6.8 Exercises
Exercise 6.1 Suppose that a customer arriving to the drive through pharmacy will decide to balk if the number of cars waiting in line is 4 or more. A customer is said to balk if he or she refuses to enter the system and simply departs without receiving service. Model this situation using and estimate the probability that a customer will balk because the line is too long. Run your model for 1 year, with 20 replications. Use stream 1 for the arrival process and stream 2 for the service process.
Exercise 6.2 Samples of 20 parts from a metal grinding process are selected every hour. Typically 2% of the parts need rework. Let X denote the number of parts in the sample of 20 that require rework. A process problem is suspected if X exceeds its mean by more than 3 standard deviations. Simulate 30 hours of the process, i.e. 30 samples of size 20, and estimate the chance that X exceeds its expected value by more than 1 standard deviation. Use stream 1.
Exercise 6.3 Samples of 20 parts from a metal grinding process are selected every hour. Typically 2% of the parts need rework. Let X denote the number of parts in the sample of 20 that require rework. A process problem is suspected if X exceeds its mean by more than 1 standard deviations. Each time X exceeds its mean by more than 1 standard deviations all X of the parts requiring rework are sent to a rework station. Use stream 1 to determine if a sample requires rework.
Each part consists of two subcomponents, which are split off and repaired separately. The splitting process takes 1 worker and lasts U(10, 20) minutes per part (stream 2). After the subcomponents have been split, they are repaired in different processes. Subcomponent 1 takes U(5, 10) minutes (stream 3) to repair with 1 worker at its repair process and subcomponent 2 takes expo(7.5) minutes (stream 4) to repair with 1 worker at its repair process. Once both of the subcomponents have been repaired, they are joined back together to form the original part. The joining process takes 5 minutes with 1 worker. The part is then sent back to the main production area, which is outside the scope of this problem. Simulate 8 hours of production and estimate the average time that it takes a part to be repaired. Use 20 replications.
Exercise 6.4 TV sets arrive at a two-inspector station for testing. The time between arrivals is exponential with a mean of 15 minutes (stream 1). The inspection time per TV set is exponential with a mean of 10 minutes (stream 2). On the average, 82 percent of the sets pass inspection (stream 3). The remaining 18% are routed to an adjustment station with a single operator. Adjustment time per TV set is uniform between 7 and 14 minutes (stream 4). After adjustments are made, sets are routed back to the inspection station to be retested. We are interested in estimating the total time a TV set spends in the system before it is released.
Develop a model for this situation. Report the average system time for the TV sets based on 20 replications of 4800 minutes. Also report statistics for the average number of times a given TV is adjusted.
Exercise 6.5 A simple manufacturing system is staffed by 3 operators. Parts arrive according to a Poisson process with a mean rate of 15 per hour (stream 1) to a workstation for a drilling process at one of three identical drill presses. The parts wait in a single queue until a drill press is available. Each part has a particular number of holes that need to be drilled. Each hole takes a Lognormal time to be drilled with an approximate mean of 1 minute and a standard deviation of 30 seconds (stream 2). Once the holes are drilled, the part goes to the grinding operation. At the grinding operation, one of the 3 available operators grinds out the burrs on the part. This activity takes approximately 5 minutes plus or minus 30 seconds (stream 3). After the grinding operation the part leaves the system. Assume that the parts have a discrete uniform number of holes to drill over the range from 2 to 4 (stream 4)
Develop model for this situation. Report the average system time for the parts based on 20 replications of 4800 minutes.
Exercise 6.6 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.
Build a KSL model to simulate this situation. Run the model for 20,000 minutes with a warm up period of 5,000 minutes. Based on 30 replications of your simulation, estimate the following performance measures.
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?
Use stream 1 for the time between arrivals and stream 2 for the service times.
Exercise 6.7 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.
Build a KSL model to simulate this situation. Run the model for 20,000 hours with a warm up period of 5,000 hours Based on 30 replications of your simulation, estimate the following performance measures.
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?
Use stream 1 for the time between arrivals and stream 2 for the service times.
Exercise 6.8 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%.
Build a KSL model to analyze this situation. Run the model for 20,000 minutes with a warm up period of 5,000 minutes Based on 30 replications of your simulation, estimate the following performance measures.
Use stream 1 for the time between arrivals and stream 2 for the service times.
Exercise 6.9 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.
Build a KSL model to analyze this situation. Run the model for 20,000 minutes with a warm up period of 5,000 minutes Based on 30 replications of your simulation, estimate the following performance measures.
What is the probability that a passenger will have to wait before being screened?
On average, how many passengers are waiting in line to enter the checkpoint?
On average, how long will a passenger spend at the checkpoint?
Use stream 1 for the time between arrivals and stream 2 for the service times.
Exercise 6.10 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 with a lognormal distribution.
Build KSL models to analyze this situation. Run the models for 20,000 minutes with a warm up period of 5,000 minutes Based on 30 replications of your simulation, which machine is better in terms of the average waiting time of the jobs?
Use stream 1 for the time between arrivals and stream 2 for the service times.
Exercise 6.11 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.
Build a KSL model to analyze this situation. Run the model for 20,000 minutes with a warm up period of 5,000 minutes Based on 30 replications of your simulation, estimate the following performance measures.
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%.
Use stream 1 for the time between arrivals and stream 2 for the service times.
Exercise 6.12 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.
Build a KSL model to analyze this situation. Run the model for 20,000 days with a warm up period of 5,000 days Based on 30 replications of your simulation, estimate the following performance measures.
What is the chance that Joe will not have any jobs to work on?
What is the average value of the waiting jobs if Joe gets about $25 per job?
Use stream 1 for the time between arrivals and stream 2 for the service times.
Exercise 6.13 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.
Build a KSL model to analyze this situation. Run the model for 20,000 minutes with a warm up period of 5,000 minutes Based on 30 replications of your simulation, answer the following questions.
What is the minimum number of tellers that should be available in order for the system to be stable (i.e. not have an infinite queue)?
If the system has 3 tellers, what is the probability that there will be no one in the bank?
What is the expected total cost of the system per hour, when there are 2 tellers?
Use stream 1 for the time between arrivals and stream 2 for the service times.
Exercise 6.14 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 is 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.
Build KSL models to analyze this situation. Run the models for 20,000 minutes with a warm up period of 5,000 minutes Based on 30 replications of your simulation, answer the following questions.
For the first proposal, what is the required number of docks so that at least 80% of the arriving trucks can be served?
For the second proposal, will the specified number of spaces have least 80% of the arriving trucks can be served?
Use stream 1 for the time between arrivals and stream 2 for the service times.
Exercise 6.15 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.
Build a KSL model to analyze this situation. Run the model for 20,000 minutes with a warm up period of 5,000 minutes Based on 30 replications of your simulation, answer the following questions.
What is the percentage of cars that will seek business elsewhere?
What is the utilization of the pumps?
Use stream 1 for the time between arrivals and stream 2 for the service times.
Exercise 6.16 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.
Build a KSL model to analyze this situation. Run the model for 20,000 minutes with a warm up period of 5,000 minutes Based on 30 replications of your simulation, answer the following questions.
What is probability of losing a potential customer?
What is the probability that an arriving phone call will not start service immediately but will be able to wait on a hold line?
Use stream 1 for the time between arrivals and stream 2 for the service times.
Exercise 6.17 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.
Build a KSL model to analyze this situation. Run the model for 20,000 weeks with a warm up period of 5,000 weeks Based on 30 replications of your simulation, what is the expected downtime cost per week.
Use stream 1 for the time between arrivals and stream 2 for the service times.
Exercise 6.18 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.
Build a KSL model to analyze this situation. Run the model for 20,000 minutes with a warm up period of 5,000 minutes Based on 30 replications of your simulation, answer the following questions.
What is the average number of patients being attended by the nurses?
What is the average time that a patient spends waiting for one of the nurses to arrive?
Use stream 1 for the time between arrivals and stream 2 for the service times.
Exercise 6.19 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.
Build a KSL model to analyze this situation. Run the model for 20,000 weeks with a warm up period of 5,000 weeks Based on 30 replications of your simulation, what is the expected number of repairs that are outsourced per week?
Use stream 1 for the time between arrivals and stream 2 for the service times.
Exercise 6.20 The Hog BBQ Joint is interested in understanding the flow of customers for diner (5 pm to 9 pm). Customers arrive in parties of 2, 3, 4, or 5 with probabilities 0.4, 0.3, 0.2, 0.1, respectively (stream 1). The time between arrivals is exponentially distributed with a mean of 1.4 minutes (stream 2). Customers must arrive prior to 9 pm in order to be seated. The dining area has 50 tables. Each table can seat 2 people. For parties, with more than 2 customers, the tables are moved together. Each arriving group gets in line to be seated. If there are already 6 parties in line, the arriving group will leave and go to another restaurant. The time that it takes to be served is triangularly distributed with parameters (14, 19, 24) in minutes (stream 3). The time that it takes to eat is lognormally distributed with a mean of 24 minutes and a standard deviation of 5 minutes (stream 4). When customers are finished eating, they go to the cashier to pay their bill. The time that it takes the cashier to process the customers is gamma distributed with a mean of 1.5 minutes and a standard deviation of 0.5 minutes (stream 5).
Develop an model for this situation. Simulate 30 days of operation. Make a table like the following to summarize your results.
| Average | Half-width | |
|---|---|---|
| Number of customers served | ||
| Number of busy tables | ||
| Number of waiting parties | ||
| Number of parties that depart without eating | ||
| Utilization of cashier | ||
| Customer System Time (in minutes) | ||
| Probability of waiting to be seated \(>\) 5 minutes |
Exercise 6.21 In the Tie-Dye T-Shirt model, the owner is expecting the business to grow during the summer season. The owner is interested in estimating the average time to produce an order and the utilization of the workers if the arrival rate for orders increases. Re-run the model for 30 eight hour days with the arrival rate increased by 20, 40, 60, and 80 percent. Will the system have trouble meeting the demand? Use the statistics to justify your conclusions.
Use the following stream allocations:
- Stream 1, time between orders
- Stream 2, order type
- Stream 3, order size
- Stream 4, shirt making time
- Stream 5, paper work time
- Stream 6, packaging time
Exercise 6.22 Suppose that the inspection and packaging process has been split into two processes for the Tie-Dye T-Shirt system and assume that there an additional worker to perform inspection. The inspection process is uniformly distributed between 2 and 5 minutes. After inspection there is a 4 percent chance that the whole order will have to be scrapped (and redone). If the order fails inspection, the scrapped order should be counted and a new order should be initiated into the system. If the order passes inspection, it goes to packaging where the packaging time is distributed according to a triangular distribution with parameters (2, 4, 10) all in minutes. Re-run the model for 30, 8-hour days, with the arrival rate increased by 20, 40, 60, and 80%. Will the system have trouble meeting the demand? In other words, how does the throughput (number of shirts produced per day) change in response to the increasing demand rate?
Use the following stream allocations:
- Stream 1, time between orders
- Stream 2, order type
- Stream 3, order size
- Stream 4, shirt making time
- Stream 5, paper work time
- Stream 6, packaging time
- Stream 7, inspection time
- Stream 8, scrapping process
Exercise 6.23 Hungry customers arrive to a Mickey R’s drive through restaurant at a mean rate of 10 per hour according to a Poisson process (stream 1). Management is interested in improving the total time spent within the system (i.e. from arrival to departure with their food).
Management is considering a proposed system that splits the order taking, payment activity and the order delivery processes. The first worker will take the orders from an order-taking speaker. This takes on average 1 minute plus or minus 20 seconds uniformly distributed (stream 2). When the order taking activity is completed, the making of the order will start. It takes approximately 3 minutes (plus or minus 20 seconds) to make the customer’s order, uniformly distributed (stream 3). Meanwhile, the customer will be instructed to drive to the first window to pay for the order. Assume that the time that it takes the customer to move forward is negligible. The first worker accepts the payment from the customer. This takes on average 45 seconds plus or minus 20 seconds uniformly distributed (stream 4). After paying for the order the customer is instructed to pull forward to the second window, where a second worker delivers the order. Assume that the time that it takes the customer to move forward is negligible.
If the order is not completed by the time the customer reaches the second window, then the customer must wait for the order to be completed. If the order is completed before the customer arrives to the 2nd window, then the order must wait for the customer. After both the order and the customer are at the 2nd window, the 2nd worker packages the customer’s order and gives it to the customer. This takes approximately 30 seconds with a standard deviation of 10 seconds, lognormally distributed (stream 5). After the customer receives their order they depart.
Simulate this system for the period from 10 am to 2 pm. Report the total time spent in the system for the customers based on 30 days.
Exercise 6.24 The city is considering improving its hazardous waste and bulk item drop off area to improve service. Cars arrive to the drop off area at a rate of 10 per hour according to a Poisson process (stream 1). Each car contains items for drop off. There is a 10% chance that the car will contain 1 item, a 50% chance that the car will contain 2 items, and a 40% chance that the car will contain 3 items (stream 2). There is an 80% chance that an item will be hazardous (e.g. chemicals, light bulbs, electronic equipment, etc.) and a 20% chance that the item will be a bulk item, which cannot be picked up in the curbside recycling program (stream 3). Of the 80% of items that have hazardous waste, about 10% are for electronic equipment that must be inspected and taken apart (stream 4).
A single worker assists the citizen in taking the material out of their car and moving the material to the recycling center. This typically takes between 0.5 to 1.5 minutes per item (uniformly distributed (stream 5)) if the item is not a bulk item. If the item is a bulk item, then the time takes a minimum of 1 minute, most likely 2.5 minutes, with a maximum of 4 minutes per item triangularly distributed (stream 6). The worker finishes all items in a car before processing the next car.
Another worker will begin sorting the items immediately after the item is unloaded. This process takes 1-2 minutes per item uniformly distributed (stream 7). If the item is electronic equipment, the items are placed in front of a special disassembly station to be taken apart.
The same worker that performs sorting also performs the disassembly of the electronic parts. Items that require sorting take priority over items that require disassembly. Each electronic item takes between 8 to 16 minutes uniformly distributed to disassemble (stream 8).
The hazardous waste recycling center is open for 7 hours per day, 5 days per week. Simulate 12 weeks of performance and estimate the following quantities:
Utilization of the workers
Average waiting time for items waiting to be unloaded
Average number of items waiting to be unloaded
Average number of items waiting to be sorted
Average waiting time of items to be sorted
Average number of items waiting to be disassembled
Average waiting time for items waiting to be disassembled.
Exercise 6.25 Orders for street lighting poles require the production of the tapered pole, the base assembly, and the wiring/lighting assembly package. Orders are released to the shop floor with an exponential time between arrival of 20 minutes (stream 1). Assume that all the materials for the order are already available within the shop floor.
Once the order arrives, the production of the pole begins. Pole production requires that the sheet metal be cut to a trapezoidal shape. This process takes place on a cutting shear. After cutting, the pole is rolled using a press brake machine. This machine rolls the sheet to an almost closed form. After rolling, the pole is sealed on an automated welding machine. Each of these processes are uniformly distributed with ranges \([3, 5]\), \([6,10]\), and \([4,8]\) minutes, (streams 2, 3, 4) respectively.
While the pole is being produced, the base is being prepared. The base is a square metal plate with four holes drilled for bolting the place to the mounting piece and a large circular hole for attaching the pole to the base. The base plates are in stock so that only the holes need to be cut. This is done on a water jet cutting machine. This process takes approximately 20 minutes plus or minus 2 minutes, triangularly distributed (stream 5). After the holes are cut, the plate goes to a grinding/deburring station, which takes between 10 minutes, exponentially distributed (stream 6).
Once the plate and the pole are completed, they are transported to the inspection station. Inspection takes 20 minutes, exponentially distributed (stream 7) with 1 operator. There could be a quality problem with the pole or with the base (or both). The chance that the problem is with the base is 0.02 (stream 8) and the chance that the problem is with the pole is 0.01 (stream 9). If either or both have a quality issue, the pole and base go to a rework station for rework. Rework is performed by a single operator and typically takes on average 100 minutes, exponentially distributed (stream 10). After rework, the pole and base are sent to final assembly. If no problems occur with the pole or the base, the pole and base are sent directly to final assembly.
At the assembly station, the pole is fixed to the base plate and the wiring assembly is placed within the pole. This process takes 1 operator approximately 30 minutes with a standard deviation of 4 minutes according to a lognormal distribution (stream 11). After assembly, the pole is sent to the shipping area for final delivery.
The shop is interested in taking on additional orders which would essentially double the arrival rate. Estimate the utilization of each resource and the average system time to produce an order for a lighting pole. Assume that the system runs 5 days per week, with two, eight hours shifts per day. Any production that is not completed within 5 days is continued on the next available shift. Run the model for 10 years assuming 52 weeks per year to report your results.
Exercise 6.26 Patients arrive at an emergency room where they are treated and then depart. Arrivals are exponentially distributed with a mean time between arrivals of 0.3 hours (stream 1). Upon arrival, patients are assigned a rating of 1 to 5, depending on the severity of their ailments. Use stream 2 to determine the category. Patients in Category 1 are the most severe, and they are immediately sent to a bed where they await medical attention. All other patients must first wait in the receiving room until a basic registration form and medical record are completed. They then proceed to a bed.
The emergency room has three beds, one registration nurse, and two
doctors. In all cases, the priority for allocating these resources is
based on the severity of the ailment. Hint: Read the documentation for the Queue class and rank the queue by a severity attribute represented by the QObject priority property.
The registration time for patients in Categories 2 through 5 is Uniform (0.1, 0.2) hours (stream 3). The treatment time for all patients is triangularly distributed with the minimum, most likely, and maximum values differing according to the patient’s category. The distribution of patients by category and the corresponding minimum, most likely, and maximum treatment times are summarized below. Use streams 4-8 for the treatment times of the categories of patients.
| Category | 1 | 2 | 3 | 4 | 5 |
|---|---|---|---|---|---|
| Percent | 6 | 8 | 18 | 33 | 35 |
| Minimum | 0.8 | 0.7 | 0.4 | 0.2 | 0.1 |
| Most Likely | 1.2 | 0.95 | 0.6 | 0.45 | 0.35 |
| Maximum | 1.6 | 1.1 | 0.75 | 0.6 | 0.45 |
The required responses for this simulation include:
Average number of patients waiting for registration
Utilization of beds
System time of each type of patient and overall (across patient types)
Using a run length of 30 days, develop a model to estimate the required responses. Report the responses based on estimating the system time of a patient regardless of type based on 50 replications.
Exercise 6.27 Customers enter a fast-food restaurant according to an exponential inter-arrival time with a mean of 0.7 minutes (use stream 1). Customers have a choice of ordering one of three kinds of meals: (1) a soft drink, (2) fries, or (3) soft drink, fries, and a burger. Upon arrive to the restaurant, the customer enters a single queue, awaits the availability of a cashier, gives the order to the cashier, then the customer pays the cashier. After the order is placed, the cooking can begin. The customer then waits until their order is ready. After receiving the order, the customer exits. A cashier may not take any additional orders until the current customer has paid. In this system, there are two cooks and two cashiers. The time to order and pay is represented by a triangular distribution with parameters (0.4, 0.8, 1.2) minutes and (0.2, 0.4, 0.6) minutes, respectively. The cooking time depends on the order as follows:
| Type | Percentage | Cooking Time |
|---|---|---|
| 1 | 30% | Uniform(0.3,0.8) |
| 2 | 15% | Uniform(0.8,1.1) |
| 3 | 55% | Uniform(1.0, 1.4) |
Use stream 2 for the ordering distribution, stream 3 for the paying distribution, streams 4, 5 and 6, for the type 1, 2, and 3 cooking distributions, respectively. Use stream 7 for the distribution across the order types.
Model the system for 8 hours of operation with 30 replications. Make a table like the following to summarize your answers for your replications.
| Average | Half-width | |
|---|---|---|
| Type 1 Throughput | ||
| Type 2 Throughput | ||
| Type 3 Throughput | ||
| Utilization of cashiers | ||
| Utilization of cooks | ||
| Customer System Time (in minutes) | ||
| Customer Waiting Time (in minutes) | ||
| Probability of wait \(>\) 5 minutes |
Exercise 6.28 Jobs arrive in batches of ten items each. The inter-arrival time is exponentially distributed with a meand of 2 hours (stream 1). The machine shop contains 2 milling machines and one drill press. About 30% of the items require drilling before being processed on the milling machine (stream 2). Drilling time per item is uniformly distributed between (10, 15) minutes (stream 3). The milling time is exponentially distributed (stream 4) with a mean of 15 minutes for items that do not require drilling, and uniformly distributed (stream 5) between (15,20) minutes for items that do. Assume that the shop has two 8-hour shifts each day and that you are only interested in the first shift’s performance. Any jobs left over at the end of the first shift are left to be processed by the second shift. Estimate the average number of jobs left for the second shift to complete at the end of the first shift to within plus or minus 5 jobs with 95% confidence. What is your replication length? Number of replications? Determine the utilization of the drill press and the milling machines as well as the average time an item spends in the system.
Exercise 6.29 A repair and inspection facility consists of two stations, a repair station with two technicians, and an inspection station with 1 inspector. Each repair technician works at a rate of 7.5 items per hour (stream 2), while the inspector can inspect 15 items per hour (stream 3) each exponentially distributed. Approximately 10% (stream 5) of all items fail inspection and are sent back to the repair station (this percentage holds even for items that have been repaired two to three times). If an item fails inspection three times then it is scrapped. When an item is scrapped, the item is sent to a disassembly station to recover the usable parts.
At the disassembly station, the items wait in a queue until a technician is available. The disassembly time is quite complex and is distributed according to a Lognormal distribution with a mean of 200 minutes and a standard deviation of 10 minutes (stream 4). Assume that items arrive according to a Poisson arrival process with a rate of 12 per hour (stream 1).
The weekly performance of the system is the key objective of this simulation analysis. Assume that the system starts empty and idle on Monday mornings and runs continuously for 2 shifts per day for 5 days. Each shift has 420 minutes of actual work time available. Any items not completed by the end of \(2^{nd}\) shift are carried over to the \(1^{st}\) shift of the next day. Any items left over at the end of the week are handled by a separate weekend staff that is not of concern to the current study.
Estimate the following:
- The average system time of items that pass inspection on the first attempt. Measure this quantity such that you are 95% confident to within +/- 3 minutes.
- The average number of items in the system at any time.
- The system time (for completed items) regardless of the number of times the item failed inspection.
- The probability that an item will be scrapped.
- The average number of items completed per week.
- The average number of items scrapped per week.
- The average number of items left for the weekend staff.
Sketch an activity diagram for this situation.
Assume that there are 2 technicians at the repair station, 1 inspector at the inspection station, and 1 technician at the disassembly station. Develop a model for this situation.
Exercise 6.30 Reconsider Exercise 6.29 Assume that there are 2 technicians at the repair station and 1 inspector at the inspection station. The disassembly station is also staffed by the 2 technicians that are assigned to the repair station. That is, the disassembly and repair stations share their technicians. The repair station should have priority over work at the disassembly station. Develop a model for this situation. Compare your results to Exercise 6.29, which design do you recommend?
Exercise 6.31 As part of a diabetes prevention program, a clinic is considering setting up a screening service in a local mall. They are considering two designs: Design A: After waiting in a single line, each walk-in patient is served by one of three available nurses. Each nurse has their own booth, where the patient is first asked some medical health questions, then the patient’s blood pressure and vitals are taken, finally, a glucose test is performed to check for diabetes. In this design, each nurse performs the tasks in sequence for the patient. If the glucose test indicates a chance of diabetes, the patient is sent to a separate clerk to schedule a follow-up at the clinic. If the test is not positive, then the patient departs.
Design B: After waiting in a single line, each walk-in is served in order by a clerk who takes the patient’s health information, a nurse who takes the patient’s blood pressure and vitals, and another nurse who performs the diabetes test. If the glucose test indicates a chance of diabetes, the patient is sent to a separate clerk to schedule a follow-up at the clinic. If the test is not positive, then the patient departs. In this configuration, there is no room for the patient to wait between the tasks; therefore, a patient who as had their health information taken cannot move ahead unless the nurse taking the vital signs is available. Also, a patient having their glucose tested must leave that station before the patient in blood pressure and vital checking can move ahead.
Patients arrive to the system according to a Poisson arrival process at a rate of 9.5 per hour (stream 1). Assume that there is a 5% chance (stream 2) that the glucose test will be positive. For design A, the time that it takes to have the paperwork completed, the vitals taken, and the glucose tested are all log-normally distributed with means of 6.5, 6.0, and 5.5 minutes respectively (streams 3, 4, 5). They all have a standard deviation of approximately 0.5 minutes. For design B, because of the specialization of the tasks, it is expected that the mean of the task times will decrease by 10%.
Assume that the clinic is open from 10 am to 8 pm (10 hours each day) and that any patients in the clinic before 8 pm are still served. The distribution used to model the time that it takes to schedule a follow up visit is a Weibull distribution with shape 2.6 and scale 7.3 using stream 6.
Make a statistically valid recommendation as to the best design based on the average system time of the patients. We want to be 95% confident of our recommendation to within 2 minutes.
Exercise 6.32 A copy center has one fast copier and one slow copier. The copy time per page for the fast copier is thought to be lognormally distributed with a mean of 1.6 seconds and a standard deviation of 0.3 seconds (stream 2). A co-op Industrial Engineering student has collected some time study data on the time to copy a page for the slow copier. The times, in seconds, are given in the data set associated with Exercise B.13. Use stream 3 for the fitted distribution.
The copy times for the slow and fast copiers are given on a per page basis. Thus, the total time to perform a copy job of N pages is the sum of the copy times for the N individual pages. Each individual page’s time is random.
Customers arrive to the copy center according to a Poisson process with a mean rate of 1 customer every 40 seconds (stream 1). The number of copies requested by each customer is equally likely over the range of 10 and 50 copies (stream 4). The customer is responsible for filling out a form that indicates the number of copies to be made. This results in a copy job which is processed by the copying machines in the copy center. The copying machines work on the entire job at one time.
The policy for selecting a copier is as follows: If the number of copies requested is less than or equal to 30, the slow copier will be used. If the number of copies exceeds 30, the fast copier will be used, with one exception: If no jobs are in queue on the slow copier and the number of jobs waiting for the fast copier is at least two, then the customer will be served by the slow copier. After the customer gives the originals for copying, the customer proceeds to the service counter to pay for the copying. Assume that giving the originals for copying requires no time and thus does not require action by the copy center personnel. In addition, assume that one cashier handles the payment counter only so that sufficient workers are available to run the copy machines. The time to complete the payment transaction is lognormally distributed with a mean of 20 seconds and a standard deviation of 10 seconds (stream 5). As soon as both the payment and the copying job are finished, the customer takes the copies and departs the copying center. The copy center starts out a day with no customers and is open for 10 hours per day.
Management has requested that the co-op Industrial Engineer develop a model because they are concerned that customers have to wait too long for copies. Recently, several customers complained about long waits. Their standard is that the probability that a customer waits longer than 4 minutes should be no more than 10%. They define a customer’s waiting time as the time interval from when the customer enters the store to the time the customer leaves the store with their completed copy job. If the waiting time criteria is not met, several options are available: The policy for allocating jobs to the fast copier could be modified or the company could purchase an additional copier which could be either a slow copier or a fast copier.
Develop a model for this problem. Based on 25 replications, report in table form, the appropriate statistics on the waiting time of customers, the daily throughput of the copy center, and the utilization of the payment clerk. In addition estimate the probability that a customer spends in the system is longer than 4 minutes.
Exercise 6.33 Passengers arrive at an airline terminal according to an exponential distribution for the time between arrivals with a mean of 1.5 minutes (stream 1). Of the arriving passengers 7% are frequent flyers (stream 2). The time that it takes the passenger to walk from the main entrance to the check-in counter is uniform between 2.5 and 3.5 minutes (stream 3). Once at the counter the travellers must wait in a single line until one of four agents is available to serve them. The check-in time (in minutes) is Gamma distributed with a mean of 5 minutes and a standard deviation of 4 minutes (stream 4). When their check-in is completed, passengers with carry-on only go directly to security. Those with a bag to check, walk to another counter to drop their bag. The time to walk to bag check is uniform(1,2) minutes (stream 9). Since the majority of the flyers are business, only 25% of the travellers have a bag to check (stream 5). At the baggage check, there is a single line served by one agent. The time to drop off the bag at the bag check stations is lognormally distributed with a mean of 2 minutes and a standard deviation of 1 minute (stream 6). After dropping off their bags, the traveller goes to security. The time to walk to security (either after bag check or directly from the check in counter) is exponentially distributed with a mean of 8 minutes (stream 10). At the security check point, there is a single line, served by two TSA agents. The TSA agents check the boarding passes of the passengers. The time that it takes to check the boarding pass is triangularly distributed with parameters (2, 3, 4) minutes (stream 7). After getting their identity checked, the travellers go through the screening process. We are not interested in the screening process. However, the time that it takes to get through screening is distributed according to a triangular distribution with parameters (5, 7, 9) minutes (stream 8). After screening, the walking time to the passenger’s gate is exponentially distributed with a mean of 5 minutes (stream 11).
We are interested in estimating the average time it takes from arriving at the terminal until a passenger gets to their gate. This should be measured overall and by type (frequent flyer versus non-frequent flyer).
Assume that the system should be studied for 16 hours per day.
- Report the average and 95% confidence interval half-width on the following based on the simulation of 10 days of operation. Report all time units in minutes.
| Statistic | Average | Half-Width |
|---|---|---|
| Utilization of the check-in agents | ||
| Utilization of the TSA agents | ||
| Utilization of the Bag Check agent | ||
| Frequent Flyer Total time to Gate | ||
| Non-Frequent Flyer Total time to Gate | ||
| Total time to Gate regardless of type | ||
| Number of travellers in the system | ||
| Number of travellers waiting for check-in | ||
| Number of travellers waiting for security | ||
| Time spent waiting for check-in | ||
| Time spent waiting for security |
- Based on the results of part (a) determine the number of replications necessary to estimate the total time to reach their gate regardless of type to within \(\pm\) 1 minute with 95% confidence.
Exercise 6.34 Consider the soccer mom and daughter example of Section 6.4.5. Use the Signal class to implement the example. Use print statements to prove that the interaction is the same as the implementation with the Suspension class.
Exercise 6.35 Consider the soccer mom and daughter example of Section 6.4.5. Update the Blockage example to implement the example. Use print statements to prove that the interaction is the same as the implementation with the Suspension class.
Exercise 6.36 Consider the soccer mom and daughter example of Section 6.4.5. Update the BlockingActivity example to implement the example. Use print statements to prove that the interaction is the same as the implementation with the Suspension class.
Exercise 6.37 Consider the Tie-Dye T-Shirt Example 6.9.
- Redo the example using the
HoldQueueclass to model the process interaction. - Redo the example using blockages to model the process interaction.