MHC 503 Jacksonville University Chapter 12 Probability Problems

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DISCUSSION QUESTIONS AND PROBLEMS
471
(d) What is the average time spent waiting in line to
get to the ticket window?
(e) What is the probability that there are more than
two people in the system? More than three peo-
ple? More than four?
of clerks to have on duty each Saturday to minimize
the store’s total expected cost.
• 12-11 The Rockwell Electronics Corporation retains a
service crew to repair machine breakdowns that oc-
cur on an average of = 3 per day (approximately
Poisson in nature).
The crew can service an average of u = 8
machines per day, with a repair time distribution that
resembles the exponential distribution.
(a) What is the utilization rate of this service sys-
tem?
(b) What is the average downtime for a machine
that is broken?
(c) How many machines are waiting to be serviced
at any given time?
(d) What is the probability that more than one
machine is in the system? Probability that more
than two are broken and waiting to be repaired
or being serviced? More than three? More than
four?
: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 dis-
tribution. The single cashier ringing up sales takes
about 12 seconds per customer, following an expo-
nential distribution.
(a) 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?
(c) How long will the average student have to wait
before reaching the cashier?
(d) What is the expected number of students in the
queue?
(e) 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 character-
istics computed in parts (b), (c), (d), and (e)
change? Assume that customers wait in a single
line and go to the first available cashier.
2.12-12 From historical data, Harry’s Car Wash estimates that
dirty cars arrive at the rate of 10 per hour all day
Saturday. With a crew working the wash line, Harry
figures that cars can be cleaned at the rate of one
every 5 minutes. One car at a time is cleaned in this
example of a single-channel waiting line.
Assuming Poisson arrivals and exponential service
times, find the
(a) average number of cars in line.
(b) average time a car waits before it is washed.
(c) average time a car spends in the service system.
(d) utilization rate of the car wash.
(e) probability that no cars are in the system.
12-13 Mike Dreskin manages a large Los Angeles movie
theater complex called Cinema I, II, III, and IV.
Each of the four auditoriums plays a different film;
the schedule is set so that starting times are stag-
gered to avoid the large crowds that would occur
if all four movies started at the same time. The the-
ater has a single ticket booth and a cashier who can
maintain an average service rate of 280 movie pa-
trons per hour. Service times are assumed to follow
an exponential distribution. Arrivals on a typically
active day are Poisson distributed and average 210
Q: 12-15 The wheat harvesting season in the American Mid-
west 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 the receiving bin. The central bin
is owned cooperatively, and it is to every farmer’s
benefit to make the unloading/storage process as
efficient as possible. The cost of grain deterioration
caused by unloading delays, the cost of truck rental,
and idle driver time are significant concerns to the
cooperative members. Although farmers have dif-
ficulty 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 unloading 35 trucks per hour
according to an exponential 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) average number of trucks in the unloading system.
(b) average time per truck in the system.
per hour.
To determine the efficiency of the current ticket
operation, Mike wishes to examine several queue
operating characteristics.
(a) Find the average number of moviegoers waiting
in line to purchase a ticket.
(b) What percentage of the time is the cashier busy?
(c) What is the average time that a customer spends
in the system?
Case Study
New England Foundry
For more than 75 years, New England Foundry, Inc., has
manufactured wood stoves for home use. In recent years, with
increasing energy prices, George Mathison, president of New
England Foundry, has seen sales triple. This dramatic increase
in sales has made it even more difficult for George to maintain
quality in all the wood stoves and related products.
Unlike other companies manufacturing wood stoves, New
England Foundry is only in the business of making stoves and
stove-related products. Their major products are the Warmglo
I, the Warmglo II, the Warmglo III, and the Warmglo IV. The
Warmglo I is the smallest wood stove, with a heat output of
30,000 Btu, and the Warmglo IV is the largest, with a heat out-
put of 60,000 Btu. In addition, New England Foundry, Inc., pro-
duces a large array of products that have been designed to be
used with one of their four stoves, including warming shelves,
surface thermometers, stovepipes, adaptors, stove gloves, triv-
ets, mitten racks, andirons, chimneys, and heat shields. New
England Foundry also publishes a newsletter and several paper-
back books on stove installation, stove operation, stove main-
tenance, and wood sources. It is George’s belief that its wide
assortment of products was a major contributor to the sales
increases.
The Warmglo III outsells all the other stoves by a wide
margin. The heat output and available accessories are ideal for
the typical home. The Warmglo III also has a number of out-
standing features that make it one of the most attractive and
heat-efficient stoves on the market. Each Warmglo III has a ther-
mostatically controlled primary air intake valve that allows the
stove to adjust itself automatically to produce the correct heat
output for varying weather conditions. A secondary air opening
is used to increase the heat output in case of very cold weather.
The internal stove parts produce a horizontal flame path for
more efficient burning, and the output gases are forced to take
an S-shaped path through the stove. The S-shaped path allows
476
CHAPTER 12 WAITING LINES AND QUEUING THEORY MODELS
FIGURE 12.3
Overview of Factory
FIGURE 12.4
Overview of Factory After Changes
Cleaning,
Grinding,
and
Preparation
Storage
and
Shipping
Cleaning,
Grinding,
and
Preparation
Storage
and
Shipping
Maintenance
Pattern
Shop
Molding
Casting
Pattern Shop
and
Maintenance
Casting
Molding
Sand
Sand
more complete combustion of the gases and better heat transfer
from the fire and gases through the cast iron to the area to be
heated. These features, along with the accessories, resulted in
expanding sales and prompted George to build a new factory to
manufacture Warmglo III stoves. An overview diagram of the
factory is shown in Figure 12.3.
The new foundry uses the latest equipment, including a
new Disamatic that helps in manufacturing stove parts. Regard-
less of new equipment or procedures, casting operations have
remained basically unchanged for hundreds of years. To begin
with, a wooden pattern is made for every cast-iron piece in the
stove. The wooden pattern is an exact duplication of the cast-
iron piece that is to be manufactured. New England Foundry has
all of its patterns made by Precision Patterns, Inc., and these
patterns are stored in the pattern shop and maintenance room.
Then a specially formulated sand is molded around the wooden
pattern. There can be two or more sand molds for each pattern.
Mixing the sand and making the molds are done in the molding
room. When the wooden pattern is removed, the resulting sand
molds form a negative image of the desired casting. Next, the
molds are transported to the casting room, where molten iron is
poured into the molds and allowed to cool. When the iron has
solidified, the molds are moved into the cleaning, grinding, and
preparation room. The molds are dumped into large vibrators
that shake most of the sand from the casting. The rough castings
are then subjected to both sandblasting to remove the rest of the
sand and grinding to finish some of the surfaces of the castings.
The castings are then painted with a special heat-resistant paint,
assembled into workable stoves, and inspected for manufactur-
ing defects that may have gone undetected thus far. Finally, the
finished stoves are moved to storage and shipping, where they
are packaged and shipped to the appropriate locations.
At present, the pattern shop and the maintenance depart-
ment are located in the same room. One large counter is used by
both maintenance personnel to get tools and parts and by sand
molders that need various patterns for the molding operation.
Peter Nawler and Bob Bryan, who work behind the counter, are
able to service a total of 10 people per hour (or about 5 per hour
each). On the average, 4 people from maintenance and 3 people
from the molding department arrive at the counter per hour.
People from the molding department and from maintenance
arrive randomly, and to be served they form a single line. Pete
and Bob have always had a policy of first come, first served.
Because of the location of the pattern shop and maintenance
department, it takes about 3 minutes for a person from the main-
tenance department to walk to the counter, and it takes about 1
minute for a person to walk from the molding department to the
pattern and maintenance room.
After observing the operation of the pattern shop and
maintenance room for several weeks, George decided to make
some changes to the layout of the factory. An overview of these
changes is shown in Figure 12.4.
Separating the maintenance shop from the pattern shop had
a number of advantages. It would take people from the mainte-
nance department only 1 minute instead of 3 to get to the new
maintenance department counter. Using time and motion stud-
ies, George was also able to determine that improving the layout
of the maintenance department would allow Bob to serve 6 peo-
ple from the maintenance department per hour, and improving
the layout of the pattern department would allow Pete to serve
7 people from the molding shop per hour.
Discussion Question
1. How much time would the new layout save?
2. If maintenance personnel were paid $9.50 per hour and
molding personnel were paid $11.75 per hour, how much
could be saved per hour with the new factory layout?
DISCUSSION QUESTIONS AND PROBLEMS
473
12-23 Bill First, general manager of Worthmore Depart-
ment Store, has estimated that every hour of cus-
tomer time spent waiting in line for a sales clerk to
become available costs the store $100 in lost sales
and goodwill. Customers arrive at the checkout
counter at the rate of 30 per hour, and the average
service time is 3 minutes. The Poisson distribution
describes the arrivals, and the service times are ex-
ponentially distributed. The number of sales clerks
can be 2, 3, or 4, with each one working at the same
rate. Bill estimates the salary and benefits for each
clerk to be $10 per hour. The store is open 10 hours
per day.
(a) Find the average time in the line if 2, 3, and 4
clerks are used.
(b) What is the total time spent waiting in line each
day if 2, 3, and 4 clerks are used?
(c) Calculate the total of the daily waiting cost and
the daily service cost if 2, 3, and 4 clerks are
used. What is the minimum total daily cost?
(a) How many customers would enter the bank on a
typical day?
(b) How much total time would the customers
spend waiting in line during the entire day if one
teller was used? What is the total daily waiting
time cost?
(c) How much total time would the customers
spend waiting in line during the entire day if two
tellers were used? What is the total daily waiting
time cost?
(d) If Billy wishes to minimize the total wait-
ing time and personnel cost, how many tellers
should be used?
12-27 Customers arrive at an automated coffee vending
machine at a rate of 4 per minute, following a Pois-
son distribution. The coffee machine dispenses a cup
of coffee in exactly 10 seconds.
(a) What is the average number of people waiting
in line?
(b) What is the average number in the system?
(c) How long does the average person wait in line
before receiving service?
12-24 Billy’s Bank is the only bank in a small town in
Arkansas. On a typical Friday, an average of 10
customers per hour arrives at the bank to transact
business. There is one single teller at the bank, and
the average time required to transact business is
4 minutes. It is assumed that service times can be
described by the exponential distribution. Although
this is the only bank in town, some people in the
town have begun using the bank in a neighboring
town about 20 miles away. If a single teller at Billy’s
is used, find
12-28 The average number of customers in the system in
the single-channel, single-phase model described in
Section 12.4 is
L =
u
Show that for m = 1 server, the multichannel queu-
ing model in Section 12.5,
(a) the average time in the line.
(b) the average number in the line.
(c) the average time in the system.
(d) the average number in the system.
(e) the probability that the bank is empty.
)”
L=
??
·Pot
1)!(mj – 1)
(m
472
CHAPTER 12 WAITING LINES AND QUEUING THEORY MODELS
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.
(c) utilization rate for the bin area.
(d) probability that there are more than three trucks
in the system at any given time.
(e) total daily cost to the farmers of having their
trucks tied up in the unloading process.
The cooperative, as mentioned, uses the storage bin
only two weeks per year. Farmers estimate that en-
larging the bin would cut unloading costs by 50%
next year. It will cost $9,000 to do so during the off-
season. Would it be worth the cooperative’s while to
enlarge the storage area?
Q: 12-16 Ashley’s Department Store in Kansas City maintains
a successful catalog sales department in which a
clerk takes orders by telephone. If the clerk is occu-
pied on one line, incoming phone calls to the catalog
department are answered automatically by a record-
ing machine and asked to wait. As soon as the clerk
is free, the party that has waited the longest is trans-
ferred and answered first. Calls come in at a rate of
about 12 per hour. The clerk is capable of taking an
order in an average of 4 minutes. Calls tend to fol-
low a Poisson distribution, and service times tend to
be exponential. The clerk is paid $10 per hour, but
because of lost goodwill and sales, Ashley’s loses
about $50 per hour of customer time spent waiting
for the clerk to take an order.
(a) What is the average time that catalog customers
must wait before their calls are transferred to the
order clerk?
(b) What is the average number of callers waiting to
place an order?
(c) Ashley’s is considering adding a second clerk to
take calls. The store would pay that person the
same $10 per hour. Should it hire another clerk?
Explain.
(a) What is the average time a car is in the system?
(b) What is the average number of cars in the system?
(c) 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?
(e) What is the probability that there are no cars in
the system?
(f) What percentage of the time are the clerks busy?
(g) What is the probability that there are exactly two
cars in the system?
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 expo-
nential distribution in service times.
Determine the operating characteristics of this
loading gate problem. What is the probability that
there will be more than three trucks either being
loaded or waiting? Discuss the results of your queu-
ing model computation.
12-20 Juhn believes that adding a second fruit loader
will substantially improve the firm’s efficiency. He
estimates that a two-person crew, still acting like a
single-server system, at the loading 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.

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