Question

1. Willow Brook National Bank operates a drive-up teller window that allows customers to complete bank transactions without getting out of their cars. On weekday mornings, arrivals to the drive-up teller window occur at random, with an arrival rate of 6 customers per hour or 0.1 customers per minute. Also assume that the service times for the drive-up teller follow an exponential probability distribution with a service rate of 54 customers per hour, or 0.9 customers per minute. Determine the probabilities of 0, 1, 2, and 3 customers in the system. Round your answers to 4 decimal places.

n |
P(n) |

0 | |

1 | |

2 | |

3 |

What is the probability that more than three customers will be in the drive-up teller system at the same time? Round your answer to 4 decimal places.

2. The reference desk of a university library receives requests for assistance. Assume that a Poisson probability distribution with an arrival rate of 11 requests per hour can be used to describe the arrival pattern and that service times follow an exponential probability distribution with a service rate of 13 requests per hour.

- What is the probability that no requests for assistance are in
the system? If required, round your answer to four decimal
places.

*P*=_{0} - What is the average number of requests that will be waiting for
service? If required, round your answer to four decimal
places.

*L*=_{q} - What is the average waiting time in minutes before service
begins? If required, round your answer to nearest whole
number.

*W*= min_{q} - What is the average time at the reference desk in minutes
(waiting time plus service time)? If required, round your answer to
nearest whole number.

*W*= min - What is the probability that a new arrival has to wait for
service? If required, round your answer to four decimal
places.

*P*=_{w}

3. Marty's Barber Shop has one barber. Customers have an arrival rate of 1.3 customers per hour, and haircuts are given with a service rate of 6 per hour. Use the Poisson arrivals and exponential service times model to answer the following questions:

- What is the probability that no units are in the system? Round
your answer to four decimal places.

*P*=_{0} - What is the probability that one customer is receiving a
haircut and no one is waiting? Round your answer to four decimal
places.

*P*=_{1} - What is the probability that one customer is receiving a
haircut and one customer is waiting? Round your answer to four
decimal places.

*P*=_{2} - What is the probability that one customer is receiving a
haircut and two customers are waiting? Round your answer to four
decimal places.

*P*=_{3} - What is the probability that more than two customers are
waiting? Round your answer to four decimal places.

*P*(More than 2 waiting) = - What is the average time a customer waits for service? Round
your answer to four decimal places.

*W*= minutes_{q}

4.

Agan Interior Design provides home and office decorating assistance to its customers. In normal operation, an average of 2.6 customers arrive each hour. One design consultant is available to answer customer questions and make product recommendations. The consultant averages 12 minutes with each customer.

- Compute the operating characteristics of the customer waiting
line, assuming Poisson arrivals and exponential service times.
Round your answers to four decimal places. Do not round
intermediate calculations.

*L*=_{q}

*L*=

*W*=_{q}

*W*=

*P*=_{w} - Service goals dictate that an arriving customer should not wait
for service more than an average of 4 minutes. Is this goal being
met? If not, what action do you recommend?

No. Firm should increase the mean service rate u for the consultant or hire a second consultant. - If the consultant can reduce the average time spent per
customer to 8 minutes, what is the mean service rate? Round your
answer to four decimal places. Do not round intermediate
calculations.

*µ*= customers per hour

*W*= minutes_{q}

Will the service goal be met?

4.

Pete's Market is a small local grocery store with only one
checkout counter. Assume that shoppers arrive at the checkout lane
according to a Poisson probability distribution, with an arrival
rate of 9 customers per hour. The checkout service times follow an
exponential probability distribution, with a service rate of 15
customers per hour. The manager’s service goal is to limit the
waiting time prior to beginning the checkout process to no more
than five minutes. Also the manager of Pete's Market wants to
consider one of the following alternatives for improving service.
Calculate the value of *W _{q}* for each
alternative.

- Hire a second person to bag the groceries while the cash
register operator is entering the cost data and collecting money
from the customer. With this improved single-server operation, the
service rate could be increased to 19 customers per hour. Round
your answer to three decimal places. Do not round intermediate
calculations.

*W*= minutes_{q} - Hire a second person to operate a second checkout counter. The
two-server operation would have a service rate of 15 customers per
hour for each server.
*Note*: Use*P*_{0}values from Table 11.4 to answer this question. Round your answer to three decimal places. Do not round intermediate calculations.

*W*=_{q}

Answer #1

1.

Arrival rate, = 0.1 customers per minute

Service rate, = 0.9 customers per minute

Utilization = / = 0.1/0.9 = 0.1111

Probability that n customers are waiting is,

P(n) = (1 - )

For n = 0, P(0) = (1 - ) = 1 - 0.1111 = 0.8889

For n = 1, P(1) = (1 - ) = (1 - 0.1111) * 0.1111 = 0.0988

For n = 2, P(1) = (1 - ) = (1 -
0.1111) * 0.1111^{2} = 0.0110

For n = 3, P(1) = (1 - ) = (1 -
0.1111) * 0.1111^{3} = 0.0012

Probability that more than three customers will be in the
drive-up teller system = P(N 4) = =
0.1111^{4} = 0.0002

Willow Brook National Bank operates a drive-up teller window
that allows customers to complete bank transactions without getting
out of their cars. On weekday mornings, arrivals to the drive-up
teller window occur at random, with an arrival rate of 30 customers
per hour or 0.5 customers per minute. In the same bank waiting line
system, assume that the service times for the drive-up teller
follow an exponential probability distribution with a service rate
of 36 customers per hour, or 0.6...

Problem 15-3 (Algorithmic) Willow Brook National Bank operates a
drive-up teller window that allows customers to complete bank
transactions without getting out of their cars. On weekday
mornings, arrivals to the drive-up teller window occur at random,
with an arrival rate of 18 customers per hour or 0.3 customers per
minute. In the same bank waiting line system, assume that the
service times for the drive-up teller follow an exponential
probability distribution with a service rate of 30 customers per...

Willow Brook National Bank operates a drive-up teller window
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out of their cars. On weekday mornings, arrivals to the drive-up
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times for the drive-up teller follow an exponential probability
distribution with a service rate of 36 customers per hour, or 0.6
customers per minute. Determine the probabilities...

Problem 11-04 (Algorithmic)
Willow Brook National Bank operates a drive-up teller window
that allows customers to complete bank transactions without getting
out of their cars. On weekday mornings, arrivals to the drive-up
teller window occur at random, with an arrival rate of 24 customers
per hour or 0.4 customers per minute. Also assume that the service
times for the drive-up teller follow an exponential probability
distribution with a service rate of 36 customers per hour, or 0.6
customers per minute....

Exponential/Poisson
Megacity Bank operates a drive-up teller window that
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get out of their car. Megacity Bank waiting line times follow an
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A.What is the probability that the time to service is 1 minute
or less? (show your formula with the numbers plugged in, as well as
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excel to...

Marty's Barber Shop has one barber. Customers have an arrival
rate of 2.3 customers per hour, and haircuts are given with a
service rate of 4 per hour. Use the Poisson arrivals and
exponential service times model to answer the following
questions:
What is the probability that no units are in the system? If
required, round your answer to four decimal places.
P0 = _____
What is the probability that one customer is receiving a haircut
and no one is...

Problem 15-9 (Algorithmic)
Marty's Barber Shop has one barber. Customers have an arrival
rate of 2.3 customers per hour, and haircuts are given with a
service rate of 4 per hour. Use the Poisson arrivals and
exponential service times model to answer the following
questions:
What is the probability that no units are in the system? If
required, round your answer to four decimal places.
P0 =
What is the probability that one customer is receiving a
haircut and no...

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What is the probability that no units are in the system? If
required, round your answer to four decimal places.
P0 =
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The National Bank of Germany is opening a new bank teller.
Management estimates that customers will arrive at the rate of 10
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at the rate of one every four minutes.
Assuming Poisson arrivals and exponential service, find:
Average number of customers in the system.
Average time in the system.
Average waiting time in the line.
Average number of customers in the line.
Utilization of the teller.

Agan Interior Design provides home and office decorating
assistance to its customers. In normal operation, an average of 3.2
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Compute the operating characteristics of the customer waiting
line, assuming Poisson arrivals and exponential service times.
Round your answers to four decimal places. Do not round
intermediate calculations.
Lq =
L =
Wq = minutes
W = minutes...

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