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# Operations Problem - Queuing

See attached file for full problem description.

3. The neighborhood Clinkos copier store has three copiers: C1, C2, and C3. Customers that come to the store can be categorized into two types of jobs: large volume and small volume jobs. Currently, Copiers C1 and C2 are only assigned to customers with large volume copy jobs, and Copier C3 only serves customers with smaller volume jobs. Customers with large volume jobs arrive according to a Poisson process with mean of 25 customers per hour. When customers arrive, they joint a single queue in front of Copier C1 and Copier C2. Customers with smaller jobs arrive at a rate of 40 customers per hour, with a standard deviation of interarrival time of 0.75 minutes. When this type of arrive, they joint another separate queue in front of Copier C3. Copier C1 has been observed to serve on an average 40 customers per hour during the peak periods when it is never idle; the service time of this copier is exponentially distributed. Copier C2 is the same model as Copier C1 so that C2 has the same service rate as C1. Finally, service time for Copier C3 is exponentially distributed with average service time of 1 minute for each customer.

a.) Suppose that the manager temporarily closes Copier C2 (i.e., Copier C1 is the
only machine serves customers with large volume jobs). What are the utilizations for
Copier C1 and Copier C3?

Utilization for C1: ___________________%

Utilization for C3: ___________________%

b.) What are the expected waiting time and number of customers in the store for
each type of customers? (Continued from part a, assume that C2 is closed.)

Expected waiting time for customers with large volume jobs ________ minutes.

Expected waiting time for customers with small volume jobs ________ minutes.

Expected number of customers with large volume jobs ________ customers.

Expected number of customers with small volume jobs ________ customers.

c.) For Copier C1, what is the probability that at any given time that there is more
than 1 customer in the queue? For Copier C3, what is the probability that at any given
time that this unit is not being used? (Continued from part a, assume that C2 is closed.)

Probability that more than one customer in the queue for C1: .

Probability that Copier C3 is not being used: .

d.) Let's denote the utilization that you calculated for Copier C1 in part (a) as X
%. Suppose the manager has noticed that the utilization for Copier C1 has increased from
X % to (X+10) %, and as a result the average total amount of time a customer spends
there (in line and in service) has increased by Y %. If the utilization further increases
from (X+10) % to (X+20) %, would you expect the additional increase in the average
total time a customer spends there to be: (Circle one)

Less than Y %
Exactly Y %
More than Y %

Now, use the parameters that given in the question statement to answer parts e and f
(i.e., don't worry about the change of utilization in part d).

e.) Suppose the manager observes some customers waiting in the queue for the
large volume jobs. He decides to turn on Copier C2 so both C1 and C2 can serve the
customers with large volume jobs. What is the utilization if both copiers are working?
What is the expected waiting time that a customer with large volume jobs spends in the
store? What is the expected number of customers with large volume jobs in the store?

Utilization: _____________ %

Expected waiting time in the store: _________ minutes

Expected number of customers in the store: _________ customers

f.) The manager has to make a decision whether he should purchase a new
machine, "2Xfaster", to replace Copier C1 and Copier C2. As its name suggests, 2Xfaster
can work twice faster than C1 (or C2). However, 2Xfaster is more complicated, and the
manager needs to hire one worker to operate this machine. The manager estimates that
the hourly cost for a large-volume-job customer waiting in the store is \$30. The worker
hourly salary is \$6. Suppose the manager aims to (1) minimize the waiting time that the
customer spends in the queue, or (2) minimize the total hourly cost (the sum of customer
waiting cost and labor cost per hour), should the manager purchase 2Xfaster? (Assume
that Copier C1 and C2 are self-service machines, i.e., there is no cost for the labor. Also,
the service time for 2Xfaster is exponential distributed.)

If the manager aims to minimize the expected waiting time in the queue: YES /NO

If the manager aims to minimize the total hourly cost: YES / NO

#### Solution Summary

Solution contains answers of queuing problems.

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