IE 339/429
Homework for Chapter 6
Due Friday, 2006-03-24

Draw and turn in a CTMC diagram for each problem,  in addition
to doing the analytical work that the problem specifies.

Everyone:
Problem 6.2: your CTMC diagram should be a nice neat 2-dimensional
lattice.

Problem 6.4: suppose alpha_n = 0.8 for all n>=1, then set up and solve
the birth-death system using ctmc_solve.xls, using mu=1 and lambda=0.9.
Once you have that solved, compute the average number of people in
the system.

Problem 6.13.  Hints:
part (b) asks for the proportion, which is just 1-Pr{system is full}.
It is NOT lambda*(1-Pr{system is full}); that is the rate at which
customers enter the system.
For part (c), re-solve the system using the new service rate.
Suppose each entering customer pays $10.  At what rate ($/hour) does the
barber earn money under the two scenarios (mu=4 and mu=8)?  What
is the percent increase?


Q1: We have disallowed self-transitions for convenience in forming the
R matrix.  But, let's suppose that we have a CTMC that should have
self-transitions.  For example, let the Q matrix be
Q = [ 3  2
      4 10]
(a) Find its steady-state distribution (p vector) by using View #1 of a
CTMC (as a DTMC along with a vector "nu" of rates)

(b) Find a way to convert this Q matrix to an equivalent CTMC that doesn't
have self-transitions.  Write down the R matrix.  Find the steady-state
vector (p vector) to verify that it matches part (a).



Q2: Consider the barbershop system from problem 6.13.  Suppose
the shop operates 24 hours a day, always with one server on duty.
Instead of a steady 3 per hour arrival rate as the book specifies,
we will set it to be
lambda(t) = 3 * (1 - 0.7*cos(2*pi*t/24) )
where pi=3.14159 etc. as usual, as opposed to being a probability.
The 2*pi/24 factor converts t from hours to radians.
(a) Graph the arrival rate for t=[0,1,...24]
(b) Suppose the system starts with 1 person in it at midnight.
Using the Transient CTMC spreadsheet demonstrated in class,
you will need to enter something like
=3*(1-0.7*cos(2*PI()*$B$3/24)
in the proper cells of the matrix.
Try a step size of 0.1 hours. 
Calculate and graph Pr{system empty} and Pr{system full} as a function
of time for one day.  Compare to the constant arrival rate answer.
(c) What happens if the step size is too large?

---------------------------------------------------------
Three problems for IE 429: 
Problem 6.22
Problem 6.25  (Hint: let rho_1 = lambda / mu_1, and let rho_2 = lambda/mu_2,
		just to save typing--then, how do rho_1 and rho_2 relate
		to alpha and beta?)

Note: Problem 6.22 and 6.25 end up being "guess-and-check" problems, after
you do the initial setup.  We're not asking you to derive the
steady-state distribution, just show that the guess does indeed satisfy
the steady-state balance equations (once you determine the appropriate
parameters). Each should be just a few lines of algebra.

Q3:  (again, just for IE 429)
(a) Suppose you have a DTMC that has a P matrix, as usual.  Now let
R=P-I where I is the Identity matrix.  R is now a perfectly valid
matrix for a CTMC.  How does this CTMC
relate to the original DTMC?  Are their stationary probabilities
the same?  Can you say anything else about them?
(b) If I start with an R matrix for a CTMC and then let
P=R+I (the reverse of the above), do I always get a valid P matrix
for a DTMC?  Why or why not?