Check One: MATH428
or
MATH828
Exam 2
Spring Semester, 2016
Print Your Name Legibly:
Score:
1(25pts) The birth-death diagram below represents a queuing system of one server that has a finite calling population
with a constant input rate λ for each customer and whose customers tend to renege with a constant reneging
rate for each customer. Assuming the arrival, the service, and the renege processes are mutually independent and
exponentially distributed.
4.5
3
1.5
0
(a) What is the total population N and what is the value of λ?
0
1
2
2
3
4
4
3
5
(b) What is the service rate µ and what is the reneging rate θ?
(c) Find the probability that one customer is waiting in the queue.
2(25pts) Consider a staged bipartite graph as shown with the assumption that each edge weight is some type of cost.
(a) The dynamical programming backward iterative method to find the optimal path or paths from state A to
state G looks like below for the first two iterations. Fill in the blanks below for the f2 function.
S3
E
F
f3∗ (S3 )
2
1
S4∗
G
G
S2 \S3
B
C
D
f2 (S2 , S3 )
= CS2 S3 + f3∗ (S3 )
E
F
7
5
6
3
B
f2∗ (S2 )
5
S3∗
F
4
2
A
5
4 4
E
2
C 3
3
D
0
2
G
F
1
(b) Complete the remaining iterative function f1 and then find the optimal solution.
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Cheat Sheet: Useful performance parameters for M/M/1 model:
ρ = λ/µ, L =
λ
λ2
, Lq =
µ−λ
µ(µ − λ)
n
Pn = (1 − ρ)ρ
P {Wq > t} = ρe−µ(1−ρ)t for t ≥ 0
P0 = 1 − ρ
P {Wq = 0} = P0
P {W > t} = e−µ(1−ρ)t for t ≥ 0
3(25pts) On-line orders for a product come in according to a Poisson process at a mean rate of 30 per hour. Two
candidates are applying for the job to fill the mail-out orders. Both candidates have an exponential distribution for
service time, with candidate X having a mean of 1.2 minutes and candidate Y having a mean of 1.5 minutes.
(a) Find the expected waiting time (in minutes) for an order in the system before it is ready for shipping if X is
hired.
(b) Find the same expected waiting time if Y is hired.
(c) If the management has decided that for each minute saved for an order’s waiting time the handler/server
should make $1 more per hour, how much more should handler X make per hour than handler Y should?
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4(25pts) A presidential candidate is to buy three prime-time commercials on TV stations located in three different areas.
Based on polling, an estimate below has been made of the number of additional votes in thousands that can be
won in the different broadcasting areas depending on the number of commercials run. Use the graphical method
for dynamic programming to determine the optimal ways to distribute the commercials in order to maximize the
estimated number of votes won.
Area
Commercials
1
2
3
0
0
0
0
4
6
5
1
2
7
8
9
9
10
11
3
5(10pts) (For Math828 Students Only) Consider the M/M/1 model with λ < µ. Determine the steady-state probability that a customer’s random waiting time W in the system is longer than the expected waiting time W in the
system, i.e., P {W > W }. (Use the cheat sheet formulas from page 2.)
2 Bonus Points: True or False: The Nebraska State Book is the Bible.
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