simulation 6644 HW3

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Feb 20, 2024

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(Lesson 3.1: Solving a Differential Equation.) Suppose that f(z) = €**. We know that if is small, then f(e+h)—f(z) fl(z) = _— Using this expression with A = 0.01, find an approximate value for f'(1). a.l c.7.38 d. 14.93 We have / o JEth)—flo) _ edeth) ¢k f(z) = —n T T n So using h = 0.01, we have 202 )~ Sst = 1493 Thus, the answer is (d).
1 Question 2 | (Lesson 3.1: Solving a Differential Equation.) Suppose that f(z) = €**. What is the actual value of f'(1)? d.2¢? ~ 14.78 f'(z) = 2€¥*,s0 that f'(1) = 2¢?, and thus the answer is (d). e. 14.93
Question 3 (Lesson 3.1: Solving a Differential Equation.) Consider the differential equation f'(z) = (@ + 1) f(x) with f(0) = 1. What is the exact formula for f(x)? This takes a little work. The good news is that you can actually get the true answer using the technique of separation of variables. We have f@ _ o = ¢ A1 so that f@ . [ f(z)da:ffav:Jrldz Which implies In(f(z)) = & +z+C, 2 sothat f(z) = Ke7 ™, where C and K are arbitrary constants. Setting f(0) = 1 implies that K = 1, so that the exact answer is , the 2 answer is f(z) = e7 1%, i.e. choice (c). d fll\ = exp{a® + 2z}
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1/1pts Question 4 (Lesson 3.1: Solving Differential Equations.) Consider the differential equation f'(z) = (z + 1) f(z) with £(0) = 1. Solve for f(0.20) using Euler's approximation method with increment A = 0.01 for z € [0,0.20]. a. £(0.20) ~ 0.0 . £(0.20) ~ 1.0 ¢ £(0.20) ~ 1.24
¢ £(0.20) ~ 1.24 2 By previous question, the true answer is the answer is f(z) = ez ™. But our job is to use Euler to come up with an iterative approximation, so here it goes. As usual, we start with f@+h) = f(@) + h f'(2) = f(2) + h(z + 1) f() = f(@)[1 + (= +1)], from which we obtain the following table. z | Eulerapprox true f(z) 0.00 1.0000 1.0000 0.01 1.0100 1.0101 0.02 1.0202 1.0204 0.03 1.0306 1.0309 0.04 1.0412 1.0416 0.05 1.0521 1.0526 0.06 1.0631 1.0637 0.07 1.0744 1.0751 0.08 1.0859 1.0868 0.09 1.0976 1.0986 0.10 1.1096 1.01107 0.19 1.2287 1.2313 0.20 1.2433 1.2461 Wow, what a good match! In any case, the answer is (c).
Question 5 daes (Lesson 3.2: Monte Carlo Integration.) Suppose that we want to use Monte Carlo integration to approximate e f13 H;w dz. If Uy, Us, ..., U, areiid. Unif(0,1)'s, what's a good approximation fn for I? 1y 1 w 2ai=1 T30, In the notation of the lesson, the general approximation we've been using is T ‘“Zga+<b—a> ) 3-1 (3 - 1)) n = %Zm +auy) =1 _Ei; T 1+ (1+20)
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2 & 1 _H;1+(1+2U,) oy gl Thn& I so that the answer has simplified very nicely to ().
Question 6 LS (Lesson 3.2: Monte Carlo Integration.) Again suppose that we want to use Monte Carlo integration to approximate I = 1-13 H;z dzx. You may have recently discovered that the MC estimator is of the form _iyw 1 In=3Y0 oo Estimate the integral I by calculating I_n with the following 4 uniforms: U; =03 U, =0.9 Us =0.2 Uy =0.7
d.0.679 I,= % el TIU, = 0.679, so the answer is Iy= % o #U‘ = 0.679, so the answer is (d). Question 7 SAte (Lesson 3.2: Monte Carlo Integration.) Yet again suppose that we want to use Monte Carlo integration to approximate I = flz HLI dz. What is
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(Lesson 3.2: Monte Carlo Integration.) Yet again suppose that we want to use Monte Carlo integration to approximate I = f13 H;m dz. What is the exact value of I? a.0.197 b.0.693 I=1In(1+2)} =1In(4) — In(2) = 0.693. Thus, the answer is (b). I=In(1+2)|? =In(4) — In(2) = 0.693. Thus, the answer is (b).
Question 8 RS (Lesson 3.3: Making Some 7.) Inscribe a circle in a unit square and toss n = 1000 random darts at the square. Suppose that 760 of those darts land in the circle. Using the technology developed in class, what is the resulting estimate for 7? oo The estimate #n = 4 x (proportion in circle) = 4(760/1000) = 3.04 Thus, the answer is (d).
. 1/ 1 pt: Question 9 arpe (Lesson 3.4: Single-Server Queue.) Consider a single- server Q with LIFO (last-in-first-out) services. Suppose that three customers show up at times 5, 6, and 8, and that they all have service times of 4. When does customer 2 leave the system? d. 17 Let's make a version of our usual table.
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Question 10 A/AES (Lesson 3.5: (s, .S) Inventory Model.) Consider our numerical example from the lesson. What would the third day's total profits have been if we had used a (4,10) policy instead of a (3,10)? Day | begin sales order hold penalty | TO. 1 stock D; I, Z;| rev cost cost cost pro LaTeX 1 10 5 5 0 50 0 =5 0 4 2 5 2 3 7 20 —(2+4(7) -3 0 = 3 10 8 2 8 80 —(2+4(8)) -2 0 4 Thus, the answer is (c).
Question 11 Lilre (Lesson 3.6: Simulating Random Variables.) If U is a Unif(0,1) random number, what is the distribution of —0.5¢n(U)? 2 Who knows? b. Exp(2) By the Inverse Transform Theorem, we know that — iln(l — U) ~ Exp()). Butsince U and 1 — U are both Unif(0,1) (why?), we also have —+(U) ~ Exp()). In particular, —0.5¢n(U) ~ Exp(2), so that the answer is (b).
By the Inverse Transform Theorem, we know that 7§2n(1 —U) ~ Exp()). But since U and 1 — U are both Unif(0,1) (why?), we also have —3n(U) ~ Exp()). In particular, —0.5¢n(U) ~ Exp(2), so that the answer is (b). Question 12 B8
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(Lesson 3.6: Simulating Kandom Variables.) It Uy and U, are i.i.d. Unif(0,1) random variables, what is the distribution of Uy + Us? Hints: (i) | may have mentioned this in class at some point; (i) You may be able to reason this out by looking at the distribution of the sum of two dice tosses; or (iii) You can use something like Excel to simulate U; + Uy many times and make a histogram of the results. 1. Unif(0,2 . Normal c. Exponential d. Triangular By any of the hints, you get a Triangular(0,1,2) distribution, i.e., answer (d).
Question 13 s (Lesson 3.7: Spreadsheet Simulation.) | stole this problem from the Banks, Carson, Nelson and Nicol text (5th edition). Expenses for Joey's college attendance next year are as follows (in $): Tuition = 8400 Dormitory = 5400 Meals ~ Unif(900,1350) Entertainment ~ Unif(600,1200) Transportation ~ Unif(200,600) Books ~ Unif(400,800) Here are the income streams the student has for next year: Scholarship = 3000 Parents = 4000 Waiting Tables ~ Unif(3000,5000) Library Job ~ Unif(2000,3000)
Use Monte Carlo simulation to estimate the expected value of the loan that will be needed to enable Joey to go to college next year. c. $3325 An easy spreadsheet simulation (or an almost-as-easy exact analytical calculation) reveals that the expected loan amount is $3325, or answer (c). If you don't believe me, here's some Matlab code (if you happen to have Matlab)... m = 1000000; % reps Income = 7000 + unifrnd(3000,5000,[1 m]) + unifrnd(2000,3000,[1 m]); Expenses = 13800 + unifrnd(900,1350,[1 m]) + unifrnd(600,1200,[1 m]) + unifrnd(200,600,(1 m]) + unifrnd(400,800,[1 m]); Totals = Income - Expenses; hist(Totals,100) mean(Totals) var(Totals)
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Question 14 LS UIE (Lesson 4.1: Steps in a Simulation Study.) Which steps are regarded as essential for a successful simulation study? (There may be more than one correct answer.) a. Problem formulation b. Model validation c. Model verification d. Experimental design e. Output analysis f. Attendance at a Justin Bieber concert
Question 15 — (Lesson 4.1: Steps in a Simulation Study.) Suppose that | have modelled the arriving calls to a call center as a Poisson process. What do | have to carry out in order to determine if that’s indeed a reasonable model assumption? b. Model validation c. Model verification d. Attend a Justin Bieber concert (b).
Question 16 e (Lesson 4.2: Some Useful Definitions.) Which of the following times could be regarded as events? (There may be more than one correct answer.) | | m a. Customers arrive at Justin's concert venue b. Justin forgets a lyric c. Justin sings the wrong note d. Angry customers depart the venue e. A customeris 11 years old (a)-(d) are all events. [(e) is an attribute, not an event.]
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Question 17 e (Lesson 4.2: Some Useful Definitions.) TRUE or FALSE? Customer waiting times are activities because these are typically explicitly specified in the simulation. False FALSE. Although activities are indeed times of specified length,waiting times typically need to be calculated from the sequence of customer arrival and service times | and so are not explicitly specified beforehand.
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uUESLIVIT 10 (Lesson 4.3: Time-Advance Mechanisms.) TRUE or FALSE? The simulation clock time is a variable. True Question 19 APE (Lesson 4.3: Time-Advance Mechanisms.) TRUE or FALSE? The simulation clock time always equals real time. False
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| Question 20 Lo (Lesson 4.3: Time-Advance Mechanisms.) TRUE or FALSE? In this class, time always moves forward. ! Question 21 1/1pts [ (Lesson 4.3: Time-Advance Mechanisms.) TRUE or FALSE? A fixed-increment time-advance mechanism is used primarily in continuous-time models such as those involving differential equations.
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Question 22 i (Lesson 4.3: Time-Advance Mechanisms.) TRUE or FALSE? A next-event time-advance mechanism is typically used in queueing models involving customer arrivals, services, and departures. True Question 23 /A (Lesson 4.3: Time-Advance Mechanisms.) TRUE or FALSE? The future events list contains all known upcoming events, including arrival times, departure times, and machine breakdown times.
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e True False Question 24 BYEES (Lesson 4.3: Time-Advance Mechanisms.) TRUE or FALSE? The FEL can be updated any time an event occurs. True Fals
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Question 25 /A (Lesson 4.3: Time-Advance Mechanisms.) TRUE or FALSE? It is possible for the system state to change between consecutive event times. Question 26 e (Lesson 4.3: Time-Advance Mechanisms.) TRUE or FALSE? In a simulation using the “next-event” time- advance mechanism, the simulation clock moves to the most-imminent event.
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Question 27 SASEE (Lesson 4.3: Time-Advance Mechanisms.) TRUE or FALSE? When a new event occurs, the simulation may update the chronological order of the FEL's events by inserting new events, deleting events, moving them around, or even doing nothing.
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Question 28 d/AEE (Lesson 4.3: Time-Advance Mechanisms.) TRUE or FALSE? Almost every discrete-event computer simulation language maintains a FEL somewhere. e Question 29 S4LPE (Lesson 4.3: Time-Advance Mechanisms.) TRUE or FALSE? In Arena, you are responsible for maintaining the language’s FEL.
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D - Question 30 S4te (Lesson 4.4: Two Modeling Approaches.) Which is generally the easier simulation modeling approach — Event-Scheduling or Process-Interaction? a. Event-Scheduling b. Process-Interaction (b).
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Quuestion 51 modeling approach adopted by Arena — Event- Scheduling or Process-Interaction? a. Event-Scheduling | | (Lesson 4.4: Two Modeling Approaches.) Which is the m b. Process-Interaction (b). Question 32 e (Lesson 4.4: Two Modeling Approaches.) TRUE or FALSE? A simulation language incorporating the P-1
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approach considers the events that a generic customer undergoes as it passes through the system, and then automatically does the bookkeeping to keep track of how all such customers interact with each other. True False Question 33 8/8CS (Lesson 4.5: Simulation Languages.) How many simulation languages are there? w
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d. >>50. (d). Question 34 SAPE (Lesson 4.5: Simulation Languages.) Where can you learn about simulation languages? (There may be more than one correct answer.) b. Simulation language textbooks a. Right here, right now! | [Fome— |
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d. Vendor short courses 1e Justin Bieber School of Hard Ki spelling, Justin.) @), (b), (c) and (d). Question 35 simulation language, what characteristics do you have | | | | (Lesson 4.5: Simulation Languages.) When selecting a to take into consideration?
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b. Ease o c. Modeling “world view” (e.g., event-scheduling or process-interaction) d. Random variate generation c f. All of the above (f).
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