1.  The CPU that had been accessing module 1 finishes, then makes
another request for that module.  The CPU that had been accessing module
3 finishes, then makes a request for module 4.  The CPU that had been
accessing module 4 finishes, then makes another request for that module.
The probability is q_1 q_4^2.

2.  The product Ab of a matrix A and a vector b is a linear combination
of the columns of A, with the coefficients of that linear combination
being the elements of b.  Here, take A = ! - P' and b = pi =
(0.37,0.45,0.18).

3.  q_{ijm} = p_{ij} 1 + S_k p_{ik} q_{kj,m-1}

4.  An unnecessary replacement occurs when we receive a 0 in state
(k-1,1) but the receiver does not go down.  The long-run proportion of
the time this occurs is

pi_(k-1,1) 0.5 (1-rho)

The long-run average time between such events is the reciprocal of that
quantity.

5.

getPMat <- function(n)
{
   P <- matrix(0,nrow=n,ncol=n)
   for (i in 0:(n-2)) {
      P[i+1,1] <- 0.5
      P[i+1,i+2] <- 0.5
   }
   P[n,n] <- 0.5
   P[n,1] <- 0.5
   P
}


meanDistPerStep <- function(n)
{

   P <- getPMat(n)
   pi <- findpi1(P)

   lrAvg <- 0
   for (i in 0:(n-2))
      lrAvg <- lrAvg + pi[i+1] * (0.5*i + 0.5*1)
   lrAvg + pi[n] * (0.5*0 + 0.5*(n-1))
}