit the OMSI
Submit and Run button.


QUESTION 

(Text answer, 20 pts.) Consider the shared-memory multiprocessor
example, pp.217-219, with m = 4.  Find p_{(2,0,1,1),(2,0,0,2)}.  

QUESTION 

(Text answer, 20 pts.) Consider the ALOHA example, pp.123-124, with p =
0.4 and q = 0.3.  Find a linear combination (write the coefficients
here) of the columns of I-P' that is equal to 0.

QUESTION 

(Text answer, 20 pts.) Consider a Markov chain with transition
probabilities p_{ij}, i,j =1,2,...,r.  Let q_{ijm} denote P(T_{ij} <= m
| X_0 = i).  Here T_{ij}, as in our textbook, is the time needed to go
from state i to state j.  Write an equation for q_{ijm} in terms of the
the other q_{uvn}.

Use S to denote the "capital sigma" summing operator.  E.g.,
S_{i = 1}^3 i^2 means 1^2 + 2^2 + 3^2 = 14.

QUESTION 

(Text answer, 20 pts.) Consider the stuck-at-0 example, pp.213-217.
Find the long-run average "time" (number of bits) between unnecessary
replacements of the receiver, i.e. replacements of a nonbroken receiver.
Express your answer in terms of the elements of pi, for general k.  Use 
2-tuple notation for the state names, e.g. (3,1) for the state in which
we currently have 3 consecutive 0s and the receiver is not broken.

QUESTION -ext .R -run 'Rscript omsi_answer5.R'

(R code answer, 20 pts.)  Consider the following Markov chain.  Its
state space is the integers 0,1,2,...,n-1.  At state i, we move either
to i+1 or 0, with probability 1/2 each, i = 0,1,2,...,n-2.  At state n-1,
we either go to 0 or stay at n-1, with probability 1/2 each. 

Write a function with call form

meanDistPerStep(n)

which finds the long-run distance traveled per time step.  Test it by calling
meanDistPerStep(6) and printing the result.

findpi1 <- function(p) {
   n <- nrow(p)
   imp <- diag(n) - t(p)
   imp[n,] <- rep(1,n)
   rhs <- c(rep(0,n-1),1)
   solve(imp,rhs)
}

meanDistPerStep <- function(n) 
{
...
}

print(meanPerStep(6))