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ISYE 6644 FINAL EXAM 202 5 QUESTIONS
If f(x, y) = cxy for all 0 < x < 1 and 1 < y < 2, where c is whatever value makes this thing integrate to 1, then X and Y are independent random variables - ANS-TRUE. (Because f(x, y) = a(x)b(y) factors nicely, and there are no funny limits.) In our Arena Call Center example, it was possible for entities to be left in the system when it shut down at 7:00 p.m. (even though we stopped allowing customers to enter the system at 6:00 p.m.). - ANS-TRUE โ because of the small chance that a callback will occur. An entity can be scheduled to visit the same resource twice, with different service time distributions on the two visits! - ANS-TRUE Arena has a built-in Input Analyzer tool that allows for the fitting of certain distributions to data. - ANS-TRUE The Kolmogorov-Smirnov test can be used both to see (i) if data seem to fit to a particular hypothesized distribution and (ii) if the data are independent. - ANS-FALSE. (It's just a goodness-of-fit test.) Welch's method is a graphical technique to estimate truncation (initialization bias) points for steady-state simulation. - ANS-TRUE If you are using a ranking-and-selection procedure and two competitors happen to fall within the indifference-zone, then you don't really care too much which one you end up selecting. - ANS-TRUE. That's why it's called the IZ! TRUE or FALSE? Sequential ranking-and-selection procedures are designed to stop early if one alternative seems to be way out in front of the others. - ANS-TRUE Suppose, when designing a ranking-and-selection procedure, you have decided to increase the desired probability of correct selection compared to a previous run of the procedure. What can you expect? (a) Sample sizes that are about the same (b) Larger sample sizes (c) Somewhat lower achieved Pr(CS) (d) A larger indifference zone (e) Lower confidence in your selection - ANS-b In Arena, you can use a single DECIDE module to route customers to that one of four servers having the smallest queue. - ANS-TRUE TRUE or FALSE? In Arena, you can use the expression EXPO(3) to generate exponential interarrival times having a mean of 1/3. - ANS-FALSE. The mean of an EXPO(3) is 3.
TRUE or FALSE? In Arena, an alternative expression to generate exponential interarrival times having a mean of 1/3 is - (1/3)*LN(UNIF(0,1)) - ANS-TRUE. This is the Inverse Transform Method to generate an observation coming from Exp(ฮป = 3). TRUE or FALSE? In Arena, it is possible to set the maximum capacity of a queue from within a SEIZE module. - ANS-FALSE. It is only possible to set a maximum queue capacity from within a QUEUE block found in the Blocks template. 2 TRUE or FALSE? In Arena, the expression NORM(-1,2)+NORM(3,2) has the same distribution as the expression NORM(2,4) - ANS-FALSE. The above expression evaluates to NORM(2,โ 8) since, in Arena, NORM(x,2) corresponds to a normal random variate with a mean of x and a standard deviation of 2; so the variance of this random variable is 22 = 4. TRUE or FALSE? In Arena, it is only possible to use one resource at a time in a PROCESS module. - ANS-FALSE TRUE or FALSE? In Arena, once an entity is defined, it is not possible to change the priority in which it gets processed by a resource. - ANS-FALSE TRUE or FALSE? In Arena, it is not possible to use a single RELEASE module to release multiple different resources - ANS-FALSE. We observed this in the ReleaseHospitalRoomLater demo in HW What is the distribution given by the Arena expression UNIF(1,2)+UNIF(1,2)? - ANS- Tria(2,3,4). An Arena ASSIGN module can be used to assign values to attributes and variables - ANS-TRUE TRUE or FALSE? In Arena, there are several ways to terminate a simulation, for example, when a given simulation run length has been reached, or when a given number of entities have been created, or when a specific stopping criterion as be met. - ANS-TRUE Suppose X is a discrete random variable that can take on four possible values: โ0.1, 0, 2 , and 4. The associated probabilities are, respectively: 0.3, 0.1, 0.2, and 0.4. Find an expression in Arena to generate one observation coming from this p.m.f. - ANS- DISC(0.3, โ0.1, 0.4, 0, 0.6, 2, 1, 4) DISC(F(x1), x1, F(x2), x2,.. .) Suppose there are 3 people in the line called joe.queue and 5 people in the line called tom.queue. What is the value of the following Arena expression? (NQ(joe.queue) > NQ(tom.queue)) + (TNOW >= 0 - ANS-The logical expression (NQ(joe.queue) > NQ(tom.queue)) is false (because 3 โค 5) and therefore 0. The logical
What is the distribution given by the Arena expression UNIF(0,2)+UNIF(0,2)? - ANS- Tria(0,2,4) TRUE or FALSE? An Arena ASSIGN module can be used to assign values to attributes but not variables. - ANS-FALSE. It can be used for both Suppose X is a discrete random variable that can take on four possible values: โ2, 1, 4, and 7. The associated probabilities are, respectively: 0.1, 0.3, 0.2, and 0.4. Find an expression in Arena to generate one observation coming from this p.m.f. - ANS- DISC(โ2, 0.1, 1, 0.4, 4, 0.6, 7, 1) Suppose there are 3 people in the line called joe.queue and 2 people in the line called tom.queue. What is the value of the following Arena expression? NQ(tom.queue) โ NQ(joe.queue) + (TNOW < โ1) - ANS-The logical expression (TNOW < - 1) is false (since time is always at least 0). Thus, NQ(tom.queue) โ NQ(joe.queue) โ (TNOW < โ1) = 6 + 0 = 6. TRUE or FALSE? The Arena Call Center model discussed in class restricted all servers to one specialty product each. - ANS-FALSE. The use of resource sets allowed certain servers to work on multiple products. The runs "up and down" test is most often used to test for goodness-of-fit of observations. - ANS-FALSE. It's a test for independence. Draw an Arena DECIDE block - ANS-It's a sideways diamond. In what Arena template would you find the Sequence spreadsheet? - ANS-Advanced Transfer. In Arena, you can LEAVE a station. What is the analogous block to use to get into a station? - ANS-ENTER Which method is not a viable way for an entity to LEAVE a resource? (i) Route; (ii) Connect; (iii) Move; (iv) Transport; (v) Convey. (There may be more than one correct answer.) - ANS-Move TRUE or FALSE? Suppose that U1, U2,... are truly i.i.d. U(0,1) random variables. Then a 95% chi-square goodness-of-fit test for uniformity will incorrectly reject uniformity of the observations about 5% of the time. - ANS-TRUE. (That's what Type I error is.) TRUE or FALSE? Suppose Z1,... , Z6 are i.i.d. standard normal random variables obtained by the Box-Muller method. Then P6 i=1 Z 2 i โผ Erlang3(1/2). - ANS-TRUE. The sum of 6 i.i.d. Z 2 i random variables is a ฯ 2 (6), which is itself an Erlang3(1/2).
TRUE or FALSE? If we want to generate X โผ Pois(4.5), then it's better to use a normal approximation than acceptance-rejection. - ANS-FALSE. (The normal approximation requires larger ฮป for the asymptotics to work and the efficiency to kick in.) TRUE or FALSE? The acceptance-rejection's majorizing function t(x) is usually a p.d.f. - ANS-FALSE. (It integrates to something > 1.) If X1,... , Xn are i.i.d. Exp(1/9), what is the expected value of the sample variance S 2 ? - ANS-E[S^2 ] = ฯ^2 = 81. Find x such that e 2x = 1/x. (Get within two decimals) - ANS-Bisection search quickly reveals that x is about 0. TRUE or FALSE? The square root of the sample variance is unbiased for the standard deviation - ANS-FALSE. (E[S 2 ] = ฯ 2 ; E[S] = ฯ.) Which is usually a better way to deal with initialization bias in steady-state simulation analysis: (i) make an extremely long run to overwhelm the bias, or (ii) perform truncation? - ANS-Truncate If W(t) is a standard Brownian motion process and a < b, find Pr(W(a) < W(b)). - ANS- W(a) โ W(b) is normally distributed with mean 0. This immediately yields a probability of 1/2. Suppose I use the method of overlapping batch means with sample size n = 10000 and batch size m = 500. Approximately how many degrees of freedom will the resulting variance estimator have? - ANS-Denote b = n/m = 20. You get approximately 3 2 (b โ 1) = 28.5 d.f. (Will also accept 3b/2 = 30 or anything reasonably close.) Suppose that I'm interested in selecting the most popular television show during a particular time period. What kind of selection problem is this โ (a) normal, (b) multinomial, or (c) Bernoulli? - ANS-b
