Simulation. Inverse Functions Actually, we’ve already done this with the normal distribution.

Slides:

Advertisements

Similar presentations

Waiting Line Management

Advertisements

Random Number Generation. Random Number Generators Without random numbers, we cannot do Stochastic Simulation Most computer languages have a subroutine,

Lab Assignment 1 COP 4600: Operating Systems Principles Dr. Sumi Helal Professor Computer & Information Science & Engineering Department University of.

CDA6530: Performance Models of Computers and Networks Chapter 5: Generating Random Number and Random Variables TexPoint fonts used in EMF. Read the TexPoint.

Simulation of multiple server queuing systems

Random Number Generators. Why do we need random variables? random components in simulation → need for a method which generates numbers that are random.

Engineering Probability and Statistics - SE-205 -Chap 4 By S. O. Duffuaa.

Simulation A Queuing Simulation. Example The arrival pattern to a bank is not Poisson There are three clerks with different service rates A customer must.

Descriptive statistics Experiment  Data  Sample Statistics Sample mean Sample variance Normalize sample variance by N-1 Standard deviation goes as square-root.

Random-Variate Generation. Need for Random-Variates We, usually, model uncertainty and unpredictability with statistical distributions Thereby, in order.

Simulation Modeling and Analysis

Probability theory 2010 Main topics in the course on probability theory  Multivariate random variables  Conditional distributions  Transforms  Order.

Today Today: Chapter 5 Reading: –Chapter 5 (not 5.12) –Suggested problems: 5.1, 5.2, 5.3, 5.15, 5.25, 5.33, 5.38, 5.47, 5.53, 5.62.

1 Analysis Of Queues For this session, the learning objectives are:  Learn the fundamental structure of a queueing system.  Learn what needs to be specified.

Probability theory 2011 Main topics in the course on probability theory  The concept of probability – Repetition of basic skills  Multivariate random.

Random Number Generation

7/3/2015© 2007 Raymond P. Jefferis III1 Queuing Systems.

1 Simulation Modeling and Analysis Pseudo-Random Numbers.

Lecture 4 Mathematical and Statistical Models in Simulation.

Internet Queuing Delay Introduction How many packets in the queue? How long a packet takes to go through?

McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2008 CD12s.1 Generating Random Observations from a Probability Distribution The method for generating.

Random-Number Generation. 2 Properties of Random Numbers Random Number, R i, must be independently drawn from a uniform distribution with pdf: Two important.

Fall 2011 CSC 446/546 Part 6: Random Number Generation.

ETM 607 – Random Number and Random Variates

Standard Statistical Distributions Most elementary statistical books provide a survey of commonly used statistical distributions. The reason we study these.

The Poisson Process. A stochastic process { N ( t ), t ≥ 0} is said to be a counting process if N ( t ) represents the total number of “events” that occur.

Exponential and Chi-Square Random Variables

Statistical Distributions

Copyright ©: Nahrstedt, Angrave, Abdelzaher, Caccamo1 Queueing Systems.

Ch5 Continuous Random Variables

WOOD 492 MODELLING FOR DECISION SUPPORT Lecture 27 Simulation.

© 2003, Carla Ellis Simulation Techniques Overview Simulation environments emulation exec- driven sim trace- driven sim stochastic sim Workload parameters.

Queueing Theory What is a queue? Examples of queues: Grocery store checkout Fast food (McDonalds – vs- Wendy’s) Hospital Emergency rooms Machines waiting.

CIS 2033 based on Dekking et al. A Modern Introduction to Probability and Statistics Michael Baron. Probability and Statistics for Computer Scientists,

Example simulation execution The Able Bakers Carhops Problem There are situation where there are more than one service channel. Consider a drive-in restaurant.

Lecture 19 Nov10, 2010 Discrete event simulation (Ross) discrete and continuous distributions computationally generating random variable following various.

MAT 1235 Calculus II Section 8.5 Probability

Monte Carlo Methods So far we have discussed Monte Carlo methods based on a uniform distribution of random numbers on the interval [0,1] p(x) = 1 0  x.

Solutions Simulation ) The weather can be considered a stochastic system, because it evolves in a probabilistic manner from one day to the next.

COMP155 Computer Simulation September 8, States and Events  A state is a condition of a system at some point in time  An event is something that.

ETM 607 – Random-Variate Generation

WOOD 492 MODELLING FOR DECISION SUPPORT Lecture 25 Simulation.

MATH 3033 based on Dekking et al. A Modern Introduction to Probability and Statistics Slides by Sean Hercus Instructor Longin Jan Latecki Ch. 6 Simulations.

1 Lecture 9: The Poisson Random Variable and its PMF Devore, Ch. 3.6.

Arrival & Service Times for Assignment 3 Byung-Hyun Ha

Structure of a Waiting Line System Queuing theory is the study of waiting lines Four characteristics of a queuing system: –The manner in which customers.

1 1 Slide Continuous Probability Distributions n The Uniform Distribution  a b   n The Normal Distribution n The Exponential Distribution.

ENGM 661 Engineering Economics for Managers Risk Analysis.

MONTE CARLO METHOD DISCRETE SIMULATION RANDOM NUMBER GENERATION Chapter 3 : Random Number Generation.

Managerial Decision Making Chapter 13 Queuing Models.

CIS 2033 based on Dekking et al. A Modern Introduction to Probability and Statistics B: Michael Baron. Probability and Statistics for Computer Scientists,

Queuing Theory. Model Customers arrive randomly in accordance with some arrival time distribution. One server services customers in order of arrival.

Simulation of Inventory Systems

Random number generation

Main topics in the course on probability theory

Queuing Theory Non-Markov Systems

South Dakota School of Mines & Technology Introduction to Probability & Statistics Industrial Engineering.

Internet Queuing Delay Introduction

Introduction to Probability & Statistics Inverse Functions

Demo on Queuing Concepts

Internet Queuing Delay Introduction

Birth-Death Process Birth – arrival of a customer to the system

Our favorite simple stochastic process.

MATH 3033 based on Dekking et al

South Dakota School of Mines & Technology Introduction to Probability & Statistics Industrial Engineering.

Introduction to Probability & Statistics Inverse Functions

Introduction to Probability & Statistics Exponential Review

Waiting Line Models Waiting takes place in virtually every productive process or service. Since the time spent by people and things waiting in line is.

Uniform Probability Distribution

Presentation transcript:

Simulation

Inverse Functions Actually, we’ve already done this with the normal distribution.

Inverse Normal Actually, we’ve already done this with the normal distribution. x 3.0  0.1 x =  +  z = x =    X Z

0 Inverse Exponential Exponential Life Time to Fail Density a fxe x ()   f(x) FaXa()Pr{}     edx x a 0   e xa   1e a

Inverse Exponential xF X e   1 - )( F(x) x

Inverse Exponential aF a e  = 1 - )( F(x) x F(a) a

Inverse Exponential Suppose we wish to find a such that the probability of a failure is limited to 0.1. F(x) x F(a) a

Inverse Exponential Suppose we wish to find a such that the probability of a failure is limited to = 1 - a e  F(x) x F(a) a

Inverse Exponential Suppose we wish to find a such that the probability of a failure is limited to = 1 - ln(0.9) = - a F(x) x F(a) a a e 

Inverse Exponential Suppose we wish to find a such that the probability of a failure is limited to = 1 - ln(0.9) = - a a e  F(x) x F(a) a a = - ln(0.9)/

Inverse Exponential Suppose a car battery is governed by an exponential distribution with = We wish to determine a warranty period such that the probability of a failure is limited to 0.1. a = - ln(0.9)/ = - ( )/0.005 = hrs. F(x) x F(a) a

Model Customers arrive randomly in accordance with some arrival time distribution. One server services customers in order of arrival. The service time is random following some service time distribution.

M/M/1 Queue M/M/1 Queue assumes exponential interarrival times and exponential service times Ae i A i   Se i S i    

M/M/1 Queue M/M/1 Queue assumes exponential interarrival times and exponential service times Ae i A i   Se i S i     Exponential Review Expectations

M/M/1 Queue

M/M/1 Queue

M/M/1 Queue.347

M/M/1 Queue.347

M/M/1 Event Calendar

M/M/1 Queue

M/M/1 Event Calendar

M/M/1 Queue

M/M/1 Queue

M/M/1 Event Calendar

M/M/1 Queue

M/M/1 Event Calendar

M/M/1 Queue.795

M/M/1 Event Calendar

M/M/1 Queue

M/M/1 Queue

M/M/1 Event Calendar

M/M/1 Queue 1.349

M/M/1 Event Calendar

M/M/1 Queue

M/M/1 Queue

M/M/1 Event Calendar

M/M/1 Queue

M/M/1 Event Calendar

M/M/1 Performance Measures

Applications; Financial

Generation of Random Variables Random Numbers Table of Random Numbers Built in Functions - Rand() Congruential Mixed Mulitiplicative Additive Random Obs. from a Prob. Distribution Inverse Transformation Acceptance / Rejection Special Cases

Random Generation Rule: Start anywhere and read sequentially down or across to required significant digits.

Random Generation Rule: Start anywhere and read sequentially down or across to required significant digits.

Mixed Congruential xaxcm nn   1 ()(mod) X 0 =seed a and c < m

Mixed Congruential xaxcm nn   1 ()(mod)

Mixed Congruential xaxcm nn   1 ()(mod)

Mixed Congruential xaxcm nn   1 ()(mod)

Mixed Congruential xaxcm nn   1 ()(mod) Cycle Length < m

Mixed Congruential xaxm nn   1 ()(mod)xxcm nn   1 ( ) Multiplicative Generator Additive Generator