QAM Queuing Models

1.

The new loan officer of a bank interviews all a ll customers for new accounts. The customers desiring to open new accounts arrive at the rate of 4 per hour according to a Poisson distribution, and the accounts offi cer spends an average of 12 minutes with each customer setting up a new account. Determine the operating characteristics for this system.Add an additional accounts officer to this system and determine the operating characteristics.

2.

The ticket booth on the tech campus is operated by one person who is selling the tickets. The ticket seller can serve on the average of 12 persons/hour. On average 10 persons arrive to purchase the ticket. Determine the average time a ticket buyer must wait and the portion of time the ticket seller is busy.

3.

A service station has one pump for unleaded gas which with a n attendant can service 10 customers per hour. Cars arrive at a rate of 6/hour. Determine the average queue length, the average time a car in the system, and the average time the car must wait. If the arrival rate increases to 12 cars / hour what will be the effect on the average queue length.

4.

A bank has one outside teller counter. It takes an average of 4 minutes to serve a bank customer. Customers arrive at the window at the rate of 12/hour. The bank operation officer is currently analyzing the possibility of adding a nd

2  window at an annual cost of Rs20,000. It is assumed that the arriving customers would be equally divided between both windows. The officer estimates that each minutes reduction in customers waiting time would nd increase the banks revenue by 2000 annually. Should the 2  window needs to be installed ? 5.

An Airport has a single runway and one air traffic controller to land planes. It takes an airplane 12 mins to l and and clear the run way. Planes arrive at the Airport at the rate of 4/hour.Determine the average number of planes that will stack up waiting to land.Find the average time a plane must wait in line before it can land. Calculate the average time it takes a plane to clear the runway once it has notified the airport that it is in the vicinity and wants to land.The airport has a rule that an air ai r traffic controller can on the average land planes a maximum of 45 minutes out of every hour. There must be 15 mins of idle time available avail able to relieve the tension. Will this airport needs to hire an extra air traffic controller ?

6.

A computer maintenance contract is to by signed by a co. At an average 3 computers per month go off road due to various defects. The cost of a computer being unavailable is Rs.8000 per month. Two companies have bid for the contract. Alfa computers have quoted Rs.3000 per month where as Beta Bytes at Rs.5000 per month for f or the contract. Enquires reveal that Alfa computers has average repair capability of 5 computers per month and Beta computers can repair 6 computers per month at an average. Who should be given the contract ?

7.

A TV repairman finds that the time spent on his jobs has ha s an exponential distribution with mean 30 minutes. If he repair sets on FCFS basis and if the arrival of the sets with an average of 10/8Hrs day what is the repairman's expected idle time each day? Also obtain the average number of sets in the system.

8.

In a factory the average breakdown is 3/Hr. The idle time cost is estimated to be Rs16/Hr. The factory works 8 Hrs/day. The factory manager is considering two mechanics for repairing the machines. Mechanic A, takes on average 15 mins in repairing a machine and charges Rs 8 per hour. Mechanic B takes on average 10 minutes for repairing a machine and Charges Rs10/ Hr. Assuming machine breakdown is Poisson and repairing time exponential , which of the two mechanics should be engaged?

9.

In a bank with a single server, there are 2 chairs f or waiting customers. On an average one customer arrives in

every 10 minutes and each customer takes 5 minutes for getting served. Find The probability that an arrival will get a chair to sit on ,The probability that an arrival will have to stand, Expected waiting time of a customer. 10. A small car repair shop can repair cars at rate of 10 per day, repair time expoentially distributed. Cars arrive at the shop following Poisson distribution at an average rate of 8 per day. The shop can accommodate only a maximum of 3 cars in the system due to space shortage. Cars, which arrive when the system is full, are not taken up for service and the potential loss is estimated at Rs.50 per car. The shop owner has two alternatives to reduce the customer loss. Alternative A: He can rent out an additional adjoining area at monthly rental of Rs 300 which would enable increasing the capacity of the system to 4 cars.

Alternative B: He can appoint a trained mechanic who has to be paid an extra monthly salary of Rs.350 and the service rate will increase to 12 cars per day. Which alternative is to be chosen? 11. Automata car wash facility with only one bay. Cars arrive according to Poisson distribution with a mean of 4 cars/Hr. and may wait in the facilities parking lot if the bay is busy. The time for washing and cleaning a car is exponential with a mean of 10 mins. Cars that cannot park in the l ot can wait in the street bordering the wash facility. The manager of the facility wants to determine the size of the parking lot such that an arriving car will find a space 90% of the time. 12. A machine shop has 5 machines that periodically breakdown and require service. The average time between breakdowns is 4 days, distributed according to exponential distribution. The average time to repair a machine is one day, distributed according to exponential distribution. One mechanic repairs the machines in the order in which they are breakdown. Determine that the probability that the mechanic being idle.Determine the mean number of machines waiting to be repaired.Determine the mean time machines wait to be repaired.Find the probability that 3 machines are not operating 13. At a Railway station , only one train is handled at a time. The railway yard is sufficient for 2 trains to wait while the other is given signal to leave the station. Trains arrive at the station at an average rate of 6/hour and the railway station can handle them on an average of 12/hour. Assuming Poisson arrivals and exponential service distribution, find the service characteristics. Also find the average waiting time of a new train coming into the ward. 14. A manufacturing co. operates a shop that includes 20 M/C s. Due to the type of work performed in the shop , they require frequent repair. The co. has one senior repair person with an assistant. They repair the M/C s in the same order as they break down. M/c break down are according to Poisson and service is exponential. Each M/c operates an average of 200 hours before breaking down. The average time to repair a M/c is 3.6 Hours. The co. would like an analysis of M/c idle time due to break downs to determine if the present repair staff is sufficient. 15. A survey of customer service dept. for a 12 month period shows that the arrival and service rates in a bank are 10 and 4 respectively. Let the bank has 3 counters. Calculate the Operational characteristics. 16. The Bank management has observed that customers are frustrated by the relatively long waiting time of 21 minutes and 0.703 probability of waiting. To try to improve matters , management has decided to consider addition of an extra counter. Find the new operational characteristics. 17. The Dupit-corp is a long time leader in the office photo copier marketplace. One reason for this leadership position is the service the co. provides its customers. The service is provided at the customers’ site by th e co.’s service tech reps. Each tech is given responsibility for a specified territory. Each tech rep’s territory should be

assigned enough machines so that the tech reps will be busy 75% of the time. When working continuously each tech rep should able to repair an average of 4 machines per day. Since the co.’s machines now are averaging 50 work days between needing repairs, the target to assign approximately 150 machines to each tech rep’s

territory.Under this policy the co. has nearly 10,000tech reps with a total payroll of approximately $600 million per year. 18. Modify the current policy by decreasing the percentage of time that the tech reps are expected to be active in repairing machines. This involves simply decreasing the number of machines assigned to each tech rep and adding more tech reps. By assigning 100 machines to each tech rep we can get a new set of performance characteristics. =

(Number of machines assigned to each tech rep)/ 50