CSci 426/526 — Simulation — Fall 2008
Midterm Exam
Take-Home Portion
This exam is due at the beginning of class (2:00pm) on Tuesday, November 4 and will not be accepted late
without prior approval. Graduate students must work all three problems; undergraduate students must work
problem 1 and either of the remaining two problems.
You may not seek or receive assistance on this test from any person other than me. Failure to observe this
rule will be treated as an honor code violation.
Problem 1: [20 points] Mr. K is the manager of a new Snowball Express sno-cone stand which will be
opening for business. The stand is operated by a single attendant who is behind a window. People come up
to the window, tell the attendant which flavor sno-cone they want, wait for the attendant to fill their order,
then pay and leave. When there is more than one customer at the window, a line forms so that customers
are served in the order that they arrived. People always buy one and only one sno-cone at $3 each. The
cost of materials to make sno-cones is negligible, so Mr. K considers all of the money taken in to be profit,
except that the attendant has to be paid from the money received.
After interviewing several applicants for the attendant’s position, Mr. K has narrowed his choice to two
people: Fran and Bill. Fran can complete a transaction (take the order, make the sno-cone, take payment
and make change) in 20 seconds on the average. Bill can perform the same job in 30 seconds, on the average.
Fran is faster than Bill but demands $12 per hour, whereas Bill will work for $6 per hour.
Market research has revealed that, on the average, Mr. K can expect one customer per minute to come up
to the window. The research also shows that if a customer comes to the window and there are already N
people in line, then the customer will promptly turn about in a huff and storm off to Ben and Jerry’s.
Studies of real sno-cone stands have further shown that, even though the average customer arrival rate is
one per minute, usually the amount of time between customers is less than that. It’s the occasional lull in
business that makes the average time between customers longer than that which is usually observed. The
same is true of the time it takes to make sno-cones. Thus, assume that the time between customer arrivals
and the time it takes a person to make a sno-cone are exponentially distributed random values.
(a) For both Fran and Bill, and based on 1 000 000 served customers, construct a table of the estimated
steady-state probability that a customer will go to Ben and Jerry’s when the number of people in line N
is 1, 2, 3, 4, and 5. (b) Construct similar tables if the time to serve a customer for Fran is changed to be
Uniform(5, 35) and the time to serve a customer for Bill changed to be Uniform(15,45). (c) Comment on how
the probability of rejection (i.e., the probability to go to Ben and Jerry’s) depends on the service process.
(d) Assuming exponentially distributed service times for both Fran and Bill, if Nis equal to 3, which person
is the more cost-effective to hire: Fran or Bill? For this part of the problem provide also analytical solutions.
(e) Discuss what you did to convince yourself that your results are correct.
Problem 2: [20 points] According to the ANSI C standard, the type unsigned long is guaranteed to
support integers in the range 0,1,...,232 −1.
•Modify algorithm 2.2.1 such that it never produces an integer outside of the range (0,1,...,m).
•How many full-period multipliers are there for m= 232 −135?
•For m= 232 −135, how many full–period, modulus compatible multipliers are there? What are the
ten smallest?
•Find the smallest full–period, modulus compatible multiplier a≥100,000 for mo dulus m= 232 −135.
For this (a,m) pair, construct a table of maximal modulus-compatible jump multipliers (similar to
example 3.2.6) for 1024, 512, 256, 128, 64, and 32 streams.
