ISYE6644 (Simulation) Midterm 1

What are some characteristics of simulation models?

1. Discrete (vs. continuous)

2. Stochastic (vs. deterministic)

3. Dynamic (vs. static)

What is simulation?

Simulation is the imitation of a real-world process of system over time.

Simulation involves the generation of an artificial history to draw inferences concerning the operating characteristics of the real system that is represented.

What is simulation good for?

1. Describe / analyze real or conceptual system behavior.

2. Ask "what if" questions.

3. Aid in system design and optimization

4. Can simulate almost anything

What are the reasons to simulate?

1. Will the system accomplish its goals?

2. Current system won't accomplish its goals, now what?

3. Need incremental improvement.

4. Create a specification or action plan

5. Solve a problem, like a bottleneck.

6. Resolve disputes

7. Sell an idea

What are the advantages of simulation?

1. Can study models too complicated for analytical / numerical treatment

2. Study detailed relations that might be lost in the analytical or numerical treatment

3. Use as a basis for experimental studies of systems

4. Use to check results and give credibility to conclusions obtained by other methods.

5. Reduce design blunders

6. Really nice demo method

7. (sometimes) very easy

What are the disadvantages of simulation?

1. Sometimes not so easy

2. Sometimes very time consuming / costly

3. Simulations give "random" output (and lots of misinterpretation of results is possible)

4. To do a certain problem, better methods than simulation may exist

We are interested in modeling the arrival and service process at the local McBurger Queen burger joint. Customers come in every once in a while, stand in line, eventually get served, and off they go. Generally speaking, what kind of model are we talking about here? (More than one answer below may be right.)

A) Discrete

B) Continuous

C) Stochastic

D) Deterministic

(a) (because events such as arrivals and service completions only happen once in a while, as opposed to continuously); and (c) (because customer arrival times, service times, shift changes, etc., are all random).

Which of the following can be regarded as advantages of simulation? (More than one answer below may be right.)

A) Simulation enables you to study models too complicated for analytical or numerical treatment.

B) Simulations can serve as very pretty demos that even University of Georgia graduates can understand.

C) Simulation can be used to study detailed relations that might be lost in an analytical or numerical treatment.

D) Simulations are often tedious and time-consuming to produce.

A) Simulation enables you to study models too complicated for analytical or numerical treatment.

B) Simulations can serve as very pretty demos that even University of Georgia graduates can understand.

C) Simulation can be used to study detailed relations that might be lost in an analytical or numerical treatment.

Who is William Gosset?

A) He invented the t distribution that is used ubiquitously in statistics.

B) He invented the s distribution that is used ubiquitously in statistics.

C) He invented tea.

D) He invented the word "ubiquitous".

E) He is the brother of Louis Gossett Jr., best known for his fine acting in many films, including An Officer and a Gentleman.

A) He invented the t distribution that is used ubiquitously in statistics.

What was the first use of a computer simulation?

Stan Ulam and Johnny von Neumann in 1946 after world war 2. They used simulation in the context of the development of the H bomb.

The simulated thermonuclear reactions actually, chain reactions.

What was the main development in the simulation field in the 1960s?

Industrial applications

-manufacturing

-queueing models

Who was Harry Markowitz?

He was one of the first guys to develop a language called SIMSCRIPT. He also won a nobel prize for some of the work he did in optimizing financial portfolios (portfolio theory).

In terms of simulations origins in manufacturing, what are some of the ways that simulation was/is used?

1. Calculated the movement of parts and interaction of system components.

2. Evaluates part flow thru the system

3. Examines conflicting demand for resources

4. Studies contemplated changes before their introduction

5. Prevent design blunders

YES or NO? Has anyone closely related to the field of computer simulation ever won a Nobel Prize?

Yes - Harry Markowitz won the 1990 Nobel Prize in Economics for his work in the field of portfolio optimization. He is highly regarded in the simulation community for his conception and development of the general-purpose simulation language SIMSCRIPT.

Who is Dr. Harold Shipman?

He killed his patients using heroin overdoses

Caught after carelessly revising a patient's will, leaving all her assets to himself

Doctored his records to show his patients had needed morphine but the software recorded the dates of these modifications

hung himself in prison

Which of the following are areas where simulation has found substantial applica- tion? (More than one answer below may be correct.)

A) Inventory and Supply Chain Analysis

B) Financial Analysis

C) Manufacturing

D) Health Systems

E) Transportation Systems

All of em

Why might simulation be a good tool to analyze supply chains? (More than one answer below may be correct.)

A) Supply chains are always deterministic systems.

B) Supply chains often have complicated network structures, making exact analysis difficult.

C) Supply chains are stochastic systems, with random travel times, lead times, and order patterns.

D) Supply chain simulations can be programmed in a matter of minutes.

B) Supply chains often have complicated network structures, making exact analysis difficult.

C) Supply chains are stochastic systems, with random travel times, lead times, and order patterns.

Suppose there are 40 random people in a room. What is the probability that at least two of them will have the same birthday?

A) Close to 0

B) A bit less than 1/2

C) Almost exactly 1/2

D) Somewhat greater than 1/2

D) Somewhat greater than 1/2

In fact, you only need 23 people in the room to achieve a probability of 1/2.

Inscribe a circle in a unit square and toss 1000 random darts at the square. Suppose that 800 of those darts land in the circle. Using the technology developed in this lesson, what is the resulting estimate for 𝜋

A) -3.2

B) 2.8

C) 3.0

D) 3.2

E) 4.0

D) 3.2

Since the estimate 𝜋̂ = 4 x (proportion in circle). Note that (a) is the University of Georgia answer, and is completely incorrect.

What is a characteristic of a bad random number generator?

It must generate a random number distribution for every seed imaginable.

What is a random walk simulation?

Take a normal step up or down every time unit and plot where you are as time progresses.

This "random walk" converges to Brownian motion

Einstein and Black+Scholes won nobel Prizes for this research.

TRUE or FALSE? All random number generators perform pretty much the same.

False

Suppose customers to a barber shop show up at times 4 and 11. Moreover, suppose that it takes the barber 12 minutes to serve customer 1 and then 14 minutes to serve customer 2. When does customer 2 leave the barber?

A) 18

B) 25

C) 30

D) 40

C) 30 - Since customer 2's service starts only when customer 1 leaves, which happens at time 4 + 12 = 16.

At a high level, how are random numbers generated in a computer?

1. Generate pseudo-random numbers (PRNs) using a deterministic algorithim (not really random but appears to be)

2. Generate other random variables by starting with the PRN and applying transformations to get any other type of random variable.

Suppose we are using the (terrible) pseudo-random number generator 𝑋𝑖=(5𝑋𝑖−1+3)mod(8), with starting value ("seed") 𝑋0=1. Find the second PRN, 𝑈2=𝑋2/𝑚=𝑋2/8.

A) 0

B) 1/8

C) 3/8

D) 3

C) 3/8

𝑋1=(5𝑋0+3) mod(8) = 8 mod(8) = 0, and then 𝑋2=(5𝑋1+3) mod(8) = 3 mod(8) = 3. So 𝑈2=𝑋2/8=3/8

Suppose that we generate a pseudo-random number U = 0.728. Use this to generate an Exponential(𝜆=3) random variate.

A) -0.106

B) 0.106

C) -0.952

D) 0.952

B) 0.106

𝑋=−(1/𝜆)ℓ𝑛(𝑈)=−(1/3)ℓ𝑛(0.728)=0.1058. So the answer is (b). Note: It turns out that 𝑋=−(1/𝜆)ℓ𝑛(1−𝑈)=−(1/3)ℓ𝑛(0.272)=0.4340 would also have been an acceptable answer. Can you see why?

When analyzing randomness, what are the two general cases to consider?

Terminating Solutions

- Interested in short-term behavior

- Example: Avg customer waiting time in a bank over the course of a day

- Example: Avg # of infected victims during a pandemic

Steady-State Simulations

- Interested in long-term behavior

- Example: Long-running assembly line

How are terminating solutions analyzed?

Usually analyzed via independent replications.

1. Make independent runs (replications) of the simulation, each under identical conditions

2. Sample means from each replication are assumed to be approximately i.i.d. normal

3. Use classical statistics techniques on the i.i.d. sample means (not on the original observations)

How are steady-state simulations analyzed?

First deal with initialization (start-up) bias

- usually "warm up" simulation before collecting data

- failure to do so can ruin statistical analysis

Many methods for dealing with steady-state

- ex: batch means

What is the method of batch means in analyzing steady-state simulations?

-Make one long run (vs. many shorter reps)

-warm up simulation before collecting data

-chop remaining observations into contiguous batches

-sample means from each batch are approximately i.i.d. normal

-use classical statistics on the i.i.d. batch means

TRUE or FALSE? Simulation outputs such as consecutive customer waiting times are almost always independent and identically distributed normal random variables.

False - Output is pretty much never i.i.d. normal

Let's simulate a bank that closes at 4:30 p.m. What kind of simulation approach would you take?

A) Steady-state simulation

B) Terminating simulation

C) Arnold SChwarzenegger simulation

D) I'm from the university of georgia. What is simulation? and what is a bank?

B) Terminating Simulation

What is the derivative of X^k

kx^k-1

What is the derivative of e^x

e^x

What is the derivative of sin(x)

cos(x)

What is the derivative of cos(x)

-sin(x)

What is the derivative of ln(x)

1 / x

If the f(x) is position, the first derivative is the velocity, what is the second derivative?

acceleration

Which of the following methods cannot be used to find the zeroes of a complicated function?

A) trial-and-error

B) bisection

C) Newton's method

D) Newmans method acting

D) Newmans method acting

What is the opposite of a derivative?

Gosh Dern Integral

Which of the following is not an integration method discussed in this lesson?

A) Reimann Sums

B) Newmann Sums

C) Trapezoid Rule

D) The Monte Carlo method

B) Newmann Sums

What is the conditional probability of A given B?

The probability of some event A, given event B, is equal to the probability of the intersection of the two events A and B divided by the probability of B all by itself.

Regarding probability, events are independent when what definition is met?

If the probability of A intersect B equals P of a times P of B then A and B are independent events.

Toss a 4-side die twice (you know, one of those goofy Dungeons and Dragons pyramid dice things). Assuming the die is numbered 1,2,3,4, what's the probability that the sum will equal 3?

A) 0

B) 1/2

C) 13/16

D) 1/8

D) 1/8

P(sum=3)=P((1,2)or(2,1))=P(1,2)+P(2,1)=2(1/16)=1/8.

TRUE or FALSE? f(x)=3e−xforx>0 is a legitimate probability density function.

FALSE

Correct:

In order to be a legit p.d.f., f(x) must integrate to 1; but lo and behold. . .∫Rf(x)dx=∫∞03e−xdx=3.☹

Suppose X is a continuous random variable with cumulative distribution function F(x). What is the distribution of the nasty random variable F(X)?

A) Normal

B) Unif (0,1)

C) Exponential

D) Weibull

Unif (0, 1) - this is the Inverse Transform Theorem

Suppose U is a Unif (0,1) random variable. Name the distribution of X=−ℓn(1−U).

A) Normal

B) Unif (0, 1)

C) Exponential

D) Weibull

C) Exponential

The abbreviation "m.g.f." stands for...

Moment Generating Function

What is the concept of double expectation?

Idea: the average expected value of all of the conditional expected values is the overall population average.

TRUE or FALSE? If 𝑋 and 𝑌 are uncorrelated, then they're independent.

False

What is the most-important theorem in the universe?

Central limit theorem

What is the central limits theorem?

In probability theory, the central limit theorem establishes that, in some situations, when independent random variables are added, their properly normalized sum tends toward a normal distribution even if the original variables themselves are not normally distributed.

Let's take a bunch of independent observations from a "well-behaved" distribution. The Central Limit Theorem says that the standardized sample mean of those observations converges to what distribution?

Normal

What are three things that help define what a statistic is?

1) a statistic is a function of the observations X1 through Xn and not dependent on any unknowns. So basically something like the mean, cause you know all the parameters

2) statistics are random variables

3) a statistic is usually used to estimate some unknown parameter from the underlying probability distribution of the X's

What is unbiasedness?

the expected value of Xbar (the sample mean) equals the actual mean (mu)

TRUE or FALSE? The sample mean is always unbiased for the true mean. And, while we're at it, the sample variance is always unbiased for the true variance.

True

Suppose that we are using some estimator 𝑇 to estimate an unknown parameter 𝜃. Further suppose that 𝑇 has a bias of 3 and a variance of 5. What is 𝑇's mean squared error?

MSE = 𝖡𝗂𝖺𝗌2 + Variance = 14.

TRUE or FALSE? The length of a confidence interval increases as you demand higher confidence (larger 1 - ∞).

True

TRUE or FALSE? A 95% confidence interval means that you are 95% sure that the true parameter lies somewhere in the interior of the interval.

True

Which step is 𝑛𝑜𝑡 essential for a successful simulation study?

A). Problem formulation

B). Model validation

C) experimental design

D) output analysis

E) attendance at the university of georgia

E) attendance at the university of georgia

What is an entity in a simulation?

Entities can be permanent (like a machine), or temporary like customers, and can have various properties or attributes (priority of a customer or averaged speed of a server).

What is a system?

A system is a collection of entities that interact together to accomplish a goal.

What is a model?

A model is an abstract representation of a system.

What is the system state?

A set of variables that contains enough information to describe the system. Think of the state as a snapshot of the system?

ex knowing how many people are in the queue and if the server is busy for a single server queue model

What is a list (or queue)?

an ordered list of associated entities (for instance, a linked list, or a line of people).

What is an event?

An event is a point in time at which the system state changes (and which can't be predicted with certainty beforehand).

Ex: an arrival event, a departure event, a machine breakdown event.

What is an activity?

An activity is a duration of time of specified length (aka an unconditional wait)

ex: constant service times, customer interarrival times.

What is a conditional wait?

A conditional wait is a duration of time of unspecified length.

ex: customer waiting time.

TRUE or FALSE? The 𝑠𝑦𝑠𝑡𝑒𝑚𝑠𝑡𝑎𝑡𝑒 is a set of variables that contains enough information to describe the system.

True

Characteristics such as the priority or speed of a customer are known as ...?

A) variables

B) function values

C) entities

D) attributes

E) Activities

D) attributes

What is the simulation clock?

The simulation clock is a variable whose value represents simulated time (which doesn't equal real time).

What is a time-advance mechanism?

Basically how does the clock move in a simulation.

The clock always moves forward (never goes back in time).

What is the fixed-increment time advance approach?

Update the state of the system at fixed times, nh, n = 0, 1, 2, ... where H is chosen appropriately.

This is used in continuous-time models and models where data are only available at fixed times (like end of the month)

this is not emphasized in this course.

What is a next-event time advance approach?

The clock is initialized at 0. All known future event times are determined and placed in the future events list (FEL), ordered by time.

The clock advances to the most imminent event, then to the next most imminent event, etc.

At each event, the system state and FEL are updated.

TRUE or FALSE? The simulation clock and future event list form the ♡ of any discrete-event simulation system.

True

What are we allowed to do on the future event list?

A) Insert new events

B) delete events

C) move events around

D) all of the above

D) all of the above

What is a linked list?

Singly and doubly linked lists intelligently store the events in an array that allows the chronological order of the events to be accessed.

Such lists easily accommodate insertion, deletion, switching of events, etc.

What is the event-scheduling approach?

Concentrate on the events and how they affect the system state.

Help the simulation evolve over time by keeping track of every freaking event in increasing order of time of occurrence.

This is a book keeping hassle.

What is the process-interaction approach?

1. Create customers every once in a while.

2. Process (serve) them, maybe after waiting in line.

3. Dispose of the customers after they're done being processed.

What is the name of the primary modeling approach that we will be using in this class, especially when we do Arena?

Event-scheduling

Process-interaction

continuous modeling

mixed modeling

event-interaction

process-interaction

How many simulation languages are there?

More than 100 commercial languages in the ether.

When selecting a simulation language, what characteristics do you have to take into consideration?

Cost

Ease of use

Modeling "world view" (e.g., event-scheduling or process-interaction)

Random variate generation capabilities

output analysis capabilities

all of the above

all of the above