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# Queuing problems

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### Queuing problems

1. 1. OPERES3 Problems in Queuing Theory1. (Case 1) A computing system has a single printer attached to print out the output of the users. The operating system software sends an average of 20 requests per hour to the printer. The printer is capable of printing out 35 jobs per hour on the average. Assume that the job arrival rate and the printing rate are Poisson distributed. a. On the average, how many jobs are in the output queue waiting to be printed out? b. How many jobs are in the “printer output system” on the average? c. How long can a user expect to wait to have his or her job printed out once it has been sent to the printer? d. Once a job enters the output queue, how long does it take before printing begins on the job? e. What percentage of time will the printer be busy doing print jobs? f. (Case 2) Assume that a second printer is added to the system. Answer questions 1.1 to 1.5, assuming that the two printers are identical.2. (Case 1) Telephone calls arrive at a particular small-scale paging system at the Poisson rate of 20 per hour. The calls are transferred to an operator that can handle an average of 25 calls per hour, following an exponential distribution. The operator can handle only one caller at a time; if the operator is busy, arriving calls are put on hold and the callers are assumed to be patient. a. Assume that no limit is placed on the number of calls that can be put on hold: 1) What percentage of the time is the operator busy? 2) On the average, how many callers are on hold? 3) On the average, how long can a caller expect to spend on the telephone when calling this office? b. (Don’t do this) Suppose that with the single operator, only a maximum of two calls can be put on hold. How does this affect your answers in 2.1? c. (Case 2) Suppose that an additional operator is hired, with no limit on the number of calls on hold: 1) What percentage of the time are both operators busy? 2) On the average, how many callers are on hold? 3) On the average, how long can a caller expect to spend on the telephone when calling this office? d. (Case 2) Suppose that the office management feels that, in order to provide good customer service, a caller should not have to wait more than 10 seconds on hold. How many telephone operators should be hired to answer calls?3. Orders for a firm’s products arrive at the rate of 200 a week. A single clerk is assigned to process the orders at an average of 250 orders per week. The cost of delaying an average shipment for 1 week is P500. If a clerk costs P300 per week, how many clerks should be assigned to process the paperwork?4. Moonlight Travel Reservation Service has one agent and 3 telephone lines. When the agent is busy, the next caller is placed on hold on one of the remaining lines. When all three lines are busy, callers receive a busy signal and call elsewhere for reservations. Service appears to be exponentially distributed, with a mean of three minutes. Calls arrive randomly at a Poisson rate of 15/hour. a. What is the probability that a caller receives a busy signal? b. What is the average number of customers put on hold at any given time? c. What is the probability that a customer is placed on hold? d. How long, on the average, will a customer be on hold before the agent can attend to him? e. If a caller makes three attempts before finally calling elsewhere for reservations, what fraction of the customers will be able to do business with Moonlight Travel Reservation Service?
2. 2. 5. (Case 1) Customers arrive at a one-person barber shop with an average interarrival time of 20 minutes. The average time for a haircut is 12 minutes, exponentially distributed. a. The owner wishes to have enough seats in the waiting area so that on the average no more than 5% of the arriving customers will have to stand. How many seats should be provided? b. Suppose that there is only sufficient space in the waiting area for five seats. What is the probability that an arriving customer will not find a seat?6. A plant distributes its products by truck. The exponential average loading time per truck at the loading facility is 20 minutes. Trucks arrive at a rate of 2/hour. Management feels that the existing loading facility is more than adequate. However, the drivers complain that they have to wait more than half of the time. Analyze the situation and find how much money the company can save if the waiting time of a truck is figured at P10/hr and the plant is in operation 8 hours per day. An automatic device that can load 10 trucks per hour is available at a cost of P90/day over the cost of the existing facility.7. Campers arrive at the County Park at a Poisson rate of 100 per month. The time spent in the park by a camper varies according to an exponential distribution with a mean of 0.10 month. Since the park is very large, campers have no problem finding a campsite. However, the state recreation officials have a policy of maintaining a certain level of forest-ranger workforce for the convenience and safety of the campers. The current policy is to have one ranger per 10 campers at any given time. a. How many campers are using the park on the average? b. What percentage of the time is the park empty? c. How many forest rangers should the state employ at the County Park?8. A gasoline station is served by 1 employee who is capable of serving 30 customers per hour, varying according to an exponential distribution. There is a maximum space for five cars in the station, including those served and in line. Cars arrive at the station at an average rate of every 3 minutes. Cars that do not have parking spaces leave and do not return. Find: a. The average number of cars waiting for service. b. The probability of finding no car in the station. c. The average waiting time in line for each car. d. The probability of finding 3 cars in the station at any given time. e. It was proposed to increase the space so that 10 cars would be accommodated. The investment would entail P5 per car space per hour. A car that leaves the station means a loss of P10. Should the space be enlarged or not?9. A shop utilizes 10 identical machines. The profit per machine is P40 per hour of operation. Each machine breaks down on the average once every 7 hours. One person can repair a machine in 4 hours on the average, but the actual repair time varies according to a Poisson distribution. The repairman’s salary is P60/hour. Determine: a. The number of repairmen that will minimize cost. b. The number of repairmen needed so that the expected number of broken machines at any given time is less than 4. c. The number of repairmen needed so that the expected delay time until a machine is repaired is less than 4 hours.10. (Case 2) Students arriving at the computing center can wait either at a small table in the center (which will seat four students) or if the table is full they can go to a nearby student lounge or they can wait in the hall. What percentage of time will the waiting area inside the computing center be sufficient to accommodate the waiting students?11. The city of Manila operates a large fleet of police squad cars. When a car breaks down due to a minor malfunction it is sent to a single-garage repair facility where it is repaired by a crew of
3. 3. mechanics. The crew foreman, Jim, must put in a request to hire her team mechanics. If n mechanics work on a single car together, repair time for minor repairs is approximately constant at 20/n minutes. Disabled squad cars arrive at the repair facility at the Poisson rate of five cars per hour. The police chief tells Joanne that on the average he does not want a squad car to be out of commission for longer than 15 minutes. a. What is the minimum number of mechanics Joanne should hire? b. On the average, how many squad cars are out of commission?12. Solve Joanne’s facility design problem (exercise 11 above) under the assumption that the reapir time of 20/n minutes is a mean exponential time. Answer both parts (a) and (b) of the problem.13. The manufacturing Supervisor of Hardpress Industries must decide between three makers of forklift trucks to use on the plant floor. The truck will be used to move raw materials from storage bins to work centers in a section of the plant. Request from these work centers occur according to a poisson process at a mean rate of eight per hour. The time to move raw materials is exponentially distributed. The service rates per hour for each of the truck under consideration is given in the table along with estimated costs per hour (operating costs plus depreciation). For each hour a work center is idle waiting for raw materials, an average cost of P100 is incurred (wages plus lost productivity). Table Truck Service Rate (jobs/hr) Cost (P/hr) 1 10 30 2 15 50 3 18 40 Which truck would you purchase based on queueing analysis? (Be sure to specify clearly the queueing model you are using.)14. In exercise 13, would your purchase choice change if the Manufacturing Supervisor felt that the service times of the forklift trucks were (more or less) constant times? That is, Truck 1 can service work centers at the constant rate of 10 jobs per hour, Truck 2 at 15 per hour, and Truck 3 at 18 per hour. (What would be the total cost per hour of using each of the trucks?)15. (Case 1) The state operates a weigh station for trucks traveling on the highway. Every truck must pull off the highway, enter the weigh station, and undergo state inspection procedures before counting on. Trucks arrive at this station at a poisson rate of seven per hour. The time to inspect/weigh a truck varies, having an exponential probability density with a mean of 7 minutes. a. How many trucks are detained at the station on the average? b. How long can a truck expect to be detained? c. How many trucks are lined up in front of the station on the average? d. How long on the average does a truck driver have to wait in line for the truck to be inspected?16. Refer to exercise 15, the truck driver traveling the route passing the weigh station have complained of long waiting time at the inspection point. The major phase of the inspection procedure consists of weighing-in the truck. The state highway commissioner has decided to look into ways of reducing the average time a trucker must spend at the weigh station to 25 minutes or less. The commissioner is considering the purchase of new digital solid waste weighing scale that should reduce the weighing time. Three manufacturer’ products are available: (1) model 1 costs P10000 and could check 10 trucks per hour, (2) model 2 costs P12000 and would have an average service time of 5.5 minutes, and (3) model 3 costs P8500 and would have an average service time of 6.5 minutes. a. Determine which model to purchase. b. For the decision made in part (a), determine system operating characteristics P0, L, Lq, W and Wq.17. A recent increase in consumer complaints at Jersey Electric Authority has led to a decision to assign