IEOR 6711: Stochastic Models, I: Final Exam Fall 2007, SOLUTION NOTES
1. The GIN Barbershop (20 points) Ioannis Giannakakis, Gisli Ingimundarson and Behzad Nouri have joined together to form the GIN barbershop. The GIN barbershop has room for at most five
function v = stationary(P) % % This is a MATLAB function that calculates the stationary probability vector v % of a Markov chain transition matrix P, i.e., we solve v = v*P . % We assume the existence of a unique stationary vector. % For a finite-state Ma
function v = stat(P) % % This is a MATLAB function that calculates the stationary probability vector v % of a Markov chain transition matrix P, i.e., we solve v = v*P . % We assume the existence of a unique stationary vector. % For a finite-state Markov c
C:\work\stationary.m July 11, 2003
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function v = stationary(P) % % This is a MATLAB function that calculates the stationary probability vect or v % of a Markov chain transition matrix P, i.e., we solve v = v*P . % We assume the existence
C:\work\stat.m July 11, 2003
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function v = stat(P) % % This is a MATLAB function that calculates the stationary probability vect or v % of a Markov chain transition matrix P, i.e., we solve v = v*P . % We assume the existence of a unique
C:\work\absorbing.m July 11, 2003
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function absorbing(Q, R) % % This is a MATLAB function that calculates important characteristics of an absorbing Markov chain. % % The full Markov chain transition matrix P is assumed to be (k+m) by (k+m
IEOR 6711: Stochastic Models I SOLUTIONS to Second Midterm Exam, Chs. 3-4, November 20, 2007
Justify your answers; show your work. 1. Selling Flour: The Grainery. (30 points) The Grainery is a store that sells flour to bakers. Suppose that bakers come to
IEOR 6711: Stochastic Models I Second Midterm Exam, Chapters 3 and 4, November 20, 2007
Justify your answers; show your work. 1. Selling Flour: The Grainery. (30 points) The Grainery is a store that sells flour to bakers. Suppose that bakers come to the G
IEOR 6711: Stochastic Models I Second Midterm Exam, Chapters 3-4, November 18, 2012 SOLUTIONS
Justify your answers; show your work. 1. Forecasting the Weather (12 points) Consider the following probability model of the weather over successive days. First,
IEOR 6711: Stochastic Models I Second Midterm Exam, Chapters 3-4, November 18, 2012 There are four questions, each with multiple parts.
Justify your answers; show your work. 1. Forecasting the Weather (12 points) Consider the following probability model o
IEOR 6711: Stochastic Models I Second Midterm Exam, Chapters 3 & 4, November 21, 2010 SOLUTIONS
Justify your answers; show your work. 1. Random Walk on a Graph (25 points)
Random Walk on a Graph
C A 2
E 5 1 D
2 3 G 1 F
1 B
3
Figure 1: A random walk on a g
IEOR 6711: Stochastic Models I Second Midterm Exam, Chapters 3 & 4, November 21, 2010
Justify your answers; show your work. 1. Random Walk on a Graph (25 points)
Random Walk on a Graph
C A 2
E 5 1 D
2 3 G 1 F
1 B
3
Figure 1: A random walk on a graph. Cons
SOLUTIONS to the First Midterm Exam, October 7, 2012 IEOR 6711: Stochastic Models I,
1. Poisson Process and Transforms (30 points) [grading scheme: On all questions, partial credit will be given. Up to 4 points off for errors on parts (a)-(g); up to 8 poi
IEOR 6711: Stochastic Models I SOLUTIONS to First Midterm Exam, October 10, 2010
Justify your answers; show your work. 1. Exponential Random Variables (23 points) Let X1 and X2 be independent exponential random variables with means E[X1 ] 1/1 and E[X2 ] 1
IEOR 6711: Stochastic Models I First Midterm Exam, Chapters 1-2, October 10, 2010
Justify your answers; show your work. 1. Exponential Random Variables (23 points) Let X1 and X2 be independent exponential random variables with means E[X1 ] 1/1 and E[X2 ]
IEOR 6711: Stochastic Models I First Midterm Exam, Chapters 1-2, October 7, 2008
Justify your answers; show your work. 1. A sequence of Events. (10 points) Let cfw_Bn : n 1 be a sequence of events in a probability space (, F, P ). (a) Explain what that me
function y = laplace2(which,z,Ps1,Ps2,delta,mu,p2not,p2,r,s,x,q,xnot,qnot,beta,mndpevt,littlegam a) % laplace transform of response time tail cdf if which=1 y=Ps1*mu/(mu+z); d=(mu/(mu+z); y=y+p2not*d*(s*mu+delta(1,1)/(s*mu+delta(1,1)+z); for k=1:r-1 d=(mu
IEOR 6711: Professor Whitt
Notes on Laplace Transforms and Their Inversion
The shortest path between two truths in the real domain passes through the
complex domain; Jacques Hadamard (1865-1963).
1. Basic Denition
Let X be a nonnegative real-valued random
IEOR 6711: Stochastic Models I
Fall 2012, Professor Whitt
Numerical Transform Inversion Homework: Tuesday, September 11
You have FIVE WEEKS: Due Tuesday, October 16.
1. Write a program implementing the algorithm Euler, which does the Fourier-series
algori
IEOR 6711: Stochastic Models I
Fall 2012, Professor Whitt
Solutions to Homework Assignment 13 due on Tuesday, December 4
Problem 5.12 (a) Since
P0 =
it follows that
1/
=
1/ + 1/
+
N (t)
0
1
=
+
.
t
t
+ +
lim
(b) The expected total time spent in state 0
IEOR 6711: Stochastic Models I
Fall 2012, Professor Whitt
Homework Assignment 13, Tuesday, November 27
Chapter 5: Continuous-Time Markov Chains
Due on Tuesday, December 4.
Problems from Chapter 5 of Stochastic Processes, second edition, by Sheldon Ross.
P
IEOR 6711: Stochastic Models I
Fall 2012, Professor Whitt
Solutions to Homework Assignment 12 due on Thursday, November 29
Problem 5.3 (a) Let N (t) denote the number of transitions be t. It is easy to show in this case
that
(M t)j
P(N (t) n)
eM t
j!
j =
IEOR 6711: Stochastic Models I
Fall 2012, Professor Whitt
Homework Assignment 12, Tuesday, November 20
Chapter 5: Continuous-Time Markov Chains
Due on Thursday, November 29. (not to be due immediately after
Thanksgiving)
Problems from Chapter 5 of Stochas
IEOR 6711: Stochastic Models I
Fall 2012, Professor Whitt
Solutions to Homework Assignment 11 on DTMCs
Problem 4.40 Consider a segment of a sample path beginning and ending in state i, with
no visit to i in between, i.e, the vector (i, j1 , j2 , j3 , . .
IEOR 6711: Stochastic Models I
Fall 2012, Professor Whitt
Homework Assignment 11, Tuesday, November 13
Chapter 4: Even More Markov Chains
Need not be turned in.
Problems from Chapter 4 of Stochastic Processes, second edition, by Sheldon Ross. You
only nee