Follow the links to see homework assignments.
Homework #1: (Due Thursday, March 31,
2011.)
This is a reading and computer exploration assignment.
There is nothing to turn in for this assignment.
_________________________________________
Problems are from S. Karlin
and H. Taylor (An Introduction to
Stochastic Modeling, 3rd edition). The convention here is that
II.1.E1 refers to Exercise 1 of section 1 of Chapter II. While II.3.P4 refers to
Problem 4 of section 3 of Chapter II.
Homework #2: (Due Thursday, April 7,
2011.)
 Look at but do not hand in: V.2.E1, V.2.P8

 Hand in:
∙ V.1.E1, V.1.E4, V.1.E9, V.1.P2, V.1.P6, V.1.P7
∙ V.2.P4, V.2.P5
∙ V.3.E8, V.3.E9*, V.3.P9
*You only need find the distribution of W_{r} using the ideas in the
last two lines of the problem. You did the first part of the problem
last quarter. 
_________________________________________
Homework #3: (Due Thursday,
April 14, 2011.)
 Look at but do not hand in: V.6.E2

 Hand in:
∙ V.4.E2, V.4.E3, V.4.E5, V.4.P4, V.4.P8
∙ V.5.E3, V.5.P4*, V.5.P6**
∙ V.6.E3, V.6.E5, V.6.P9 
* Suggestion: Let M>0 be a large
number. First find the distribution of the number, K_{M}, of spheres with
centers at most M  units from the origin which cover the origin. Then take the
limit as M> infinity.
** This problem really wants us to fix a
point in space, say 0, and then compute the distribution of R  the distance to
the star nearest 0.
_________________________________________
Test #1: Monday 04/18/2011
(Will cover Poisson Processes and
related material.)
_________________________________________
Homework #4: (Due Thursday,
April 21, 2011.)
 Read and hand in the 5 exercises in the Matrix
Exponential work sheet: Matrix
Exponential Work Sheet. 
 Read through VI.1 of Karlin and Taylor, pages
333340.

 Play around with the Excel sheet
continuous time homogeneous
Markov Chain with 4 sites, either use; 
Continuous MC.xls or
Continuous MC.xlsx.
_________________________________________
Homework #5: (Due Thursday,
April 28, 2011.)
 Hand in:
(p. 342 ): VI.1.E1, VI.1.E2, VI.1.E5, VI.1.P3,
VI.1.P5*, VI.1.P8**
(p. 353 ): VI.2.E1, VI.2.P2*** 
* Please show directly (without appeal to the general theory) that W₁ and
W₂W₁ are independent exponentially distributed random variables by computing
P(W₁>t and W₂W₁>s) for all s,t>0.
**Hint: you can save some work using what we already have seen about two state
Markov chains, see the notes or sections VI.3 or VI.6 of the book.
*** Depending on how you choose to do this problem you may find one of the ODE
results in the lecture notes useful.
_________________________________________
Homework #6: (Due Thursday,
May 5, 2011.)
 Hand in:
(p. 365 ): VI.3.E3, VI.3P3, VI.3P4
(p. 377 ): VI.4.E2, VI.4E4, VI.4P1, VI.4.P3, VI.4P4, VI.4P6
(p. 405 ): VI.6.P2, VI.6.P4** 
** Another badly worded
problem. Moreover, the process in VI.6.P4
is not going to be a Markov process unless we choose the
interpretation of the problem carefully. So please modify the problem so
as to;
Assume the service center can fix both machines simultaneously when
necessary, i.e. when both are broken. The y value now represents
the number of severely broken machines.
For example the state (0,1) now means both machines are broken with one being
severely broken. Both of these machines are being worked on
simultaneously. The chain can now transition to (1,1) if the lightly
broken machine is fixed first and to (1,0) if the severely broken machine
is fixed first.
_________________________________________
Homework #7: (Due Thursday,
May 12, 2011.)
 Hand in:
(p. 392 ):
VI.5.5.P2
(p. 424426): VII.1.E2, VII.1.E3, VII.1.P1 (second part only, i.e.
the part starting with; Carry out ...)
(p. 431432): VII.2 .E1
(p. 435437): VII.3.E1*, VII.3.E3**, VII.3.P2 
*Write the event {N(t)=n and W_{N(t)+1}>t+s} purely in terms of the Poisson
process, N. Then use your knowledge of N in order to do the computations.
**Use facts you know about Poisson processes and make use of VII.3.E1.
_________________________________________
Test #2: Wednesday, May 18,
2011.
(Will cover continuous time Markov
Chains and just a bit of Renewal theory.)
_________________________________________
Homework #8: (Due Thursday,
May 19, 2011.)
 Study for the test on Wednesday and start your final review. No
other homework is due this week. 
_________________________________________
Homework #9: (Due Thursday,
May 26, 2011.)
 Hand in:
(p. 424426): VII.1.P3
(p. 435437): VII.3.P4
(p. 445447): VII.4.E2, VII.4.E3, VII.4.E5, VII.4.P1, VII.4.P5*
(p. 455457): VII.5.E1, VII.5.P1 
*See comments in the lecture notes.
 Look at but do not hand in:
(p. 455457): VII.5.P4 
_________________________________________
Homework #10: (Due Thursday,
June 2, 2011.)
The remaining problems are from Chapter 24 (p.233 237) of the
lecture notes: Final Version of
Notes
 Hand in from Lecture Notes (Final
Version of Notes): Exercises: 24.1  24.4
and 24.8  24.10. (The latest version of the notes now contains
the correct formula in Exercise 24.2.) 
_________________________________________
Final Exam,
Thursday 06/9/2011 