ISYE 6644 Final Exam Study Guide: Questions &
Answers
Which of the following problems might best be characterized by a
finite-horizon simulation?
(Ans- Simulating the operations of a bank from 9:00 a.m. until 5:00
p.m.
Let's run a simulation whose output is a sequence of daily inventory
levels for a particular product. Which of the following statements is
true?
(Ans- The consecutive daily inventory levels may not be identically
distributed.
Suppose that X 1 , X 2 , ... is a stationary (steady-state) stochastic
process with covariance function R k ≡ C o v ( X 1 , X 1 + k ), for k = 0 , 1 ,
.... We know from class that the variance of the sample mean can be
represented asV a r ( X ¯ n ) = 1 n [ R 0 + 2 ∑ k = 1 n − 1 ( 1 − k n ) R k ]
.We also know from class that for a simple AR(1) process, we have R k =
ϕ k, k = 0 , 1 , 2 , ... Compute V a r ( X ¯ n ) for an AR(1) process with n =
3 and ϕ = 0.8.
(Ans- 0.831
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 4.2
Find a 90% confidence interval for the mean average waiting time for
the first 100 customers.
(Ans- [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?
(Ans- 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.
(Ans- b=6
, 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.
(Ans-
66.3, 69.0
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
(Ans- TRUE
We often distinguish between two general types of simulations with
regard to output analysis. What are they called?
(Ans- Finite-horizon and steady-state
Answers
Which of the following problems might best be characterized by a
finite-horizon simulation?
(Ans- Simulating the operations of a bank from 9:00 a.m. until 5:00
p.m.
Let's run a simulation whose output is a sequence of daily inventory
levels for a particular product. Which of the following statements is
true?
(Ans- The consecutive daily inventory levels may not be identically
distributed.
Suppose that X 1 , X 2 , ... is a stationary (steady-state) stochastic
process with covariance function R k ≡ C o v ( X 1 , X 1 + k ), for k = 0 , 1 ,
.... We know from class that the variance of the sample mean can be
represented asV a r ( X ¯ n ) = 1 n [ R 0 + 2 ∑ k = 1 n − 1 ( 1 − k n ) R k ]
.We also know from class that for a simple AR(1) process, we have R k =
ϕ k, k = 0 , 1 , 2 , ... Compute V a r ( X ¯ n ) for an AR(1) process with n =
3 and ϕ = 0.8.
(Ans- 0.831
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 4.2
Find a 90% confidence interval for the mean average waiting time for
the first 100 customers.
(Ans- [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?
(Ans- 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.
(Ans- b=6
, 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.
(Ans-
66.3, 69.0
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
(Ans- TRUE
We often distinguish between two general types of simulations with
regard to output analysis. What are they called?
(Ans- Finite-horizon and steady-state
