Industrial Engineering 536
Stochastic Models In Operations Research I
Spring 2013
Lecture 4: Discrete Random Variables
Problem 3
There are three coins in a box. One is a two-headed coin, another is a fair coin, and third is a biased
coin that comes up hea
Industrial Engineering 536
Stochastic Models In Operations Research I
Spring 2013
Lecture 21: Hidden & Continuous-Time Markov Chains
Hidden Markov Chains
- Let cfw_Xn , n = 1, 2, , be a Markov chain with Pi,j .
- The initial states probabilities are pi =
Industrial Engineering 536
Stochastic Models In Operations Research I
Spring 2013
Lecture 18: Limiting Probabilities
Transient vs. Recurrent States
Conclusion state i is recurrent if and only if, starting in state i, the expected number of time periods th
Industrial Engineering 536
Stochastic Models In Operations Research I
Spring 2013
Lecture 17: Classication of States
Chapman-Kolmogorov Equation
- For an n-step transition probability
n
Pij P (cfw_Xn+k = j |Xk = i), n 0, ij 0.
- The Chapman-Kolmogorov equ
Industrial Engineering 536
Stochastic Models In Operations Research I
Spring 2013
Lecture 15: Bose-Einstein Statistics
Connected Components
Def. A Graph consists of r connected components, if its node can be partitioned into r subsets, s.t. each
subset is
Industrial Engineering 536
Stochastic Models In Operations Research I
Spring 2013
Lecture 16: Markov Chains
Exampl 1: rain, revisited
- Suppose that the probability of rain tomorrow depends on whether or not it is raining today, but not in the
past.
- If
Industrial Engineering 536
Stochastic Models In Operations Research I
Spring 2013
Lecture 14: Random Graphs
Properties
- Analog to before, the conditional expectations given Y = y satisfy all the properties of ordinary expectations:
w E [X |W = w]P (cfw_
Industrial Engineering 536
Stochastic Models In Operations Research I
Spring 2013
Lecture 12: Stochastic Processes
Stochastic Processes
Def. A collection of random variables cfw_X (t), t T is called a stochastic process.
i.e. For each t T , there exists
Industrial Engineering 536
Stochastic Models In Operations Research I
Spring 2013
Lecture 11: Review 1
Probability
Denition of probability: a number assigned to an event E and satisfying:
Cond. i) 0 P (E ) 1
Cond. ii) P (S ) = 1
N
Cond. iii) En Em = , n =
Industrial Engineering 536
Stochastic Models In Operations Research I
Spring 2013
Lecture 20: Markov Chain Monte Carlo Methods
Stationary Probability
- The long-run probabilities j , j 0 are often called stationary probabilities
- If the initial state is
Industrial Engineering 536
Stochastic Models In Operations Research I
Spring 2013
Lecture 22: Continuous-Time Markov Chains 2
Expected Time of transition
- The expected time of transition is:
E [Tn+1 | Xn = i] =
i ei d =
0
1
i
We can interpret i as the
Industrial Engineering 536
Stochastic Models In Operations Research I
Spring 2013
Lecture 26: Queueing Theory 3 & Reliability Theory
Generalization (continued)
- The average number of customers in the system is
k
L=
average number at server j
j =1
k
=
j =
Industrial Engineering 536
Stochastic Models In Operations Research I
Spring 2013
Lecture 3: Random Variables
Recap
If F occurs, in order for E to occur, it must be in cfw_E F .
Conditional Probability :
P (E |F ) =
Bayes Formula:
P (Fj |E ) =
P (E F )
P
Industrial Engineering 536
Stochastic Models In Operations Research I
Spring 2013
Lecture 25: Queueing Theory 2
Queueing System with Bulk Service
- Consider a single-server exponential queueing system
- The server is able to serve two customers at the sam
Industrial Engineering 536
Stochastic Models In Operations Research I
Spring 2013
Exercise Problems 3
Please use the midterms as a reference for previous topics!
1. Empty Taxis pass by a street corner at a Poisson rate of two per minute and pick up passen
Industrial Engineering 536
Stochastic Models In Operations Research I
Spring 2013
Lecture 27: Review (nal)
Continuous-Time Markov Chains
- Let the state space be S = cfw_1, . . . , m
- We introduce a new set of random variables :
- Xn : the state right af
Industrial Engineering 536
Stochastic Models In Operations Research I
Spring 2013
Lecture 23: Continuous-Time Markov Chains 3
Example 1, continued
010
P = 1 0 1 ,
2
2
100
010
5
Q= 2 0 5
2
300
- Neglecting o( ) terms, the transition probability matrix for
Industrial Engineering 536
Stochastic Models In Operations Research I
Spring 2013
Lecture 24: Queueing Theory
- So far, we discussed a queueing model in the context of continuous-time Markov chain
Now we discuss models in which customers arrive in some ra
Industrial Engineering 536
Stochastic Models In Operations Research I
Spring 2013
Lecture 19: Review 2
Conditional Probability and Expectation
Conditional mass function: pX |Y =
P (cfw_X =x , Y =y )
P (cfw_Y =y )
Conditional density function: fX |Y (x, y
Industrial Engineering 536
Stochastic Models In Operations Research I
Spring 2013
Lecture 13: Using Conditioning
Example 1
- Expectation number of accidents per week at a plant is 4.
- Number of workers injured in each accident is an independent random va
Industrial Engineering 536
Stochastic Models In Operations Research I
Spring 2013
Lecture 10: Limit Theorems
Inequalities
i) Let X 0 be some random variable and a 0.
P (cfw_X a)
E [X ]
a
Markovs Inequality
ii) Let X be a random variable with and 2 .
(X
Industrial Engineering 536
Stochastic Models In Operations Research I
Spring 2013
Lecture 5: Continuous Random Variables
Geometric Random Variables
- Independent trials, each with probability p for success
- X is the number of the trials until we have suc
10/16/13
Systems Simulation with Arena
IE 580, Fall 2013
Hong Wan
Purdue University
Fall, 2013
Hong Wan
Overview
We have introduced the design and analysis
of dynamic systems that evolve through time.
Many concepts (input modeling, logical
model, replic
9/12/13
Input Modeling
IE580, Fall 2013
Hong Wan
Purdue University
Fall 2012
Hong Wan
1
Input Modeling
Input models represent the uncertainty in a
stochastic simulation.
The fundamental requirements for an input
model are:
It must be capable of represe
9/25/13
Experiment Design & Analysis
IE580, Fall 2013
Hong Wan
Purdue University
Fall, 2013
Hong Wan
1
Outline
Interpreting standard performance measures
and graphs
Interpreting trends and sensitivity
Choosing the number of iterations
Monte Carlo vs.
Chapter 5
Statistical Models in
Simulation
Banks, Carson, Nelson & Nicol
Discrete-Event System Simulation
Purpose & Overview
The world the model-builder sees is probabilistic rather
than deterministic.
An appropriate model can be developed by sampling the
Chapter 7
Random-Number
Generation
Banks, Carson, Nelson & Nicol
Discrete-Event System Simulation
Purpose & Overview
Discuss the generation of random numbers.
Introduce the subsequent testing for
randomness:
Frequency test
Autocorrelation test.
2
Properti
What is
Simulation?
Chapter 1
Last revision June 21, 2009
Simulation with Arena, 5th ed.
Chapter 1 What Is Simulation?
Slide 1 of 23
Simulation Is
Simulation very broad term methods and
applications to imitate or mimic real systems,
usually via computer
Chapter 8
Random-Variate Generation
Banks, Carson, Nelson & Nicol
Discrete-Event System Simulation
Purpose & Overview
Develop understanding of generating samples
from a specified distribution as input to a
simulation model.
Illustrate some widely-used tec