EXAMPLE 6.10 A duplicating machine maintained for office use is operated by office assistant. The time to complete each job varies according to an exponential distribution with mean 6 minutes. Assume a Poisson input with an average arrival rate of 5 jobs per hour. If an 8-hour day is used as a base, determine (i) the percentage of idle time of the machine, and (ii) the average time a job is in the system.
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- 8. The parts department of a large automobile dealership has a counter used exclusively for mechanics’requests for parts. The time between requests can be modeled by a negative exponential distributionthat has a mean of five minutes. A clerk can handle requests at a rate of 15 per hour, andthis can be modeled by a Poisson distribution that has a mean of 15. Suppose there are two clerksat the counter.a. On average, how many mechanics would be at the counter, including those being served?b. What is the probability that a mechanic would have to wait for service?c. If a mechanic has to wait, how long would the average wait be?d. What percentage of time are the clerks idle?e. If clerks represent a cost of $20 per hour and mechanics a cost of $30 per hour, what numberof clerks would be optimal in terms of minimizing total cost?5. The DMV Licensing office has a single line for customers waiting for the next available clerk. There are two clerks who work at the same rate. On average customers arrive every 8 minutes and the average service rate is 5 per hour for each of the two clerks. The arrival rate of customers follows a Poisson distribution, while the service time follows an exponential distribution. d.) Because the state has received less than expected tax receipts, the DMV is thinking of cutting one position. Is this a good idea? Why?A new mall is considering the number of employees on desk dutyinformation, currently only one employee is employed. Based on the informationobtained, it is known that customers will come to the information desk with an average of 20people per hour. It takes an average of 3 minutes to answer the questions. Assumedthat the customer arrival rate follows the Poisson distribution and service timefollowing an exponential distribution. Assume that the employee at the information desk earns Rs50.000,- per hour. The cost of waiting time, if the customer is not satisfied, is IDR 35,000,-per hour of time spent in queue. a. The mall management has decided to consider using two peopleemployee at the information desk. Propose to mall management whether to use aor two employees at the information desk. Explain your answer.
- 20. A small bank is trying to determine the number of tellersto employ. The total cost of employing a teller is $100per day, and a teller can serve an average of 160 customers per day. On average, 210 customers arrive per day atthe bank, and both service times and interarrival times areexponentially distributed. If the delay cost per customerday is $200, how many tellers should the bank hire?The parts department of a large automobile dealership has a counter used exclusively for mechanics’ requests for parts. The time between requests can be modeled by a negative exponential distribution that has a mean of five minutes. A clerk can handle requests at a rate of 15 per hour, andthis can be modeled by a Poisson distribution that has a mean of 15. Suppose there are two clerksat the counter.a. On average, how many mechanics would be at the counter, including those being served?b. What is the probability that a mechanic would have to wait for service?c. If a mechanic has to wait, how long would the average wait be?d. What percentage of time are the clerks idle?e. If clerks represent a cost of $20 per hour and mechanics a cost of $30 per hour, what numberof clerks would be optimal in terms of minimizing total cost?Consider an M/G/1/GD/∞/∞ queuing system in which anaverage of 10 arrivals occur each hour. Suppose that eachcustomer’s service time follows an Erlang distribution, withrate parameter 1 customer per minute and shape parameter 4.a Find the expected number of customers waiting inline.b Find the expected time that a customer will spend inthe system.c What fraction of the time will the server be idle?
- Currently, Commercial Banks have 1 unit of ATM machines to meet customer needs. It is known that there is an average of 12+x customers in the system. Then it is also known that the average time spent by someone in the system is 10 minutes. Assume that the system follows M/M/1. The Commercial Bank operational manager is considering whether it is necessary to add 1 unit of ATM machine. What are your considerations? To strengthen the consideration, compare the queue length, queue length, and utilities before and after adding an ATM machine? Note: x denoted by the last number of my ID Number.Example, if my ID number is 2602296545, then the number used are “5”Paula Caplin is manager of a major electronics repair facilityowned by Fisher Electronics. Recently, top management ex-pressed concern over the growth in the number of repair jobs in process at the facility. The average arrival rate is 120 jobsper day. The average job spends four days at the facility.a. What is the current work-in-process level at the facility?b. Suppose that top management has put a limit of one-halfthe current level of work-in-process. What goal mustPaula establish, and how might she accomplish it?