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ETHE No.1 Until 30.1.2009 9am Question 1 Jonathan sells lottery tickets at a small stand in a shopping mall. The average time between two customer arrivals equals 19 minutes (Poisson arrival process). On average, Jonathan can sell tickets to 6 customers per hour (negative exponentially distributed service times). What is the probability that there are exactly 2 customers at Jonathan's stand (waiting plus being served)? Express the probability as a number between 0 and 1. 0.13 Your answer to question 1: Question 2 Consider a fast-food outlet with only one employee. All customers wait in a queue until they are served by the employee. On average, 12 customers arrive per hour (Poisson distributed). The employee can help exactly 14 customers per hour. What is the average number of customers at the fastfood outlet (waiting plus being served)? 3.43 Your answer to question 2: Question 3 Donna owns a small shop in the city centre, where she sells coffee mugs. Donna is the only person working in this shop. On average, 4 customers enter the shop every hour (Poisson distributed). Donna helps every customer that enters the shop. She needs on average 10.65 minutes (negative exponentially distributed) to help a customer. What is the probability that there are 5 customers or less in Donna's shop (waiting plus being served)? Express the probability as a number between 0 and 1. 0.87 Your answer to question 3: Question 4 Ramon's Barber Shop has one barber. Customers enter the shop with an average inter-arrival time of 23 minutes (Poisson arrival process). The barber is able to give exactly 4 haircuts per hour. What is the average time a customer will spend in the barber shop (waiting plus service time), expressed in minutes? 29.06 Your answer to question 4: Question 5 The QuickMoney bank has a single Automated Teller Machine (ATM). The time between two arrivals of clients equals on average 2.6 minutes (Poisson arrival process). It takes a client on average 2.4 minutes (negative exponentially distributed) to get money from the ATM. What is the average waiting time of a client (excluding the time the client is actually using the ATM), expressed in minutes? 28.8 Your answer to question 5: Question 6 Consider a paint shop for small steel products, where arriving products are spray painted one by one. On average 70 products arrive per hour at the paint shop (Poisson distributed). Painting of a product requires exactly 0.59 minutes per product. What is the average number of products waiting to be painted (so excluding products that are actually being painted)? 0.76 Your answer to question 6: Question 7 The company "New Internet Enterprises" has one internet server. The internet server receives many requests, with an average of 166 requests per minute (Poisson distributed). The internet server can on average process 333 requests per minute (negative exponentially distributed service times). What is the average number of request being processed (or stated differently: what is the expected number of requests in the system, but not in the waiting line)? 0.50 Your answer to question 7: Question 8 Amanda assembles computers at her workstation. On average there are 33 computers in the system (waiting plus being assembled by Amanda). Assume that the number of arriving computers is distributed according to a Poisson distribution and that the time Amanda needs for assembling a computer is negative exponentially distributed. What is the probability that the number of computers in the system is less than or equal to 53 at an arbitrary point in time? 0.80 Your answer to question 8: Question 9 Currently, a certain process is performed by a machine which has a processing time per product that is exactly equal for each of the products. The time products spend in the waiting line in front of the machine equals 60.7 minutes on average and the total time a product spends at this process (waiting plus processing) equals 78.7 minutes on average. The machine is rather old and the manager considers replacing the machine by a person. He estimates that this person can perform the same task with the same average speed; however, this time will now be distributed according to a negative exponential distribution. Assume that the number of products that arrive at the process is distributed according to a Poisson distribution. What is the expected time a product will spend in the new system with the person replacing the machine (waiting plus processing)? Express your answer in minutes. 139.4 Your answer to question 9: Question 10 The "FastScan" consulting company has performed a little investigation at the local grocery store. From this investigation it is apparent that customers arrive at the registers according to a Poisson distribution with an average value of 23.4 customers per hour. There are 4 registers available with identically skilled employees. There is one common waiting line for the registers. The customer who is waiting longest, goes to the first available register. One employee can help on average 9 customers per hour (negative exponentially distributed service times). What is the average time customers spend in the waiting line (expressed in minutes)? 1.69 Your answer to question 10: Question 11 Amy discovered an old report on waiting times of customers at her loan office. Unfortunately, someone spilled coffee on the report, because of which some numbers cannot be read. It is known from the report that the average length of the queue equals 2.9376 customers. The loan office has 6 tellers with identically skilled employees to serve the customers. On average an employee can serve 5.8 customers per hour (negative exponentially distributed service times). The report states furthermore that arrivals are Poisson distributed. What is the arrival rate, expressed as the number of customers arriving per hour? 29 Your answer to question 11: Question 12 Process 1 Process 2 Process 3 Consider the production system for wrapping CDs which is depicted above. Each CD that arrives goes first to process 1 where the quality of the CD is checked. Next, the CD goes to process 2 where the CD is put into a jewel case. Finally at process 3 a booklet is put into the jewel case. The following data are available for this production system: Every 5 minutes a CD arrives at the production line to be processed. Process 1 consists of a single machine, which needs 6 minutes per CD. Process 2 consists of 4 parallel machines. One machine needs 42 minutes per CD. Process 3 consists of 4 parallel machines. One machine needs 43 minutes per CD. Assume that transport from one process to the next does not require any time. What is the utilisation (expressed as a number between 0 and 1) of process 3? 1 Your answer to question 12: Question 13 Process 1 Process 2 Process 3 Consider the above printing facility for leaflets. Empty sheets of paper arrive (one by one) at the printing facility and are cut to the right size at process 1. Next, the empty leaflet goes to process 2 where the correct content is printed onto the leaflet. At process 3, the leaflet is folded. The following data are available for this printing facility: Every 0.9 seconds a sheet of paper arrives at the production line to be processed. Process 1 consists of 4 parallel cutting machines. One cutting machine needs 4.2 seconds per leaflet. Process 2 consists of 2 parallel printers. One printer needs 2 seconds per leaflet. Process 3 consists of 3 parallel folding machines. One machine needs 5.6 seconds per leaflet. Assume that transport from one process to the next does not require any time. What is the bottleneck? Process 3 Your answer to question 13: Question 14 Process 1 Process 2 Process 3 Consider a facility where desks are painted. A graphical display of the facility is given above. Complete desks arrive - one at a time - at the facility and go to process 1 where they are painted. At process 2 the paint is dried at a high temperature. Finally, the desk is covered with protective materials to prevent damage during transport. The following data are available for this facility: A desk arrives every 10 minutes at the facility. Process 1 consists of a single painting machine, which needs 9 minutes per desk. Process 2 consists of 2 parallel high-speed drying rooms. Each drying room has a capacity of one desk and requires 26 minutes per desk. Process 3 consists of a single packing employee, who needs 18 minutes per desk. Assume that transport from one process to the next does not require any time. What is the departure rate for this system (expressed in products per hour)? 3.33 Your answer to question 14: Question 15 Process 1 Process 2 Process 3 Consider the kitchen of a new pizza restaurant. A picture of the kitchen is given above. At process 1 the pizza dough is turned into a pizza bottom. Process 2 consists of putting tomatoes, cheese and other toppings onto the pizza. In process 3 the pizzas are heated in an oven. The following data are available for this kitchen: Every 7.1 minutes a pizza must be made (so the "inter-arrival time" for this process is 7.1 minutes). Process 1 consists of 2 parallel operating employees. One employee can make a pizza bottom in 5.4 minutes. Process 2 consists of 4 parallel operating employees. Each employee can put tomatoes and cheese onto a pizza in 15.4 minutes. Process 3 consists of a single pizza oven with a capacity of 2 pizzas. The usage of the oven is as follows. The employee opens the ovens, puts a batch of 2 pizzas into the oven, and then closes the oven. He waits for the pizzas to be baked, then he removes the pizzas from the oven and puts them onto plates. The entire process from putting pizzas into the oven until all pizzas are on their plates requires a fixed time of 6.3 minutes (fixed means that this time is independent of the number of pizzas in the batch) plus 1.6 minutes per pizza. Assume that the transfer from one process to the next does not require any time. What is the Work-In-Progress for this system, if calculated with the formula ? 4.27 Your answer to question 15: Question 16 Process 1 Process 2 Process 3 An insurance company is currently receiving many claims because of a recent storm. The company has set up a special line to deal with these insurance claims. Claims arrive and are first reviewed by an expert (process 1). The claim is then transferred to a secretary who writes a response to the client (process 2) based upon the expert's decision. Finally, a copy of the claim together with the decision is put into an envelope (process 3). The following data are available for this claim handling line: Every 9.6 minutes a claim arrives. Process 1 consists of a single expert, who can judge a claim 5.2 minutes. Process 2 consists of 3 parallel operating secretaries. A secretary can write a response in 28 minutes. Process 3 consists of 3 parallel operating employees. One employee can copy a claim and put a copy and the response into an envelope in 13.4 minutes. Assume that the transfer from one process to the next does not require any time. What is the deterministic throughput time for this system (expressed in minutes)? 46.6 Your answer to question 16: Question 17 Process 1 Process 2 Process 3 A repair shop for electronic equipment consists of three stages (processes). The first process consists of repairing the defective piece of equipment. The quality of the repairs and the functionality of the repaired equipment is tested at process 2. Finally, at process 3 a bill with exact specifications of the performed repairs is created. The following data are available for this repair line: Every 22 minutes a defective piece of equipment arrives. Process 1 consists of 4 parallel operating repair persons. One repair person can repair a piece of equipment in 77 minutes. Process 2 consists of 4 parallel operating quality employees. A quality employee can check the quality and functionality of a repaired piece of equipment in 78 minutes. Process 3 consists of a single clerk, who can make the bill with specifications in 14 minutes. Assume that the transfer from one process to the next does not require any time. What is the Work-In-Progress for this system, if calculated with Little's equation? 7.68 Your answer to question 17: Question 18 Interarrival time: 23 minutes Process 1 Process 2 Time required: Time required: 15 minutes 16 minutes per per product product Batch size: 8 Setup time per batch: 2 minutes Consider the processes depicted above. What is the productive utilisation (expressed as a number between 0 and 1) of process 2? 0.65 Your answer to question 18: Question 19 Interarrival time: 23 minutes Process 1 Time required: 16 minutes per product Batch size: x Setup time per batch: 81.9 minutes Consider the process depicted above. What is the minimum batch size to prevent process 1 from being a bottleneck? 12 Your answer to question 19: Question 20 At a container terminal, 252 containers arrive per hour by truck. The length of a container can be 6, 12 or 14 metres. From past experience, it is known that 38% of the arriving containers have a length of 6 metres. Another 30% have a length of 12 metres and the remaining containers are 14 metres long. Every incoming container is first checked for damage by an employee. The time to check a container depends on the length of the container and follows a normal distribution. The mean time for checking a container equals 1.3 minutes per container plus 2.8 seconds per metre length of the container. So a 6 metre long container requires 94.8 seconds for checking. Checking time has a standard deviation of 20 seconds, regardless of container length. There are 10 employees working in parallel to check containers for damage. After checking, containers must be transported to the stack (the storage area). There are 74 transport vehicles available each of which transports 5 containers at a time. It takes 1.5 minutes per container to load the transport vehicle. Transport starts when 5 containers are loaded onto the vehicle. The travel time from the checking area to the stack is 12 minutes. At the stack the transport vehicle is unloaded, which can be done 10% faster than loading. The return trip of the vehicle is two minutes faster, because the vehicle is now empty. The workload is spread evenly over the vehicles. What is the utilisation of the transport vehicles (expressed as a number between 0 and 1)? 0.41 Your answer to question 20: Question 21 Consider a gas station where cars arrive to buy fuel. The required type of fuel per car is as follow: 74% of the cars need 'Euro 95'; the remaining 26% of the cars need 'diesel'. Every 0.81 minutes a car arrives at the gas station. There are several fuel pumps where cars can get fuel. Every pump can be used to get Euro 95 and diesel. A pump can only be used by the car that is parked in front of it. A pump must be considered to be "busy" as long as a car is parked in front of it, even if the car is no longer using the pump. Newly arriving cars wait in a single queue for an empty fuel pump to become available. Filling up a car is assumed to follow a normal distribution with a mean of 2.6 minutes and a standard deviation of 20 seconds if Euro 95 is requested. Diesel tanks are usually a bit larger than Euro 95 tanks and therefore this takes longer, 4.8 minutes on average with a standard deviation of 30 seconds. After filling up, the car's driver leaves the car parked in front of the pump and walks in 44 seconds to the shop. Most drivers, to be precise 88%, go straight to the cashier to pay for the fuel (assume 5 seconds walking time in the shop to get to the cashier). The remaining drivers (12%) first select something to drink or eat from the shop, which takes on average 3.3 minutes (which includes the walking time to the cashier). All customers pay at the counter. The manager of the gas station always makes sure there are sufficient cashiers at the counter so drivers never have to wait. Some drivers (30%) pay cash which requires 36 seconds; the remaining drivers pay by PIN or creditcard, which requires 31 seconds. Thereafter, the driver walks back to the car in 44 seconds and drives off, after which the next car can use that pump. What is the minimum required number of fuel pumps to prevent the pumps from being a bottleneck? 7 Your answer to question 21: Question 22 Suppose it took 104 hours to make the first product. The learning rate has been estimated at 83%. How long will it take to make the 18th product (expressed in hours)? 47.82 Your answer to question 22: Question 23 Assume that the first unit takes 1157.05 minutes to produce and the learning rate is 89%. How many units should at least be produced in order to reduce the time per unit to 557 minutes or less? 78 Your answer to question 23: Question 24 Consider a production facility for computers. Last week, a new type of computers has been taken into production. It took the employees 243 hours to make the first computer. Just now, the 20th computer was made, which took 117 hours. Assuming that the learning rate will continue this way, how long will it take to make the 41st computer (expressed in hours)? 98.20 Your answer to question 24:

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