ISYE 6644, FINAL PREP LATEST ACTUAL EXAM WITH
VERIFIED QUESTIONS & CORRECT ANSWERS
Question 1.
In general, the majorizing function t ( x ) is itself a p.d.f. f ( x ).
ANSWER
False
the majorizing function generally integrates to a number greater than 1,
and so it cannot be a legitimate p.d.f.
Question 2.
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.
Question 3.
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!
,Question 4.
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}\}
Question 5.
Suppose H 0 is true, but you've just rejected it! What have you done?
ANSWER
You've committed a Type I
error.
Question 6.
What does GIGO mean?
ANSWER
Garbage-in-garbage-out
Question 7.
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.
, Question 8.
Suppose that is a continuous RV with p.d.f. ( )=30 ^4(1− ), for 0< <1. Why is acceptance-
rejection a good method to use to generate ?
ANSWER
Because the c.d.f. of is
very hard to invert
Question 9.
In the above problem, suppose that we take the necessary observations and we come up with the
following sample means: X ¯ 1 = 7.6, X ¯ 2 = 11.1, and X ¯ 3 = 3.6. What do we do?
ANSWER
Pick population 2 and say
that we are right with probability at least 95%
Question 10.
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
Question 11.
CRN depends on someone's ability to manipulate the underlying pseudo-random numbers - e.g., use
the same arrival times when simulating two competing simulated systems. So who ultimately controls
those PRNs??
ANSWER
You do - you are powerful!
Question 12.
How can we deal with initialization bias if we want to do a steady-state analysis?
ANSWER
Make an extremely long
run in order to overwhelm
it.
Truncate (delete) some of
the initial data.
VERIFIED QUESTIONS & CORRECT ANSWERS
Question 1.
In general, the majorizing function t ( x ) is itself a p.d.f. f ( x ).
ANSWER
False
the majorizing function generally integrates to a number greater than 1,
and so it cannot be a legitimate p.d.f.
Question 2.
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.
Question 3.
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!
,Question 4.
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}\}
Question 5.
Suppose H 0 is true, but you've just rejected it! What have you done?
ANSWER
You've committed a Type I
error.
Question 6.
What does GIGO mean?
ANSWER
Garbage-in-garbage-out
Question 7.
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.
, Question 8.
Suppose that is a continuous RV with p.d.f. ( )=30 ^4(1− ), for 0< <1. Why is acceptance-
rejection a good method to use to generate ?
ANSWER
Because the c.d.f. of is
very hard to invert
Question 9.
In the above problem, suppose that we take the necessary observations and we come up with the
following sample means: X ¯ 1 = 7.6, X ¯ 2 = 11.1, and X ¯ 3 = 3.6. What do we do?
ANSWER
Pick population 2 and say
that we are right with probability at least 95%
Question 10.
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
Question 11.
CRN depends on someone's ability to manipulate the underlying pseudo-random numbers - e.g., use
the same arrival times when simulating two competing simulated systems. So who ultimately controls
those PRNs??
ANSWER
You do - you are powerful!
Question 12.
How can we deal with initialization bias if we want to do a steady-state analysis?
ANSWER
Make an extremely long
run in order to overwhelm
it.
Truncate (delete) some of
the initial data.
