Download flashcards for homework and more Exercises Mathematics in PDF only on Docsity!
ISYE6644 (Simulation) Midterm 1
Leave the first rating
Students also studied
Terms in this set (82)
Save
ISYE 6644 EX 1
67 terms
HIMDestroyer (^) Preview
ISYE 6644 - Final prep 113 terms
Asel_Baidyldaeva (^) Preview
Simulation Homework 2 18 terms
Mister_Moderna (^) Preview
What are some characteristics of simulation models? 1. Discrete (vs. continuous)
- Stochastic (vs. deterministic)
- 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.
- Ask "what if" questions.
- Aid in system design and optimization
- Can simulate almost anything
What are the reasons to simulate? 1. Will the system accomplish its goals?
- Current system won't accomplish its goals, now what?
- Need incremental improvement.
- Create a specification or action plan
- Solve a problem, like a bottleneck.
- Resolve disputes
- Sell an idea
What are the advantages of simulation? 1. Can study models too complicated for analytical / numerical treatment
- Study detailed relations that might be lost in the analytical or numerical treatment
- Use as a basis for experimental studies of systems
- Use to check results and give credibility to conclusions obtained by other methods.
- Reduce design blunders
- Really nice demo method
- (sometimes) very easy
What are the disadvantages of simulation? 1. Sometimes not so easy
- Sometimes very time consuming / costly
- Simulations give "random" output (and lots of misinterpretation of results is possible)
- 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.
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/ C) Almost exactly 1/ D) Somewhat greater than 1/
D) Somewhat greater than 1/
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. B) 2. C) 3. D) 3. E) 4.
D) 3.
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?
- Generate pseudo-random numbers (PRNs) using a deterministic algorithim (not really random but appears to be)
- 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/ C) 3/ D) 3
C) 3/
𝑋1=(5𝑋0+3) mod(8) = 8 mod(8) = 0, and then 𝑋2=(5𝑋1+3) mod(8) = 3 mod(8) =
- So 𝑈2=𝑋2/8=3/
Suppose that we generate a pseudo-random number U = 0.728. Use this to generate an Exponential(𝜆=3) random variate.
A) -0. B) 0. C) -0. D) 0.
B) 0.
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
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/ C) 13/ D) 1/
D) 1/
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?
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
statistics are random variables
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 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.
- Process (serve) them, maybe after waiting in line.
- 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
