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ISYE 6644 SIMULATION WITH SOLUTIONS
What is a possible goal of an indifference-zone normal means selection technique? - ANSWER -Find the normal population having the largest mean, especially if the largest mean is ≫ the second-largest. TRUE or FALSE? The Bechhofer procedure for selecting the normal population with the largest mean specifies the appropriate number of observations to take from each competing population, and simply selects the competitor having the largest sample mean. - ANSWER -True TRUE or FALSE? Sometimes a single-stage procedure like Bechhofer's is inefficient. In fact, it's possible to use certain sequential procedures that take observations one-at-a-time (instead of all at once in a single stage) to make good selection decisions using fewer observations. - ANSWER -True For which scenarios(s) below might it be appropriate to use a Bernoulli selection procedure? a) Find the inventory policy having the largest profit. b) Find the drug giving the best chance of a cure. c) Find the maintenance policy having the lowest failure probability. d) Find the scheduling rule that that has the best chance of making an on-time delivery. - ANSWER -All three of (b), (c), and (d). Suppose that a Bernoulli selection procedure tells you to take 100 observations from each of two populations, A and B. It turns out
that A gets 85 successes and B gets 46 successes. What do you think? - ANSWER -1) A almost certainly has a higher success probability than B.
- We could've probably stopped sampling a bit earlier (i.e., with fewer than 100 observations) because A was so far ahead of B. For which scenarios(s) below might it be appropriate to use a multinomial selection procedure? - ANSWER -Find the most- popular political candidate. Suppose that we want to know which of Coke, Pepsi and Dr.pepper is the most popular. We would like to make the correct selection with probability of at least P=0.90 in the event that the ration of the highest-to-second-highest preference probabilities happens to be at least 0=1.4. How many people does the single- stage procedure Mbem require us to interview? - ANSWER - Go to the table in the notes and pick off the entry for k=3, P^⋆=0.90, and θ^⋆=1.4. Which of the following parameters can you get confidence intervals for? Means Variances Quantiles Differences between the means of two systems - ANSWER -All of the above If we have an iid normal sample of observations, X1, X2,...Xn, what probability distribution is most-commonly used to obtain cofidence intervals for the mean? - ANSWER -t
Statistical ranking and selection techniques have been designed to address a variety of comparison problems. Which ones from the following list? Find the population having the largest mean. Find the system with the smallest variance. Find the alternative with the highest success probability. Find the most-popular candidate. All of the above. - ANSWER -All of the above. Suppose we are dealing with i.i.d. normal observations with unknown variance. Which of the following is true about a 95% confidence interval for the mean μ? - ANSWER -We are 95% sure that our CI will actually contain the unknown value of μ. We are studying the waiting times arising from two queueing systems. Suppose we make 4 independent replications of both systems, where the systems are simulated independently of each other. replication system 1 system 1 10 25 2 20 10 3 5 40 4 30 30 Assuming that the average waiting time results from each replication are approximately normal, find a two-sided 95% CI for the difference in the means of the two systems. - ANSWER -This is a two-sample CI problem assuming unknown and unequal variances. We have [-29.76, 9.76]
This is sort of the same as Question 2, except we have now used common random numbers to induce positive correlation between the results of the two systems. Again find a two-sided 95% CI for the difference in the means of the two systems. - ANSWER -This is a paired-t CI problem assuming unknown variance of the differences. [-16.5, -3.5] Suppose A and B are two identically distributed, unbiased, antithetic estimators for the mean μ of some random variable, and let C = ( A + B ) / 2. Which of the following is true? - ANSWER -E [ C ] = μ and V a r ( C ) < V a r ( A ) / 2. Suppose that you want to pick that one of three normal populations having the largest mean. We'll assume that the variances of the three competitors are all known to be equal to σ 2 = 4. (Ya, I know that this is a crazy, unrealistic assumption, but let's go with it anyway, okey dokey?) I want to choose the best of the three populations with probability of correct selection of 95% whenever the best population's mean happens to be at least δ ⋆ = 1 larger than the second-best population's. How many observations from each population does Bechhofer's procedure N B tell me to take before I can make such a conclusion? - ANSWER -Using the notation of the notes, we want to make sure to get the right ANSWER with probability of P ⋆ = 0.95 whenever μ [ k ] − μ [ k − 1 ] ≥ δ ⋆ = 1. We simply go to NB's table with k = 3 and δ ⋆ / σ = 1 / 2 to obtain a sample size of n = 30 from each population. In the above problem, suppose that we take the necessary observations and we come up with the following sample means: X
Suppose we want to estimate the expected average waiting time for the first m = 100 customers at a bank. We make r = 4 independent replications of the system, each initialized empty and idle and consisting of 100 waiting times. The resulting replicate means are: i 1 2 3 4 Z i 5.2 4.3 3.1 4. Find a 90% confidence interval for the mean average waiting time for the first 100 customers. - ANSWER -[3.188,5.212] Consider a particular data set of 100,000 stationary waiting times obtained from a large queueing system. Suppose your goal is to get a confidence interval for the unknown mean. Would you rather use (a) 50 batches of 2000 observations or (b) 10000 batches of 10 observations each? - ANSWER -50 batches of 2000 observations because the method of batch means requires a very large batch size Consider the output analysis method of non overlapping batch means. Assuming that you have a sufficiently large batch size, it can be shown that when the number of batches b is even, the expected width of the 90% two-sided confidence interval for μ is proportional tot 0.05 , b − 1 b − 1 ( b − 1 2 ) ( b − 3 2 ) ⋯ 1 2 ( b − 2 2 )! .Using the above equation, determine which of the following values of b gives the smallest expected width. - ANSWER -b= Let h ( b ) denote the value of the above expression as a function of b. Then easy calculations reveal that h ( b ) = 3.157, h ( 4 ) = 1.019, and h ( 6 ) = 0.845. So the ANSWER is b = 6
Consider the following observations: 54 70 75 62 If we choose a batch size of 3, calculate all of the overlapping batch means for me. - ANSWER -66.3, 69. X1,3 = 1/3 ΣXi = 66.3 and X2,3 = 1/3 ΣXi = 69. TRUE or FALSE? Simulation output (e.g., consecutive customer waiting times) is almost never i.i.d. normal - and that's a big fat problem - ANSWER -TRUE We often distinguish between two general types of simulations with regard to output analysis. What are they called? - ANSWER - Finite-horizon and steady-state TRUE or FALSE? Suppose that X 1 , X 2 ,... , X n are consecutive waiting times, and we define the sample mean X ¯ = ∑ i = 1 n X i / n. Then V a r ( X ¯ ) = V a r ( X i ) / n. - ANSWER -Very FALSE! (The issue is that correlation between the observations messes up the variance of the sample mean. In fact, this is one of the main reasons why output analysis is difficult!) Which of the following scenarios might be well-suited for a finite- horizon analysis? - ANSWER -Simulate bank operations from 8: a.m. to 5:00 p.m. Simulate an inventory system until the first stock-out occurs TRUE or FALSE? The main method of attack for terminating simulations is via independent replications. - ANSWER -TRUE
c) You can use the method of batch means to obtain a confidence interval for the steady-state mean. d) The batch means estimator for the variance parameter 𝜎^2 is asymptotically unbiased as the batch size 𝑚→∞. e) All of the above - ANSWER -All of the above Which of the following methods can be used for steady-state analysis? a. Batch Means b. Independent Replications (though it might suffer some minor(?) initialization problems) c. Overlapping Batch Means (something for nothing!) d. Regeneration e. Standardized Time Series f. All of the above - ANSWER -f. All of the above It's GIGO time! Let's consider an M / M / 1 queueing system with Exp(λ) interarrivals and Exp(μ) FIFO services at a single server. You may recall from some class (either this one or stochastic processes) that the steady-state expected cycle time (i.e., the time that the customer is in the system, including wait + service) is w = 1 / ( μ − λ ). If you were to try this out in Arena, let's say with E X P O ( 10 = 1 / λ ) interarrivals and E X P O ( 8 = 1 / μ ) services (note the notation change between my usual "Exp" and Arena's " E X P O "), then we'd get w = 1 / ( 0.125 − 0.1 ) = 40. Go ahead, see for yourself in Arena, but make sure that you run the system for 100,000 or so customers so that you can be sure that you're in steady-state!
Finally, here's the GIGO question, which will show what can happen when you mis-model a component of your process: What is the (approximate) steady-state expected cycle time if you hav - ANSWER -about 23 The U N I F case has waaaay smaller tails than the E X P O, so it's reasonable to assume that the cycle times will tend to be lower for the U N I F case. In fact, after 100,000 customers in Arena, I got an average time of 23.5. Let's play Name That Distribution! The number of times a "3" comes up in 10 dice tosses. - ANSWER -Binomial Name That Distribution! The number of dice tosses until a 3 comes up. - ANSWER - Geometric Name That Distribution! The number of dice tosses until a 3 comes up for the 4th time. - ANSWER -Negative Binomial Name That Distribution! IQs - ANSWER -Normal Name That Distribution! Cases in which you have limited information, e.g., you only know the min, max, and "most likely" values that a random variable can take. - ANSWER -Triangular
Setting the derivative to 0 and solving yields λ ^ = x ¯ = 5. Duh! What a surprise! Suppose that we have a number of observations from a Pois( λ ) distribution, and it turns out that the MLE for λ is λ ^ = 5. What's the maximum likelihood estimate of Pr ( X = 3 )? - ANSWER -By invariance, Pr ^ ( X = k ) = e − λ ^ λ ^ k k! , so that we have Pr ^ ( X = 3 ) = e − x ¯ x ¯ k k! = e − 5 5 3 3! = 0.1404. Suppose that we observe X 1 = 5, X 2 = 9, and X 3 = 1. What's the method of moments estimate of E [ X 2 ]? - ANSWER -The MOM estimator for E[X^2] = ( 25 + 81 + 1 ) / 3 = 107 / 3 Consider the PRN's U 1 = 0.1 , U 2 = 0.9 , and U 3 = 0.2. Use Kolmogorov-Smirnov with α = 0.05 to test to see if these numbers are indeed uniform. Do we ACCEPT or REJECT uniformity? - ANSWER -Accept Since D < D α , n, we ACCEPT uniformity (though it's kind of a joke since it's only based on 3 observations) What does GIGO mean? - ANSWER -Garbage-in-garbage-out What's a good distribution for modeling heights of people? - ANSWER -Normal
What's a good distribution for modeling the number of random customer arrivals to a store? - ANSWER -The Poisson distribution is used for counting the number of arrivals over some interval of time. TRUE or FALSE? If the expected value of your estimator equals the parameter that you're trying to estimate, then your estimator is unbiased. - ANSWER -TRUE. (This is the definition of unbiasedness in words.) TRUE or FALSE? If X 1 , X 2 ,... , X n are i.i.d. with mean μ, then the sample mean X ¯ is unbiased for μ. - ANSWER -TRUE What is the MSE of an estimator? - ANSWER - 𝐵𝑖𝑎𝑠^2+𝑉𝑎𝑟𝑖𝑎𝑛𝑐𝑒 Suppose that X 1 = 4 , X 2 = 3 , X 3 = 5 are i.i.d. realizations from an E x p ( λ ) distribution. What is the MLE of λ? - ANSWER -Since λ^ = 1/ X ¯ = 0. TRUE or FALSE? It's possible to estimate two MLEs simultaneously, e.g., for the N o r ( μ , σ 2 ) distribution. - ANSWER -TRUE (it's possible based on taking the partial derivatives of the likelihood function with respect to each parameter). TRUE or FALSE? Sometimes it might be difficult to obtain an MLE in closed form. - ANSWER -TRUE. (Think of the gamma example that we did.)
goodness-of-fit tests - ANSWER -Kolmogorov-Smirnov Cramer-von Mises Anderson-Darling Shapiro-Wilk Which of the following problematic issues can arise in input data analysis? - ANSWER -Not enough data Data coming from strange-looking, "non-standard" distributions Nonstationary data (in which the distribution appears to change over time) Correlated data TRUE or FALSE? Arena has an Input Analyzer that automates distribution fitting for certain well-known distributions. - ANSWER -TRUE. (It's often a great time-saver! Suppose that a machine consists of components A, B, and C. If any of the components fail, then the machine breaks down. The failure times for A, B, and C are exponential (with probability 0.5), normal (w.p. 0.1), and Weibull (w.p. 0.4), respectively. YES or NO? Would the composition method be a good choice to generate the overall machine breakdown time for this scenario? - ANSWER -YES. Composition would be much more efficient than generating all three failure times for A, B, and C, and then taking the minimum. Let H and W denote a person's height and weight. Which of the following best describes the joint distribution of ( H , W )? - ANSWER -Bivariate normal distribution with positive correlation coefficient
The multivariate normal distribution's covariance matrix ∑ can be decomposed into the form ∑ = C C T. What does the C informally stand for? - ANSWER -Cholesky TRUE or FALSE? The times between Poisson( λ ) arrivals are i.i.d. Exp( λ ). - ANSWER -TRUE. (... as long as ( λ ) doesn't change over time.) TRUE or FALSE? The times between nonhomogeneous Poisson( λ ) arrivals are i.i.d. Exp( λ ). - ANSWER -FALSE. (They're independent, but the distribution changes as the arrival rate λ ( t ) changes over time.) What is the recommended method to generate nonhomogeneous Poisson arrivals? - ANSWER -Thinning Consider the time series Y i = ϕ Y i − 1 + ϵ i , i = 1 , 2 , ..., where the ϵ i's are i.i.d. normal. What is this time series called? (Give the best ANSWER .) - ANSWER -first-order autoregressive process Consider the waiting time W i Q of the i th customer in an M / M / 1 queuing system with first-in-first-out services. What relation allows you to calculate customer ( i + 1 )'s waiting time W i + 1 Q based on W i Q, customer i's service time S i, and customer ( i + 1 )'s interarrival time I i + 1? - ANSWER -Lindley's equation W^Q_{i+1} = max{0, W^Q_i + S_i - I_{i+1}} Brownian motion is also known as ... - ANSWER -Wiener process
TRUE or FALSE? If you can't find a good theoretical distribution to model a certain random variable, you might want to use the empirical distribution of the data to do so. - ANSWER -TRUE. That's the point of this lesson! TRUE or FALSE? The convolution method involves sums of random variables. - ANSWER -TRUE Support that U1 and U2 are PRNs. What's the distribution of U1 + U2? - ANSWER -Triangular (0,1,2) Suppose that I want to generate a simple Unif(2/3, 1) via A-R. Support I generate a PRN U1=0.16. Do I accept U1 as my Unif(2/3, 1)? - ANSWER -NO. In this example, we only accept if U1 ≥ 2/3; so we reject and try again until we meet that condition. TRUE or FALSE? The proof that A-R works is really easy. - ANSWER -FALSE. Super false, in fact! Suppose that 𝑋 is a continuous RV with p.d.f. (𝑥)=30𝑥^4(1−𝑥), for 0<𝑥<1. Why is acceptance-rejection a good method to use to generate 𝑋? - ANSWER -Because the c.d.f. of 𝑋 is very hard to invert True or False? The A-R algorithm for X~Pois(λ) tells us to generate PRNs until e^-λ .... Ui for the first time, and then set X=n. - ANSWER -TRUE. Yup - that's how you do it (though sometimes it's a bit tedious if λ is large).
Unif(0,1) PRNs can be used to generate which of the following random entities? - ANSWER -All of the above --- and just about anything else! If X is an Exp(λ) random variable with c.d.f. F ( x ) = 1 − e − λ x, what's the distribution of the random variable 1 − e − λ X? - ANSWER -Unif(0,1) If U is a Unif(0,1) random variable, what's the distribution of − 1 λ ln ( U )? - ANSWER -Exp(λ) Suppose that U 1 , U 2 , ... , U 5000 are i.i.d. Unif(0,1) random variables. Using Excel (or your favorite programming language), simulate X i = − ln ( U i ) for i = 1 , 2 , ... , 5000. Draw a histogram of the 5000 numbers. What p.d.f. does the histogram look like? - ANSWER -Exponential Suppose the c.d.f. of X is F ( x ) = x^3 / 8, 0 ≤ x ≤ 2. Develop a generator for X and demonstrate with U = 0.54. - ANSWER -X = 2 U^{1/3} = 1. If X is a Nor(0,1) random variate, and Φ ( x ) is the Nor(0,1) c.d.f., what is the distribution of Φ ( X )? - ANSWER -Uniform If U is a Unif(0,1) random variate, and Φ ( x ) is the Nor(0,1) c.d.f., what is the distribution of 2 Φ − 1 ( U ) + 3? - ANSWER -Nor(3,4) How would you simulate the sum of two 6-sided dice tosses? (Note that ⌈ ⋅ ⌉ is the round-up function; and all of the U's denote PRNs.) - ANSWER - ⌈ 6 U 1 ⌉ + ⌈ 6 U 2 ⌉
