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A Refresher on Probability and Statistics. What We’ll Do. Ground-up review of probability and statistics necessary to do and understand simulation Outline Probability – basic ideas, terminology Random variables, joint distributions Sampling - PowerPoint PPT Presentation
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A Refresher on Probability and Statistics
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What We’ll Do ... Ground-up review of probability and
statistics necessary to do and understand simulation
Outline— Probability – basic ideas, terminology— Random variables, joint distributions— Sampling— Statistical inference – point estimation,
confidence intervals, hypothesis testing
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Monte Carlo Simulation
Monte Carlo method: Probabilistic simulation technique used when a process has a random component
Identify a probability distribution
Setup intervals of random numbers to match probability distribution
Obtain the random numbers Interpret the results
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Probability Basics
Experiment – activity with uncertain outcome— Flip coins, throw dice, pick cards, draw balls from
urn, …— Drive to work tomorrow – Time? Accident?— Operate a (real) call center – Number of calls?
Average customer hold time? Number of customers getting busy signal?
— Simulate a call center – same questions as above Sample space – complete list of all possible
individual outcomes of an experiment— Could be easy or hard to characterize— May not be necessary to characterize
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Probability Basics (cont’d.)
Event – a subset of the sample space— Describe by either listing outcomes, “physical”
description, or mathematical description— Usually denote by E, F, G… or E1, E2, etc.— Ex: arrival of a customer, start of work on a job
Probability of an event is the relative likelihood that it will occur when you do the experiment
— A real number between 0 and 1 (inclusively)— Denote by P(E), P(E F), etc.— Interpretation – proportion of time the event occurs
in many independent repetitions (replications) of the experiment
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Some properties of probabilitiesIf S is the sample space, then P(S) = 1If Ø is the empty event (empty set), then P(Ø) = 0If EC is the complement of E, then P(EC) = 1 – P(E)P(E F) = P(E) + P(F) – P(E F)If E and F are mutually exclusive (i.e., E F = Ø), then
P(E F) = P(E) + P(F)If E is a subset of F (i.e., the occurrence of E implies the
occurrence of F), then P(E) P(F)
If o1, o2, … are the individual outcomes in the sample space, then
Probability Basics (cont’d.)
E FS
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Probability Basics (cont’d.)
Conditional probability
— Knowing that an event F occurred might affect the probability that another event E also occurred
— Reduce the effective sample space from S to F, then measure “size” of E relative to its overlap (if any) in F, rather than relative to S
— Definition (assuming P(F) 0):
E and F are independent if P(E F) = P(E) P(F)— Implies P(E|F) = P(E) and P(F|E) = P(F), i.e.,
knowing that one event occurs tells you nothing about the other
— If E and F are mutually exclusive, are they independent?
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Random Variables
One way of quantifying, simplifying events and probabilities
A random variable (RV) is a number whose value is determined by the outcome of an experiment— Assigns value to each point in the sample
space— Associates with each possible outcome of
the experiment— Usually denoted as capital letters: X, Y, W1,
W2, etc. Probabilistic behavior described by
distribution function
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Discrete vs. Continuous RVs
Two basic “flavors” of RVs, used to represent or model different things
Discrete – can take on only certain separated values— Number of possible values could be finite or
infinite Continuous – can take on any real value in some
range— Number of possible values is always infinite— Range could be bounded on both sides, just one
side, or neither ( ∞ ∞ )
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RV in Simulation
Input— Uncertain time duration (service or inter-arrival
times)— Number of customers in an arriving group— Which of several part types a given arriving part
is Output
— Average time in system— Number of customers served— Maximum length of buffer
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Discrete Distributions
Let X be a discrete RV with possible values (range) x1, x2, … (finite or infinite list)
Probability Mass Function (PMF)p(xi) = P(X = xi) for i = 1, 2, ...
— The statement “X = xi” is an event that may or may not happen, so it has a probability of happening, as measured by the PMF
— Can express PMF as numerical list, table, graph, or formula
— Since X must be equal to some xi, and since the xi’s are all distinct,
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Discrete Distributions (cont’d.)
Cumulative distribution function (CDF) – probability that the RV will be a fixed value x:
Properties of discrete CDFs0 F(x) 1 for all xAs x – , F(x) 0As x + , F(x) 1F(x) is nondecreasing in xF(x) is a step function continuous from the
right with jumps at the xi’s of height equal to the PMF at that xi
0.5
1.0
F(1)
F(2)F(3)
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Example of CDF
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Example of CDF
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Discrete Distributions (cont’d.)
Computing probabilities about a discrete RV – usually use the PMF— Add up p(xi) for those xi’s satisfying the
condition for the event
With discrete RVs, must be careful about weak vs. strong inequalities – endpoints matter!
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Discrete Expected Values Data set has a “center” – the average (mean) RVs have a “center” – expected value
— Also called the mean or expectation of the RV X— Other common notation: , X
— Weighted average of the possible values xi, with weights being their probability (relative likelihood) of occurring
— What expectation is not: The value of X you “expect” to get
E(X) might not even be among the possible values x1, x2, …
— What expectation is:Repeat “the experiment” many times, observe many X1, X2, …, Xn
E(X) is what converges to (in a certain sense) as n
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Discrete Variances andStandard Deviations
Data set has measures of “dispersion” –
— Sample variance
— Sample standard deviation RVs have corresponding measures
— Other common notation: — Weighted average of squared deviations of the
possible values xi from the mean— Standard deviation of X is — Interpretation analogous to that for E(X)
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Continuous Distributions
Now let X be a continuous RV— Possibly limited to a range bounded on left or right
or both— No matter how small the range, the number of
possible values for X is always (uncountably) infinite
— Not sensible to ask about P(X = x) even if x is in the possible range
— Technically, P(X = x) is always 0— Instead, describe behavior of X in terms of its
falling between two values
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Continuous Distributions (cont’d.)
Probability density function (PDF) is a function f(x) with the following three properties:f(x) 0 for all real values xThe total area under f(x) is 1:For any fixed a and b with a b, the
probability that X will fall between a and b is the area under f(x) between a and b:
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CDF and PDF
P = P[ X x] = Probability that the random variable X is less than or equal to a given number x
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Continuous Distributions (cont’d.)
Cumulative distribution function (CDF) - probability that the RV will be a fixed value x:
Properties of continuous CDFs0 F(x) 1 for all xAs x –, F(x) 0As x +, F(x) 1F(x) is nondecreasing in xF(x) is a continuous function with slope equal to the
PDF:f(x) = F'(x)
These four propertiesare same asdiscrete CDFs
F(x) may ormay not havea closed-formformula
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Continuous Expected Values, Variances, and Standard Deviations
Expectation or mean of X is
— Roughly, a weighted “continuous” average of possible values for X
— Same interpretation as in discrete case: average of a large number (infinite) of observations on the RV X
Variance of X is
Standard deviation of X is
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Joint Distributions
So far: Looked at only one RV at a time But they can come up in pairs, triples, …, tuples,
forming jointly distributed RVs or random vectors— Input: (T, P, S) = (type of part, priority, service
time)— Output: {W1, W2, W3, …} = output process of
times in system of exiting parts One central issue is whether the individual RVs are
independent of each other or related Will take the special case of a pair of RVs (X1, X2)
— Extends naturally (but messily) to higher dimensions
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Joint CDF of (X1, X2) is a function of two variables
— Same definition for discrete and continuous If both RVs are discrete, define the joint PMF
If both RVs are continuous, define the joint PDF f(x1, x2) as a nonnegative function with total volume below it equal to 1, and
Joint Distributions (cont’d.)
Replace“and” with “,”
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Covariance Between RVs
Measures linear relation between X1 and X2
Covariance between X1 and X2 is
— Covariance tells us whether the two random variables are related or not. If they are, whether the relationship is positive or negative.
Interpreting value of covariance – difficult since it depends on units of measurement
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Correlation Between RVs
Correlation (coefficient) between X1 and X2 is
— Always between –1 and +1 Ex: Correlation of 0.85 means strong
relationship, 0.10 means weak.— Cor (X, Y) > 0 means +ve Correlation
— X & Y move in the same direction — Cor (X, Y) = 0 means no correlation— Cor X, Y) < 0 means –ve correlation X , and Y
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Independent RVs
X1 and X2 are independent if their joint CDF factors into the product of their marginal CDFs:
— Equivalent to use PMF or PDF instead of CDF Properties of independent RVs:
— They have nothing (linearly) to do with each other— Independence uncorrelated
— But not vice versa, unless the RVs have a joint normal distribution
— Tempting just to assume it whether justified or not Independence in simulation
— Input: Usually assume separate inputs are indep. – valid?
— Output: Standard statistics assumes indep. – valid?!?!?!?
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Sampling
Statistical analysis – estimate or infer something about a population or process based on only a sample from it— Think of a RV with a distribution governing the
population— Random sample is a set of independent and
identically distributed (IID) observations X1, X2, …, Xn on this RV
— In simulation, sampling is making some runs of the model and collecting the output data
— Don’t know parameters of population (or distribution) and want to estimate them or infer something about them based on the sample
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Sampling (cont’d.)
Population parameterPopulation mean =
E(X)Population variance 2
Population proportion Parameter – need to
know whole population Fixed (but unknown)
Sample estimateSample meanSample varianceSample proportion
Sample statistic – can be computed from a sample
Varies from one sample to another – is a RV itself, and has a distribution, called the sampling distribution
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Point Estimation
A sample statistic that estimates (in some sense) a population parameter
Properties— Unbiased: E(estimate) = parameter— Efficient: Var(estimate) is lowest among
competing point estimators— Consistent: Var(estimate) decreases
(usually to 0) as the sample size increases
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Confidence Intervals
A point estimator is just a single number, with some uncertainty or variability associated with it
Confidence interval quantifies the likely imprecision in a point estimator
— An interval that contains (covers) the unknown population parameter with specified (high) probability 1 –
— Called a 100 (1 – )% confidence interval for the parameter
Confidence interval for the population mean :
CIs for some other parameters – in text book
tn-1,1-/2 is point below which is area1 – /2 in Student’s t distribution withn – 1 degrees of freedom
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Confidence Intervals in Simulation
Run simulations, get results View each replication of the simulation as a
data point Random input random output Form a confidence interval Brackets (with probability 1 – ) the “true”
expected output (what you’d get by averaging an infinite number of replications)
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Example
1.2, 1.5, 1.68, 1.89, 0.95, 1.49, 1.58, 1.55, 0.50, 1.09. Calculate the 90% confidence interval
Sample Mean = = 1.34 Sample Variance = s2 = 0.17l 90% confidence interval means: = 1 – 0.90 = 0.1 Degrees of freedom: n = 10 – 1 = 9.
1.34 t9,0.95 (0.17 / 10). Look into ‘t’ distribution table for t9,0.95 = 1.83
1.34 1.83 (0.17 / 10). = 1.34 0.24 Confidence Interval = [1.10, 1.58]
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Hypothesis Tests
Test some assertion about the population or its parameters
Null hypothesis (H0) – what is to be tested Alternate hypothesis (H1 or HA) – denial of H0
H0: = 6 vs. H1: 6H0: < 10 vs. H1: 10H0: 1 = 2 vs. H1: 1 2
Develop a decision rule to decide on H0 or H1 based on sample data
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Errors in Hypothesis Testing
Type-I error is often called the producer's risk The probability of a type-I error is the level of significance of the test of hypothesis and is denoted by . Type-II error is often called the consumer's risk for not rejecting possibly a worthless product The probability of a type-II error is denoted by . The quantity 1 - is known as the Power of a Test H0 and H1 are not given equal treatment. Benefit of doubt is given to H0
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p-Values for Hypothesis Tests
Traditional method is “Accept” or Reject H0
Alternate method – compute p-value of the test— p-value = probability of getting a test result more in
favor of H1 than what you got from your sample— Small p (< 0.01) is convincing evidence against H0
— Large p (> 0.10) indicates lack of evidence against H0
Connection to traditional method— If p < , reject H0
— If p , do not reject H0
p-value quantifies confidence about the decision
A p-value is a measure of how much evidence you have against the null hypothesis
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Goodness-of-fit Test
Chi – Square Test Kolmogorov – Smirnov test
—Both tests ask how close the fitted distribution is to the empirical distribution defined directly by the data
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Hypothesis Testing in Simulation
Input side— Specify input distributions to drive the simulation— Collect real-world data on corresponding processes— “Fit” a probability distribution to the observed real-world
data— Test H0: the data are well represented by the fitted
distribution Output side
— Have two or more “competing” designs modeled— Test H0: all designs perform the same on output, or test
H0: one design is better than another
Case Study
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Case Study: Printed Circuit Assembly Manufacturing
The company, engaged in electronic assembly contract manufacturing, wants to achieve the following goals:—Maximize equipment utilization—Minimize machine downtime—Increase inventory control accuracy—Provide material traceability—Minimize time and resources spent
looking for materials and tools on the shop-floor
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Electronics Assembly
Surface Mount Technology (SMT) or Pin Through-Hole (PTH) are used to place components on bare boards
An SMT assembly line typically include:— Screen printer - to apply solder paste on the bare board— High-speed placement machine - for chips typically— Fine-Pitch placement machine - for larger components
typically— Owen - to bake the board after components are placed.
The Company has 3 assembly lines
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Typical Reasons for Assembly Line Down Time
Poor line balance and flexibility Poor machine balance within assembly lines Large number of setups and total setup time Part shortage during the run Feeder problems Long reel changeovers Operator is not attending the machine Setup kit is not delivered on time Placing wrong parts Component data problems Process Control – 1st piece inspection Operator waiting for support Machine program changeover time
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Real-Time Performance Monitoring
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Machine Utilization
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Machine Utilization
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Assembly Line Performance Metrics
Assembly efficiency - the difference (in percentage) between the desired assembly time and the actual assembly time required to complete a board (desired time/actual time)*100; target = 95-100%
Minimum cycle time - the largest machine operation time within the assembly line
Average cycle time - the average time a board is completed, i.e. the last operation is completed
The average number of boards in the queue -between two placement machines
A Guided Tour Through Arena
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Flowchart and Spreadsheet Views
Model window split into two views— Flowchart view
— Graphics— Process flowchart— Animation, drawing— Edit things by double-clicking on them, get into a dialog
— Spreadsheet view— Displays model data directly— Can edit, add, delete data in spreadsheet view— Displays all similar kinds of modeling elements at once
— Many model parameters can be edited in either view— Horizontal splitter bar to apportion the two views— View/Split Screen to see only the most recently
selected view
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Modules
Basic building blocks of a simulation model Two basic types: flowchart and data Different types of modules for different actions,
specifications “Blank” modules are on the Project Bar
— To add a flowchart module to your model, drag it from the Project Bar into the flowchart view of the model window
— To use a data module, select it (single-click) in the Project Bar and edit in the spreadsheet view of the model window
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Relations Among Modules
Flowchart and data modules are related via names for objects— Queues, Resources, Entity types, Variables …
others Arena keeps internal lists of different kinds of names
— Presents existing lists to you where appropriate— Helps you remember names, protects you from
typos All names you make up in a model must be unique
across the model, even across different types of modules
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Create Module
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Process Module
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Queue-Length Plot
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Dispose Module
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Setting the Run Conditions Run/Setup menu dialog – five tabs
— Project Parameters – Title, your name, output statistics
— Replication Parameters – Number of Replications, Length of Replication (and Time Units), Base Time Units (output measures, internal computations), Warm-up Period (when statistics are cleared), Terminating Condition (complex stopping rules), Initialization options Between Replications
— Other three tabs specify animation speed, run conditions, and reporting preferencesTerminating your simulation:
You must specify – part of modeling Arena has no default termination If you don’t specify termination, Arena will
usually keep running forever
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Viewing the Reports Click Yes in the Arena box at the end of the run
— Opens up a new reports window (separate from model window) inside the Arena window
— Project Bar shows Reports panel, with different reports (each one would be a new window)
— Remember to close all reports windows before future runs
Default installation shows Category Overview report – summarizes many things about the run— Reports have “page” to browse Also, “table
contents” tree at left for quick jumps via Times are in Base Time Units for the model
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Types of Statistics Reported Many output statistics are one of three types:
— Tally – avg., max, min of a discrete list of numbers— Used for discrete-time output processes like waiting times
in queue, total times in system
— Time-persistent – time-average, max, min of a plot of something where the x-axis is continuous time
— Used for continuous-time output processes like queue lengths, WIP, server-busy functions (for utilizations)
— Counter – accumulated sums of something, usually just nose counts of how many times something happened
— Often used to count entities passing through a point in the model
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Homework 2
Work as a team of 2.
Problem 1: Question C4 from Appendix C Problem 2: Question 3.6
Due 9/9/03.— Electronic submission