B. Maddah
ENMG 622 Simulation
11/04/08
Discrete Event and Process Oriented Simulation (2)
Discrete event hand simulation of an (s, S) periodic review
inventory system
Consider a retailer who sells a commodity having a random
weekly demand D .
At the begi

B. Maddah
ENMG 622 Simulation
05/02/07
Output Analysis (Chapter 9, Law)
Introduction
The basic, most serious disadvantage of simulation is that
we dont get exact answers.
Two different runs of the same model (with different
random numbers) lead to differ

B. Maddah
ENMG 622 Simulation
05/02/07
Output Analysis (2, Chapters 10 &11 Law)
Comparing alternative system configuration
Since the output of a simulation is random, then comparing
different systems via simulation should be done on the basis
of a statis

Use the KS test to check if the following data is exponentially distributed with mean 1.
Use a significance level of 0.025.
0.854 0.064 0.058 8.134 1.200 0.176 1.052 3.789 1.388 0.353

S1. Consider a single-server queue. The arrival times, AT, and service times, ST, of the
first 10 customers were generated from corresponding distributions as follows:
AT 3.2, 10.9, 13.2, 14.8, 17.7, 19.8, 21.5, 26.3, 32.1, 36.6
ST 3.8, 3.5, 4.2, 3.2, 2.4

P1. Let E, F, G be three events. Find expressions for the events that out of E, F, G,
(a) only F occurs,
(b) both E and F but not G occurs,
(c) at least one event occurs,
(d) at least two events occur,
(e) all three events occur,
(f) none occurs,
(g) at m

An (s, S) Inventory Simulation (Model 5-4)
Single-item (widgets), managed by (s, S) Policy
s = 20, S = 40, Initial Inventory = 60.
Starting Net Inventory of day t, I(t), is reviewed
If I(t) < s, place an order to bring I(t) up to S
An order incurs a $32 f

A
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
B
C
D
E
F
G
Black-Schole's Option Pricing Problem
Using the Option Price to Find the Implied Volatility
Input data
Stock price, S0
Exercise price, K
Duration, T
Interest rate, r
Implied volatility
Put price
$

B. Maddah
ENMG 622 Simulation
05/09/07
Simulating Stock Prices
The geometric Brownian motion stock price model
Recall that a rv Y is said to be lognormal if X = ln(Y) is a
normal random variable.
Alternatively, Y is a lognormal rv if Y = eX, where X is a

B. Maddah
ENMG 622 Simulation
03/14/07
Generating Random Variates 3 (Chapter 8, Law)
Generating from an empirical distribution with raw data
Suppose that n observations of a random variable X are
available.
Rather than fitting a theoretical distribution

B. Maddah
ENMG 622 Simulation
10/28/08
Queueing Primer
What is a queueing system?
A queueing system consists of servers (resources) that
provide service to customers (entities).
A Customer requesting service will start service if the
required server is n

B. Maddah
ENMG 622 Simulation
02/22/10
Probability and Random Variable Primer
Sample space and Events
Suppose that an experiment with an uncertain outcome is
performed (e.g., rolling a die).
While the outcome of the experiment is not known in
advance, th

B. Maddah
ENMG 622 Simulation
02/15/09
OR and Simulation Modeling
What is special about the OR analysis approach?
(i) A primary focus on decision making. The analysis must
lead to clear suggestions to the decision maker.
(ii) An appraisal resting on econ

B. Maddah
ENMG 622 Simulation
11/11/08
Random-Number Generators (Chapter 7, Law)
Overview
All stochastic simulations need to generate IID uniformly
distributed on (0,1), U(0,1), random numbers.
1
fX ( x)
0.5
0
0
0.5
1
This is the case since all other ran

B. Maddah
ENMG 622 Simulation
05/02/07
Selecting Input Probability Distributions 1
(Chapter 6, Law)
Introduction
The objective here is to determine what probability
distributions to use as input to a simulation.
The input distributions are usually determ

B. Maddah
ENMG 622 Simulation
03/22/10
Random-Number Generators 2 (Chapter 7, Law)
Testing random number generators
Since random number generators are completely
deterministic, we need to test to see if they appear to be
random and IID uniform on [0, 1].

B. Maddah
ENMG 622 Simulation
03/01/10
Discrete Event and Process Oriented Simulation (1)
Discrete event simulation
Discrete event simulation concerns the modeling of a
system by a representation in which the state of the system
changes at discrete time

B. Maddah
ENMG 622 Simulation
05/02/07
Selecting Input Probability Distributions 2
(Chapter 6, Law)
Activity III: Determining How Representative the Fitted
Distributions Are
Having hypothesized a family of distributions and estimated
parameters, the fina

Discrete Event Hand Simulation
of a GI/GI/1 Queue
Simulation with Arena
Chapter 2 Fundamental Simulation Concepts
The System
(Server)
Arriving
customers
7
6
5
Queue (FIFO)
Departing
customers
4
customer in Service
Simulation Objectives
Claimed objective:

B. Maddah
ENMG 622 Simulation
03/29/09
Generating Random Variates (Chapter 8, Law)
Overview
We will discuss algorithms for generating observations
(variates) from non-uniform distributions (e.g.
Exponential, Weibull, etc.)
Generating random variates is a

B. Maddah
ENMG 622 Simulation
11/25/08
Generating Random Variates 2 (Chapter 8, Law)
Generating random variates from U(a, b)
Recall that a random X which is uniformly distributed on
interval [a, b], X ~ U(a, b), has the distribution function:
0,
if x a
F