## C.5 Exercises

For the exercises in this section, first start with specifying the appropriate queueing models needed to solve the exercise using Kendall’s notation. Then, specify the parameters of the model, e.g. \(\lambda_{e}\), \(\mu\), \(c\), size of the population, size of the system, etc. Specify how and what you would compute to solve the problem. Be as specific as possible by specifying the equations needed. Then, compute the quantities if requested. You might also try to use to solve the problems via simulation.

**Exercise C.1**

*True*or

*False*: In a queueing system with random arrivals and random service times, the performance will be best if the arrival rate is equal to the service rate because then there will not be any queueing.

**Exercise C.2**The Burger Joint in the UA food court uses an average of 10,000 pounds of potatoes per week. The average number of pounds of potatoes on hand is 5,000. On average, how long do potatoes stay in the restaurant before being used? What queuing concept is use to solve this problem?

**Exercise C.3**Consider a single pump gas station where the arrival process is Poisson with a mean time between arrivals of 10 minutes. The service time is exponentially distributed with a mean of 6 minutes. Specify the appropriate queueing model needed to solve the problem using Kendall’s notation. Specify the parameters of the model and what you would compute to solve the problem. Be as specific as possible by specifying the equation needed. Then, compute the desired quantities.

What is the probability that you have to wait for service?

What is the mean number of customer at the station?

What is the expected time waiting in the line to get a pump?

**Exercise C.4**Suppose an operator has been assigned to the responsibility of maintaining 3 machines. For each machine the probability distribution of the running time before a breakdown is exponentially distributed with a mean of 9 hours. The repair time also has an exponential distribution with a mean of 2 hours. Specify the appropriate queueing model needed to solve the problem using Kendall’s notation. Specify the parameters of the model and what you would compute to solve the problem. Be as specific as possible by specifying the equation needed. Then, compute the desired quantities.

What is the probability that the operator is idle?

What is the expected number of machines that are running?

What is the expected number of machines that are not running?

**Exercise C.5**SuperFastCopy wants to install self-service copiers, but cannot decide whether to put in one or two machines. They predict that arrivals will be Poisson with a rate of 30 per hour, and the time spent copying is exponentially distributed with a mean of 1.75 minutes. Because the shop is small they want the probability of 5 or more customers in the shop to be small, say less than 7%. Make a recommendation based on queueing theory to SuperFastCopy.

**Exercise C.6 **Each airline passenger and his or her carry-on baggage must be checked
at the security checkpoint. Suppose XNA averages 10 passengers per
minute with exponential inter-arrival times. To screen passengers, the
airport must have a metal detector and baggage X-ray machines. Whenever
a checkpoint is in operation, two employees are required (one operates
the metal detector, one operates the X-ray machine). The passenger goes
through the metal detector and simultaneously their bag goes through the
X-ray machine. A checkpoint can check an average of 12 passengers per
minute according to an exponential distribution.

**Exercise C.7**Two machines are being considered for processing a job within a factory. The first machine has an exponentially distributed processing time with a mean of 10 minutes. For the second machine the vendor has indicated that the mean processing time is 10 minutes but with a standard deviation of 6 minutes. Using queueing theory, which machine is better in terms of the average waiting time of the jobs?

**Exercise C.8**Customers arrive at a one-window drive in bank according to a Poisson distribution with a mean of 10 per hour. The service time for each customer is exponentially distributed with a mean of 5 minutes. There are 3 spaces in front of the window including that for the car being served. Other arriving cars can wait outside these 3 spaces. Specify the appropriate queueing model needed to solve the problem using Kendall’s notation. Specify the parameters of the model and what you would compute to solve the problem. Be as specific as possible by specifying the equation needed. Then, compute the desired quantities.

What is the probability that an arriving customer can enter one of the 3 spaces in front of the window?

What is the probability that an arriving customer will have to wait outside the 3 spaces?

How long is an arriving customer expected to wait before starting service?

How many spaces should be provided in front of the window so that an arriving customer can wait in front of the window at least 20% of the time? In other words, the probability of at least one open space must be greater than 20%.

**Exercise C.9**Joe Rose is a student at Big State U. He does odd jobs to supplement his income. Job requests come every 5 days on the average, but the time between requests is exponentially distributed. The time for completing a job is also exponentially distributed with a mean of 4 days.

What would you compute to find the chance that Joe will not have any jobs to work on?

What would you compute to find the average value of the waiting jobs if Joe gets about $25 per job?

**Exercise C.10 **The manager of a bank must determine how many tellers should be
available. For every minute a customer stands in line, the manager
believes that a delay cost of 5 cents is incurred. An average of 15
customers per hour arrive at the bank. On the average, it takes a teller
6 minutes to complete the customer’s transaction. It costs the bank $9
per hour to have a teller available. Inter-arrival and service times can
be assumed to be exponentially distributed.

**Exercise C.11 **You have been hired to analyze the needs for loading dock facilities at
a trucking terminal. The present terminal has 4 docks on the main
building. Any trucks that arrive when all docks are full are assigned to
a secondary terminal, which a short distance away from the main
terminal. Assume that the arrival process is Poisson with a rate of 5
trucks each hour. There is no available space at the main terminal for
trucks to wait for a dock. At the present time nearly 50% of the
arriving trucks are diverted to the secondary terminal. The average
service time per truck is two hours on the main terminal and 3 hours on
the secondary terminal, both exponentially distributed. Two proposals
are being considered. The first proposal is to expand the main terminal
by adding docks so that at least 80% of the arriving trucks can be
served there with the remainder being diverted to the secondary
terminal. The second proposal is to expand the space that can
accommodate up to 8 trucks. Then, only when the holding area is full
will the trucks be diverted to secondary terminal.

**Exercise C.12**Sly’s convenience store operates a two-pump gas station. The lane leading to the pumps can house at most five cars, including those being serviced. Arriving cars go elsewhere if the lane is full. The distribution of the arriving cars is Poisson with a mean of 20 per hour. The time to fill up and pay for the purchase is exponentially distributed with a mean of 6 minutes.

Specify using queueing notation, exactly what you would compute to find the percentage of cars that will seek business elsewhere?

Specify using queueing notation, exactly what you would compute to find the utilization of the pumps?

**Exercise C.13**An airline ticket office has two ticket agents answering incoming phone calls for flight reservations. In addition, two callers can be put on hold until one of the agents is available to take the call. If all four phone lines (both agent lines and the hold lines) are busy, a potential customer gets a busy signal, and it is assumed that the call goes to another ticket office and that the business is lost. The calls and attempted calls occur randomly (i.e. according to Poisson process) at a mean rate of 15 per hour. The length of a telephone conversation has an exponential distribution with a mean of 4 minutes.

Specify using queueing notation, exactly what you would compute to find the probability of losing a potential customer?

What would you compute to find the probability that an arriving phone call will not start service immediately but will be able to wait on a hold line?

**Exercise C.14**SuperFastCopy has three identical copying machines. When a machine is being used, the time until it breaks down has an exponential distribution with a mean of 2 weeks. A repair person is kept on call to repair the machines. The repair time for a machine has an exponential distribution with a mean of 0.5 week. The downtime cost for each copying machine is $100 per week.

Let the state of the system be the number of machines not working, Construct a state transition diagram for this queueing system.

Write an expression using queueing performance measures to compute the expected downtime cost per week.

**Exercise C.15 **NWH Cardiac Care Unit (CCU) has
5 beds, which are virtually always occupied by patients who have just
undergone major heart surgery. Two registered nurses (RNs) are on duty
in the CCU in each of the three 8 hour shifts. About every two hours
following an exponential distribution, one of the patients requires a
nurse’s attention. The RN will then spend an average of 30 minutes
(exponentially distributed) assisting the patient and updating medical
records regarding the problem and care provided.

What would you compute to find the average number of patients being attended by the nurses?

What would you compute to fine the average time that a patient spends waiting for one of the nurses to arrive?

**Exercise C.16 **HJ Bunt, Transport Company maintains a large fleet of refrigerated
trailers. For the purposes of this problem assume that the number of
refrigerated trailers is conceptually infinite. The trailers require
service on an irregular basis in the company owned and operated service
shop. Assume that the arrival of trailers to the shop is approximated by
a Poisson distribution with a mean rate of 3 per week. The length of
time needed for servicing a trailer varies according to an exponential
distribution with a mean service time of one-half week per trailer. The
current policy is to utilize a centralized contracted outsourced service
center whenever more than two trailers are in the company shop, so that,
at most one trailer is allowed to wait. Assume that there is currently
one 1 mechanic in the company shop.

**Exercise C.17**Rick is a manager of a small barber shop at Big State U. He hires one barber. Rick is also a barber and he works only when he has more than one customer in the shop. Customers arrive randomly at a rate of 3 per hour. Rick takes 15 minutes on the average for a hair cut, but his employee takes 10 minutes. Assume that the cutting time distributions are exponentially distributed. Assume that there are only 2 chairs available with no waiting room in the shop.

Let the state of the system be the number of customers in the shop, Construct a state transition diagram for this queueing system.

What is the probability that a customer is turned away?

What is the probability that the barber shop is idle?

What is the steady-state mean number of customers in the shop?

**Exercise C.18 **Using the supplied data set, draw the sample path for the state
variable, \(N(t)\). Give a formula for estimating the time average number
in the system, \(N(t)\), and then use the data to compute the time average
number in the system over the range from 0 to 25. Assume that the value
of \(N(t\) is the value of the state variable just after time \(t\).

\(t\) | 0 | 2 | 4 | 5 | 7 | 10 | 12 | 15 | 20 |

\(N(t)\) | 0 | 1 | 0 | 1 | 2 | 3 | 2 | 1 | 0 |

Give a formula for estimating the time average number in the system, \(N(t)\), and then use the data to compute the time average number in the system over the range from 0 to 25.

Give a formula for estimating the mean rate of arrivals over the interval from 0 to 25 and then use the data to estimate the mean arrival rate.

Estimate the average time in the system (waiting and in service) for the customers indicated in the diagram.

What queueing formula relationship is used in this problem?