Deck 16: Capacity Planning and Queuing Models Computer Simulation

Question 1

Green Valley Airport has been in operation for several years and is beginning to experience flight congestion. A study of airport operations revealed that planes arrive at an average rate of 12 per hour, with a Poisson distribution. On the single runway, a plane can land and be cleared every 4 minutes on average, and service times have a negative exponential distribution. Planes are processed on an FCFS basis, with takeoffs occurring between landings. Planes waiting to land are asked to circle the airport.
a. What is the expected number of airplanes circling the airport, waiting in queue for clearance to land?
b. A new ground-approach radar system approved by the Federal Aviation Administration is being considered as a means of reducing congestion. Under this system, planes can be processed at a constant rate of 15 per hour (i.e., the variance is zero). What would be the expected number of airplanes circling the airport, waiting in queue for clearance to land, if this system were to be used?
c. Assume that the cost of keeping an airplane in the air is approximately $70 per hour. If the cost of the proposed radar system were $100 per hour, would you recommend its adoption?

Answer 1

(a)
(b)
(c)
Waiting cost savings      $70 (3.2  -  1.6)  =  $112
Less  the radar cost of                                 - 100    yields
                   Savings of                                 $  12/hour
Recommend adopting radar

Question 2

What would be the impact on the above calculations of adding a second doctor to the clinic and sharing the doctor queue between them on a "first MD available" basis ?

Answer 2

We now have an M / M /2 system with = 13/15 =.867. 
From Appendix C, we find L q 0.2.
W q = L q (1/ ) = (0.2)(1/13) = 0.0154 hours or 0.92 minutes.
Physician idle time assuming each serves half the patients: [1- (6.5/15)]60 = 34 minutes
The result is a dramatic reduction in patient waiting time (26 min. to about 1 min.), but it is obtained at the unacceptable cost of increased idle time per hour (8 min. to 34 min.) for the physicians.

Question 3

Community Bank is planning to expand its drive-in facility. Observations of the existing single-teller window reveal that customers arrive at an average rate of 10 per hour, with a Poisson distribution, and that they are given FCFS service, with an average transaction time of 5 minutes. Transaction times have a negative exponential distribution. Community Bank has decided to add another teller and to install four remote stations with pneumatic tubes running from the stations to the tellers, who are located in a glassed-in building. The cost of keeping a customer waiting in the system is represented as a $5-per-hour loss of goodwill. The hourly cost of a teller is $10.
a. Assume that each teller is assigned two stations exclusively, that demand is divided equally among the stations, and that no customer jockeying is permitted. What is the average number of customers waiting in the entire system?
b. If, instead, both tellers work all the stations and the customer waiting the longest is served by the next available teller, what is the average number of customers in the system?
c. What hourly savings are achieved by pooling the tellers?

Answer 3

(a)   Assume we have two independent M / M /1 systems with
 for each system.
(b)   Use M/M/ 2 Model with   =  10/12  =  5/6  =  0.833  and  c   =  2.
From Appendix C, we find the value for
.
(c)   Savings from pooling:   $5(1.42 - 1.02)  =  $2 per hour.

Question 4

The clinic is considering adopting a queue priority system that is determined by the time of entry into the system at the receptionist. How might patients waiting for the doctor react to this policy ?

Question 5

Consider a one-pump gas station that satisfies the assumptions for the M / M /1 model. It is estimated that, on average, customers arrive to buy gas when their tanks are one-eighth full. The mean time to service a customer is 4 minutes, and the arrival rate is 6 customers per hour.
a. Determine the expected length of the queue and the expected time in the system.
b. Suppose that customers perceive a gas shortage (when there is none) and respond by changing the fill-up criterion to more than one-eighth full on average. Assuming that changes in
are inversely proportional to changes in the fill-up criterion, compare results when the fill-up criterion is one-quarter full with the results in part a.
c. Making the same assumption as in part b, compare the results obtained if the fill-up criterion is one-half full. Do we have the makings of a behaviorally induced gasoline panic?
d. It is reasonable to assume that the time to service a customer will decrease as the fill-up criterion increases. Under "normal" conditions, it takes, on average, 2 minutes to pump the gasoline and 2 minutes to clean the windshield, check the oil, and collect the money. Rework parts b and c if the time to pump the gasoline changes proportionally to changes in the fill-up criterion.

Question 6

Draw a process flow diagram, and identify the bottleneck operation.

Question 7

You have been asked by a retail association to develop an inventory control program for use on a PC. The program development requires the completion of the following three activities in sequence. You are concerned about the likelihood of finishing the project in the promised 10 days.

Using the Monte Carlo method, simulate 10 program development experiences. On the basis of your simulation results, what is the probability of finishing the project in 10 days?

Question 8

Calculate the expected waiting time in the order preparation queue using queuing formula I.8 found in Appendix D. Compare this value with your simulation result and comment on any difference.

Question 9

A general-purpose auto-repair garage has one mechanic who specializes in muffler installations. Customers seeking service arrive at an average rate of 2 per hour, with a Poisson distribution. The average time to install a muffler is 20 minutes, with negative exponential distribution.
a. On arrival at the garage, how many customers should one expect to find in the system?
b. The management is interested in adding another mechanic when the customer's average time in the system exceeds 90 minutes. If business continues to increase, at what arrival rate per hour will an additional mechanic be needed?

Question 10

Constructing a distribution of demand during reorder lead time is complicated if the lead time itself is variable. Consider the following distribution for a reorder point inventory system.

a. What is the range of possible demands during the variable lead time?
b. Using the Monte Carlo method and random numbers from Appendix B, simulate 10 demands during lead time that could be used to form a histogram.

Question 11

Use the ServiceModel computer simulation software and the Pronto.pkg file to determine the number of drivers that minimizes the total cost of salaries and guarantee discounts.

Question 12

Discuss how one could determine the economic cost of keeping customers waiting.

Question 13

The Dell Factory Outlet has the following daily demand distribution for laptop computer carrying cases.

Simulate 5 days of retail sales using the following inventory policy. If the ending inventory of cases is 0 or 1, order enough units such that the starting inventory on the next day will be 5 cases. Ordered units arrive the next morning and are stocked before the store opens for business. If a stock-out occurs, the sale is lost.
The cost structure is as follows:

Fill in the following simulation table using the random numbers provided.

Question 14

Based on your simulation recommended staffing level, what is the probability of paying off on the guarantee?

Question 15

A business school is considering replacing its copy machine with a faster model. Past records show that the average student arrival rate is 24 per hour, Poisson distributed, and that the service times are distributed exponentially. The selection committee has been instructed to consider only machines that will yield an average turnaround time (i.e., expected time in the system) of 5 minutes or less. What is the smallest processing rate per hour that can be considered?

Question 16

The Coast Guard maintains a lighted buoy in the harbor entrance to warn ships of a dangerous reef. The flashing beacon contains two high-intensity quartz halogen bulbs. The supplier has provided the following data on bulb life:

The estimated cost of dispatching a motor launch with a crew to the buoy to remove and replace the weatherproof cover over the bulbs is $50, and the bulbs cost $10 each. The time involved in replacing a bulb is negligible. Coast Guard regulations require that both bulbs work all the time.
a. Develop a Monte Carlo simulation model that will help the Coast Guard to decide between the following replacement policies: (1) Replace only the bulb that burns out or (2) replace both bulbs when one burns out.
b. Simulate five years of activity using random numbers from Appendix B. Discuss some questions of experimental design that this problem poses.
c. Can you think of other bulb replacement policies to test?

Question 17

What do you think of this service guarantee policy ?

Question 18

Example 16.1 presents a naïve capacity planning exercise and was criticized for using averages. Recall the concept of a "bottleneck" from Chapter 9, "Supporting Facility and Process Flows," and suggest other reservations about this planning exercise.

Question 19

The project network for building a garage is shown in Figure 16.11 and includes data on activity times and their deviations.
a. Develop a Monte Carlo simulation model that can be used to generate a distribution of project completion times (longest path from node 1 to 8).
b. Simulate 5 project completion experiences using random numbers from Appendix B. What is the expected project duration time?

Question 20

What other design or and operating suggestions could improve Pronto Pizza's performance and customer service ?

Question 21

The Lower Colorado River Authority (LCRA) has been studying congestion at the boat-launching ramp near Mansfield Dam. On weekends, the arrival rate averages 5 boaters per hour, Poisson distributed. The average time to launch or retrieve a boat is 10 minutes, with negative exponential distribution. Assume that only one boat can be launched or retrieved at a time.
a. The LCRA plans to add another ramp when the average turnaround time (i.e., time in the system) exceeds 90 minutes. At what average arrival rate per hour should the LCRA begin to consider adding another ramp?
b. If there were room to park only 2 boats at the top of the ramp in preparation for launching, how often would an arrival find insufficient parking space?

Question 22

Consider the travel network shown in Figure 16.12 in which the probabilities of selecting a route from each node are shown in parentheses.
a. Develop a Monte Carlo simulation model to determine the expected trip time from node 1 to node 7.
b. Simulate 10 trips and calculate the expected trip time.

Question 23

Assume the waiting lines at the receptionist, the nurse clinician, and the doctor are managed independently with a FCFS priority. Model this system using ServiceModel to determine the utilization of each resource (receptionist, nurse clinician, and physician) and the distribution of time in the system for an arriving patient. Begin by selecting the Build menu - Background Graphic - Behind Grid. Then select the Edit menu - Import Graphic - Pracmod. This will give you an office floor plan on which you can arrange furniture.

Question 24

For a queuing system with a finite queue, the arrival rate can exceed the capacity to serve. Use an example to explain how this is feasible.

Question 25

The number of fires occurring during a 24-hour day follows a Poisson distribution with a mean of 4 fires per day as shown in the accompanying table. In examining past records, the fire chief finds that 75 percent of all fires require only one truck and the amount of time required to extinguish a fire is normally distributed with a mean of 3 hours and a standard deviation of 0.5 hour. The other 25 percent of reported fires required 2 trucks and the time to put them out is normally distributed with a mean of 4 hours and a standard deviation of 1 hour. Assume the district has 10 trucks and use a simulation sample of 10 days to determine the average daily utilization of trucks.

Question 26

If you could hire one more person to augment the staff, what position would be your choice (receptionist, nurse clinician, or physician)? How would the clinic performance measures determined above change?

Question 27

On average, 4 customers per hour use the public telephone in the sheriff's detention area, and this use has a Poisson distribution. The length of a phone call varies according to a negative exponential distribution, with a mean of 5 minutes. The sheriff will install a second telephone booth when an arrival can expect to wait 3 minutes or longer for the phone.
a. By how much must the arrival rate per hour increase to justify a second telephone booth?
b. Suppose the criterion for justifying a second booth is changed to the following: install a second booth when the probability of having to wait at all exceeds 0.6. Under this criterion, by how much must the arrival rate per hour increase to justify a second booth?

Question 28

Electronic Car-diagram (ECG), a firm specializing in the diagnosis of automobile engine problems, is planning to open a new service center with two diagnostic bays. Based on the experience at other ECG facilities, it has been found that the diagnostic service time is a uniform distribution between 40 and 60 minutes. Based on its historical experience and the demographics of the new location, ECG believes the average time between customer arrivals will be 60 minutes with a negative exponential distribution. Simulate eight hours (480 minutes) of operations with a two-bay facility, recording the waiting time of customers. ECG expects customers will wait while their cars are being diagnosed and plans to promote an unconditional guarantee that customers who wait more than 60 minutes will receive a gift certificate. From your simulation, how many gift certificates do you expect to give out per day?

Question 29

What further advice would you give Dr. Thompson on the operation of her clinic ?

Question 30

What are some disadvantages associated with the concept of pooling service resources ?

Question 31

During periods of bad weather, as compared with periods of clear weather, how many additional gallons of fuel, on the average, should FreeEx expect its planes to consume owing to airport congestion ?

Question 32

A company has a central document-copying service. Arrivals are assumed to follow the Poisson probability distribution, with a mean rate of 15 per hour. Service times are assumed to follow the exponential distribution. With the present copying equipment, the average service time is 3 minutes. A new machine is available that will have a mean service time of 2 minutes. The average wage of the people who bring the documents to be copied is $8 an hour.
a. If the new machine can be rented for $10 per hour more than the old machine, should the company rent the new machine? Consider lost productive time of employees as time spent waiting in queue only because the copying machine is a self-serve device.
b. For the old copying machine, what is the probability when a person arrives that he or she will encounter people already waiting in line for service? (Be careful to identify properly the number of customers who might be present for this situation to arise.)
c. Suppose the new copying machine is rented. How many chairs should be provided for those waiting in line if we are satisfied when there will be enough chairs at least 90 percent of the time?

Question 33

Given FreeEx's policy of ensuring that its planes do not run out of fuel more than 1 in 20 times while waiting to land, how many reserve gallons (i.e., gallons over and above expected usage) should be provided for clear-weather flights? For bad-weather flights ?

Question 34

Discuss how the M/G/ model could be used to determine the number ofemergency medical vehicles that are required to serve a community.

Question 35

If FreeEx must pay $200 to charter a bus to transport its passengers from Dulles to Washington National, should it exercise the option of permitting its aircraft to land at Dulles during bad weather ?

Question 36

Sea Dock, a private firm, operates an unloading facility in the Gulf of Mexico for supertankers delivering crude oil for refineries in the Port Arthur area of Texas. Records show that, on average, 2 tankers arrive per day, with a Poisson distribution. Supertankers are unloaded one at a time on a first-come, first-served (FCFS) basis. Unloading requires approximately 8 hours of a 24-hour working day, and unloading times have a negative exponential distribution.
a. Sea Dock has provided mooring space for 3 tankers. Is this sufficient to meet the U.S. Coast Guard requirement that at least 19 of 20 arrivals should find mooring space available?
b. Sea Dock can increase its unloading capacity to a rate of 4 ships per day through additional labor at a cost of $480 per day. Considering the $1,000-per-day demurrage fee charged to Sea Dock for keeping a supertanker idle (this includes unloading time as well as time spent waiting in queue), should management consider this expansion opportunity?

Question 37

Assume the waiting lines at the receptionist, the nurse clinician, and the doctor are managed independently with a FCFS priority. Using queuing formulas and the assumption that patients exiting from an activity follow a Poisson distribution, estimate the following statistics:

Question 38

Last National Bank is concerned about the level of service at its single drive-in window. A study of customer arrivals during the window's busy period revealed that, on average, 20 customers per hour arrive, with a Poisson distribution, and they are given FCFS service, requiring an average of 2 minutes, with service times having a negative exponential distribution.
a. What is the expected number of customers waiting in queue?
b. If Last National were using an automated teller machine with a constant service time of 2 minutes, what would be the expected number of drive-in customers in the system?
c. There is space in the drive for 3 cars (including the one being served). What is the probability of traffic on the street being blocked by cars waiting to turn into the bank driveway?
d. Last National is considering adding tellers at the current drive-in facility. It has decided on $5 per hour as the imputed cost of customer waiting time in the system. The hourly cost of a teller is $10. The average arrival rate of customers has reached 30 per hour. On the basis of the total hourly cost of tellers and customer waiting, how many tellers do you recommend? Assume that with use of pneumatic tubes, tellers can serve customers as though there were a single queue.

Question 39

What are the key assumptions involved in your analysis above? Discuss the appropriateness of each in this situation.