Assignment 3 Modeling & Simulation
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Nirmanyu Chavan
Feb 8, 2016, 5:36:24 PM2/8/16
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Modeling and Simulation
Unit-2 (T03)
Q.1 Customers arrive at a one window drive according to a Poisson distribution with mean of 10 minutes and service time per customer is exponential with mean of 6 minutes. The space in front of the window can accommodate only three vehicles including the serviced one. Other vehicles have to wait outside this space Calculate:
(I) Probability that on arriving customer can drive directly to have space in front of the window.
(ii). Probability that on arriving customer will have to wait outside the directed space.
(III). How long on arriving customer is expected to wait before getting the service.
Q.2 Arrivals of mechanics at a tool crib are considered to be Poisson distributed at an average rate of 6 per hour. The length of time the mechanics must remain at the tool crib is exponentially distributed with the average time being 0.05 hours.
(I). what is the probability that a mechanic arriving at the tool crib will have to wait
(ii). what is the average number of mechanics at the tool crib.
(iii) What is the average length of the non-empty queues that that from time to time.
(iv) The company will install a second tool crib when convinced that the machines would have to spend 6 minutes waiting and being serviced at the tool crib. By how much time the flow of mechanics to the tool crib should increase to justify the addition of a second tool crib.
Q.3. on an average 96 patents per 24 hour day requires the service of an emergency clinic. Also on average, patient requires 10 minutes of active attention. Assume that the facility can handle only one emergency at a time. Suppose that it costs the clinic Rs. 100 per patient treated to obtain an average servicing time of 10 minutes, and the each minute of decrees in this average time would cost Rs. 10 per patient treated. How much have to be budgeted by the clinic to decrease the average size of the queue from patients to patient.
Q.4 In the central Railway station 15 computerized reservation counters are available. A customer can book his/ her ticket in any train on any day in any one of these computerized reservation counters. The average time spend per customer by each clerk is 5 minutes. Average arrivals per hour during three types of activity periods have been calculated and customers have been surveyed to determine how long they are willing to wait during each type of period
Type of period Arrivals per hour Customer’s acceptable waiting time
Peak 11015 minutes
Normal6010 minutes
Low305 minutes
Making suitable assumptions on this queuing process, determine how many counters should be kept open during each type of period.
Q.5.Warehouse has only one loading dock manned by a three- person crew. Truces arrive at the loading dock at an average rate of 4 trucks per hour and the arrival rate is Poisson distributed. The loading of a truck takes 10 minutes on an average and can be assumed to be exponentially distributed. The operation cost of a truck is Rs. 20 per hour and the members of the loading crew are paid @ Rs. 6 per each per hour. Would you advise the truck owner to add another crew of three persons?
Q.6.The average rate of arrivals at a services store is 30 per hour. At present there is one cashier who on average attends 45 customers per hour. The store proprietor estimates that each extra minute of system process time per customer means a loss of Re. 0.50 an assistant can be provided to the cashier and in that case the service unit can deal with 75 customers per hour. The wage rate of the assistant is Rs. 15 per hour. Is it worth employing an assistant?
Q.7.A road transport company has one reservation clerk on duty at a time. She handles information of bus schedules and makes reservations. Customers arrive at a rate of 8 per hour and the clerk can service 12 customers on an average per hour. After stating your assumptions’ answer the following:
(i). what is the average numbers of the customers waiting for the service of the clerk
(ii) What is the average time a customer’s has to wait before getting services.
(iii) The management is contemplating to install a computer system to handle the information and reservations. This is expected to reduce the service time from 5 to 3 minutes. The additional cost of having a customer to wait is estimated to be 12 paisa per minute spent waiting before being served, should the company install the computer system Assume 8-hour working day.
Q.8.In the factory, the machine breakdown on an average rate is 10 machines per hour. The idle time cost of the machine is estimated to be Rs. 20 per hour. The factory work 8 hours a day. The factory manager is considering 2 machines for repairing the machines. The first mechanic A takes about 5 minutes on an average to repair a machine and demands wages Rs.10 per hour. The second mechanic B takes about 4 minutes in repairing a machine and demands wages at the rate of Rs. 15 per hour. Assuming that the rate of machine breakdown is Poisson- distribution and the repair rate is exponentially distributed, which of the two mechanics should be engaged.
Q.9.A repairman is to be hired to repair machines that break down at an average rate of 4 per hour. Breakdown occur randomly (Poisson) over time. Non-productive time on any machine is considered to cost the company Rs. 5 per hour. Management has narrowed the choice to two repairmen: One slows but cheap, the other fast but expensive. The slow but cheap repairmen have a salary of Rs. 30 per hour: in return he will service broken-down machine at an average rate of 5 per hour. The fast but expensive repairmen have a salary of Rs. 50 per hour and will repair machine at an average rate of 7 per hour. Which repairmen should the company hire? Assume exponential repair times for both repairmen.
Q.10.A textile firm uses a Rs. 20/hr cost for direct and indirect lab our maintenance and estimates downtime costs on any of a large group of spinning machines at Rs. 100/hr if breakdown are distributed according to Poisson distribution with a mean of four/hr., and the mean number of units a worker can service is six breakdown/hr. (distributed exponentially), what is the optimum maintenance crew size.
Q.11.A firm has several machines and wants to install its own service facility for the repair of its machines. The average breakdown rate of the machines is 3 per day. The repair time has exponential. The loss incurred due to the lost time of an inoperative machine is Rs. 40 per day. There are to repair facilities available. Facility X has an installation cost of Rs. 20,000 and facility Y costs Rs. 40,000. The total lab our cost per year for the two facilities is Rs. 5,000 and Rs.8,000 respectively. Facility X can repair 4 machines daily while facility Y can repair 5 machines daily. The life span of both the facilities is 4 year. Which facility should be installed?
Q.12.A company which operates its own fleet of ships to import raw materials is considering rebuilding the unloading berth. The following information is given about the present and new berth:
Fixed cost Operating cost
Berth Per day Per Day** Capacity***
Present Rs. 4,000 Rs. 6,000 8,000
New 8,000 10,000 19,000
Ships to be unloaded each carry 8,000 tons of raw materials, and analysis of post records shows that the number of arrivals per week approximately follows a Poisson distribution with a mean arrival rate of 4 ships per week. Unloading times are considered to be exponentially distributed. The time spend in the unloading system (queuing plus unloading time) is estimated to cast the company Rs. 8,000 per day. Assuming a 7- day working week, estimate the average weekly cost of both the present and new berth systems Do you advise the introduction of the new berth.
Q.13.Trucks of a company arrive at a transshipment center to be unloaded in a pattern which is characterized by the Poisson distribution. The average rate of arrivals is 45 per hour. A set of attendants unload the trucks, and the level of service is 100 per hour on an average. The driver make Rs. 16 per hour and the set of attendance are paid Rs.10 per hour.
(i) How much expense, on the average, is incurred by the company for idle time on part on each driver each time he is at the transshipment centre?
(ii) Find the optimum number of sets of attendants to be employed for transshipments.
Unit-2 (T03)
Q.1 Customers arrive at a one window drive according to a Poisson distribution with mean of 10 minutes and service time per customer is exponential with mean of 6 minutes. The space in front of the window can accommodate only three vehicles including the serviced one. Other vehicles have to wait outside this space Calculate:
(I) Probability that on arriving customer can drive directly to have space in front of the window.
(ii). Probability that on arriving customer will have to wait outside the directed space.
(III). How long on arriving customer is expected to wait before getting the service.
Q.2 Arrivals of mechanics at a tool crib are considered to be Poisson distributed at an average rate of 6 per hour. The length of time the mechanics must remain at the tool crib is exponentially distributed with the average time being 0.05 hours.
(I). what is the probability that a mechanic arriving at the tool crib will have to wait
(ii). what is the average number of mechanics at the tool crib.
(iii) What is the average length of the non-empty queues that that from time to time.
(iv) The company will install a second tool crib when convinced that the machines would have to spend 6 minutes waiting and being serviced at the tool crib. By how much time the flow of mechanics to the tool crib should increase to justify the addition of a second tool crib.
Q.3. on an average 96 patents per 24 hour day requires the service of an emergency clinic. Also on average, patient requires 10 minutes of active attention. Assume that the facility can handle only one emergency at a time. Suppose that it costs the clinic Rs. 100 per patient treated to obtain an average servicing time of 10 minutes, and the each minute of decrees in this average time would cost Rs. 10 per patient treated. How much have to be budgeted by the clinic to decrease the average size of the queue from patients to patient.
Q.4 In the central Railway station 15 computerized reservation counters are available. A customer can book his/ her ticket in any train on any day in any one of these computerized reservation counters. The average time spend per customer by each clerk is 5 minutes. Average arrivals per hour during three types of activity periods have been calculated and customers have been surveyed to determine how long they are willing to wait during each type of period
Type of period Arrivals per hour Customer’s acceptable waiting time
Peak 11015 minutes
Normal6010 minutes
Low305 minutes
Making suitable assumptions on this queuing process, determine how many counters should be kept open during each type of period.
Q.5.Warehouse has only one loading dock manned by a three- person crew. Truces arrive at the loading dock at an average rate of 4 trucks per hour and the arrival rate is Poisson distributed. The loading of a truck takes 10 minutes on an average and can be assumed to be exponentially distributed. The operation cost of a truck is Rs. 20 per hour and the members of the loading crew are paid @ Rs. 6 per each per hour. Would you advise the truck owner to add another crew of three persons?
Q.6.The average rate of arrivals at a services store is 30 per hour. At present there is one cashier who on average attends 45 customers per hour. The store proprietor estimates that each extra minute of system process time per customer means a loss of Re. 0.50 an assistant can be provided to the cashier and in that case the service unit can deal with 75 customers per hour. The wage rate of the assistant is Rs. 15 per hour. Is it worth employing an assistant?
Q.7.A road transport company has one reservation clerk on duty at a time. She handles information of bus schedules and makes reservations. Customers arrive at a rate of 8 per hour and the clerk can service 12 customers on an average per hour. After stating your assumptions’ answer the following:
(i). what is the average numbers of the customers waiting for the service of the clerk
(ii) What is the average time a customer’s has to wait before getting services.
(iii) The management is contemplating to install a computer system to handle the information and reservations. This is expected to reduce the service time from 5 to 3 minutes. The additional cost of having a customer to wait is estimated to be 12 paisa per minute spent waiting before being served, should the company install the computer system Assume 8-hour working day.
Q.8.In the factory, the machine breakdown on an average rate is 10 machines per hour. The idle time cost of the machine is estimated to be Rs. 20 per hour. The factory work 8 hours a day. The factory manager is considering 2 machines for repairing the machines. The first mechanic A takes about 5 minutes on an average to repair a machine and demands wages Rs.10 per hour. The second mechanic B takes about 4 minutes in repairing a machine and demands wages at the rate of Rs. 15 per hour. Assuming that the rate of machine breakdown is Poisson- distribution and the repair rate is exponentially distributed, which of the two mechanics should be engaged.
Q.9.A repairman is to be hired to repair machines that break down at an average rate of 4 per hour. Breakdown occur randomly (Poisson) over time. Non-productive time on any machine is considered to cost the company Rs. 5 per hour. Management has narrowed the choice to two repairmen: One slows but cheap, the other fast but expensive. The slow but cheap repairmen have a salary of Rs. 30 per hour: in return he will service broken-down machine at an average rate of 5 per hour. The fast but expensive repairmen have a salary of Rs. 50 per hour and will repair machine at an average rate of 7 per hour. Which repairmen should the company hire? Assume exponential repair times for both repairmen.
Q.10.A textile firm uses a Rs. 20/hr cost for direct and indirect lab our maintenance and estimates downtime costs on any of a large group of spinning machines at Rs. 100/hr if breakdown are distributed according to Poisson distribution with a mean of four/hr., and the mean number of units a worker can service is six breakdown/hr. (distributed exponentially), what is the optimum maintenance crew size.
Q.11.A firm has several machines and wants to install its own service facility for the repair of its machines. The average breakdown rate of the machines is 3 per day. The repair time has exponential. The loss incurred due to the lost time of an inoperative machine is Rs. 40 per day. There are to repair facilities available. Facility X has an installation cost of Rs. 20,000 and facility Y costs Rs. 40,000. The total lab our cost per year for the two facilities is Rs. 5,000 and Rs.8,000 respectively. Facility X can repair 4 machines daily while facility Y can repair 5 machines daily. The life span of both the facilities is 4 year. Which facility should be installed?
Q.12.A company which operates its own fleet of ships to import raw materials is considering rebuilding the unloading berth. The following information is given about the present and new berth:
Fixed cost Operating cost
Berth Per day Per Day** Capacity***
Present Rs. 4,000 Rs. 6,000 8,000
New 8,000 10,000 19,000
Ships to be unloaded each carry 8,000 tons of raw materials, and analysis of post records shows that the number of arrivals per week approximately follows a Poisson distribution with a mean arrival rate of 4 ships per week. Unloading times are considered to be exponentially distributed. The time spend in the unloading system (queuing plus unloading time) is estimated to cast the company Rs. 8,000 per day. Assuming a 7- day working week, estimate the average weekly cost of both the present and new berth systems Do you advise the introduction of the new berth.
Q.13.Trucks of a company arrive at a transshipment center to be unloaded in a pattern which is characterized by the Poisson distribution. The average rate of arrivals is 45 per hour. A set of attendants unload the trucks, and the level of service is 100 per hour on an average. The driver make Rs. 16 per hour and the set of attendance are paid Rs.10 per hour.
(i) How much expense, on the average, is incurred by the company for idle time on part on each driver each time he is at the transshipment centre?
(ii) Find the optimum number of sets of attendants to be employed for transshipments.
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