# UVI Math Worksheet

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11
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12-19 Juhn and Sons Wholesale Fruit Distributors employs one
worker whose job is to load fruit on outgoing company trucks. Trucks
arrive at the loading gate at an average of 24 per day, or 3 per hour,
according to a Poisson distribution. The worker loads them at a rate
of 4 per hour, following approximately the exponential distribution in
service times.
problem. What is the probability that there will be more than three
trucks either being loaded or waiting? Discuss the results of your
queuing model computation.
substantially improve the firm’s efficiency. He estimates that a two-
gate will double the loading rate from 4 trucks per hour to 8 trucks
per hour. Analyze the effect on the queue of such a change and
compare the results with those found in Problem 12-19 0.
12-21 Truck drivers working for Juhn and Sons (see Problems 12-
19 and 12-20) are paid a salary of \$20 per hour on average. Fruit
queue or at the loading gate are drawing a salary but are
productively idle and unable to generate revenue during that time.
What would be the hourly cost savings to the firm associated with
12-22 Juhn (of Problem 12-19 ) is considering building a second
platform or gate to speed the process of loading the fruit trucks. This,
he thinks, will be even more efficient than simply hiring another
loader to help out the first platform (as in Problem 12-20 9).
Assume that each worker at each nlatform will be able to load 1
B
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hour. Should it hire another clerk? Explain.
11
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12-17 Automobiles arrive at the drive-through window at a post
office at the rate of four every 10 minutes. The average service time
is 2 minutes. The Poisson distribution is appropriate for the arrival
rate and service times are exponentially distributed.
2
a.
C.
What is the average time a car is in the system?
b. What is the average number of cars in the system?
What is the average time cars spend waiting to receive
service?
d. What is the average number of cars in line behind the
customer receiving service?
What is the probability that there are no cars at the
window?
f. What percentage of the time is the postal clerk busy?
g. What is the probability that there are exactly two cars in
the system?
e.
a
12-18 For the post office in Problem 12-17 , a second drive-
through window is being considered. A single line would be formed,
and as a car reached the front of the line it would go to the next
available clerk. The clerk at the new window works at the same rate
as the current one.
a.
b.
What is the average time a car is in the system?
What is the average number of cars in the system?
What is the average time care onend waiting to receive
B
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11
12-15 The wheat harvesting season in the American Midwest is
short, and most farmers deliver their truckloads of wheat to a giant
central storage bin within a 2-week span. Because of this, wheat-
known to back up for a block at the receiving bin. The central bin is
owned cooperatively, and it is to every farmer’s benefit to make the
and idle driver time are significant concerns to the cooperative
members. Although farmers have difficulty quantifying crop damage,
it is easy to assign a waiting and unloading cost for truck and driver
of \$18 per hour. The storage bin is open and operated 16 hours per
day, 7 days per week, during the harvest season and is capable of
distribution. Full trucks arrive all day long (during the hours the bin is
open) at a rate of about 30 per hour, following a Poisson pattern.
To help the cooperative get a handle on the problem of lost time
while trucks are waiting in line or unloading at the bin, find the
a.
C.
b. average time per truck in the system.
utilization rate for the bin area.
d. probability that there are more than three trucks in the
system at any given time.
total daily cost to the farmers of having their trucks tied
e.
B
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< a AA E < QUANTITATIVE ANALYSIS for MANAGEMENT Quantitative Analysis for Management Barry Render; Ralph M. Stair; Mic... а 12-14 A university cafeteria line in the student center is a self-serve facility in which students select the food items they want and then form a single line to pay the cashier. Students arrive at the cashier at a rate of about four per minute according to a Poisson distribution. The single cashier ringing up sales takes about 12 seconds per customer, following an exponential distribution. Expand | Collapse Glossary 447 a. Key Equations 447 v Solved Problems 449 C. Self-Test 451 What is the probability that there are more than two students in the system? More than three students? More than four? b. What is the probability that the system is empty? How long will the average student have to wait before reaching the cashier? d. What is the expected number of students in the queue? What is the average number in the system? f. If a second cashier is added (who works at the same pace), how will the operating characteristics computed in parts (b), (c), (d), and (e) change? Assume that customers wait in a single line and go to the first available cashier. 452 Discussion Questions and Problems e. Bibliography 459 460 Appendix 12.1: Using QM for Windows Chapter 13 Simulation Modeling 461 12-15 The wheat harvesting season in the American Midwest is short, and most farmers deliver their truckloads of wheat to a giant central storage bin within a 2-week span. Because of this, wheat-filled trucks waiting to unload and
return to the fields have been known to back up for a block at
v Chapter 14 Markov Analysis
501
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