ISYE 6644 CORRECT FINAL EXAM QUESTIONS AND
ANSWERS SURE A+
✔✔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 - ✔✔TRUE
✔✔We often distinguish between two general types of simulations with regard to output
analysis. What are they called? - ✔✔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. - ✔✔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? -
✔✔Simulate bank operations from 8:00 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. - ✔✔TRUE (Independent replications help to mitigate issues
with simulation outputs not being i.i.d.)
✔✔TRUE or FALSE? You can also conduct finite-horizon estimation for quantities other
than expected values, e.g., simulate a bank from 8:00 a.m. to 5:00 p.m., and find a
confidence interval for the 95th quantile of customer waiting times. - ✔✔TRUE (the
method of independent replications allows us to determine sample variances, which
subsequently allows us to determine confidence intervals)
✔✔How can we deal with initialization bias if we want to do a steady-state analysis? -
✔✔Make an extremely long run in order to overwhelm it.
Truncate (delete) some of the initial data.
, ✔✔Which of the following scenarios might be well-suited for a steady-state analysis? -
✔✔Simulate an assembly line working 24/7
A Markov chain simulated until the transition probabilities appear to converge
✔✔Which of the following is useful for analyzing steady-state simulation output? -
✔✔The method of batch means
✔✔Which of the following statements is true?
a) The method of batch means is easy to use.
b) Batch means chops the consecutive observations into a number of nonoverlapping,
contiguous batches
c) You can use the method of batch means to obtain a confidence interval for the
steady-state mean 𝜇 .
d) The batch means estimator for the variance parameter 𝜎^2 is asymptotically
unbiased as the batch size 𝑚→∞ .
e) All of the above - ✔✔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 - ✔✔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 - ✔✔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. - ✔✔Binomial
ANSWERS SURE A+
✔✔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 - ✔✔TRUE
✔✔We often distinguish between two general types of simulations with regard to output
analysis. What are they called? - ✔✔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. - ✔✔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? -
✔✔Simulate bank operations from 8:00 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. - ✔✔TRUE (Independent replications help to mitigate issues
with simulation outputs not being i.i.d.)
✔✔TRUE or FALSE? You can also conduct finite-horizon estimation for quantities other
than expected values, e.g., simulate a bank from 8:00 a.m. to 5:00 p.m., and find a
confidence interval for the 95th quantile of customer waiting times. - ✔✔TRUE (the
method of independent replications allows us to determine sample variances, which
subsequently allows us to determine confidence intervals)
✔✔How can we deal with initialization bias if we want to do a steady-state analysis? -
✔✔Make an extremely long run in order to overwhelm it.
Truncate (delete) some of the initial data.
, ✔✔Which of the following scenarios might be well-suited for a steady-state analysis? -
✔✔Simulate an assembly line working 24/7
A Markov chain simulated until the transition probabilities appear to converge
✔✔Which of the following is useful for analyzing steady-state simulation output? -
✔✔The method of batch means
✔✔Which of the following statements is true?
a) The method of batch means is easy to use.
b) Batch means chops the consecutive observations into a number of nonoverlapping,
contiguous batches
c) You can use the method of batch means to obtain a confidence interval for the
steady-state mean 𝜇 .
d) The batch means estimator for the variance parameter 𝜎^2 is asymptotically
unbiased as the batch size 𝑚→∞ .
e) All of the above - ✔✔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 - ✔✔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 - ✔✔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. - ✔✔Binomial
