Question

The robotics manufacturing company operates an equipment repair business where emergency jobs arrive randomly at the rate of two jobs per 8-hour day. the company's repair facility is a single-server system operated by a repair technician. the service time varies, with a mean repair time of 3.2 hours and a standard deviation of 2.0 hours. the company's cost of the repair operation is $27 per hour. in the economic analysis of the waiting line system, robotics uses $37 per hour cost for customers waiting during the repair process. (a) What are the arrival rate and service rate in jobs per hour? (Round your answers to four decimal places.)

λ=[
μ=
​
(b) Show the operating characteristics. (Round your answers to four decimal places. Report time in hours.)
Lq = ____
​L = ____
Wq = ____
​W = ____

Answer 1

The arrival rate is 0.25 jobs per hour and the service rate is 0.3125 jobs per hour. The operating characteristics are Lq = 0.0833 jobs, L = 0.25W, Wq = 0.333 hours, and W = 4 hours.

Step-by-step explanation:

To determine the arrival rate and service rate in jobs per hour, we need to convert the given information to the appropriate units. The rate of two jobs per 8-hour day can be converted to jobs per hour by dividing by 8, giving an arrival rate of 0.25 jobs per hour. The mean repair time of 3.2 hours can be converted to service rate by taking the reciprocal, giving a service rate of 0.3125 jobs per hour.

The operating characteristics can be calculated using the formulas for a single-server queue. The formulas are as follows:

• Lq: Average number of jobs in the waiting queue = (ρ^2) / (1 - ρ), where ρ = arrival rate / service rate
• L: Average number of jobs in the system (waiting queue + being serviced) = λ * W, where λ = arrival rate and W = average time spent in the system
• Wq: Average time spent in the waiting queue = Lq / λ
• W: Average time spent in the system = 1 / (μ - λ), where μ = service rate

Using the given information, we can calculate the operating characteristics:

• Lq = (0.25^2) / (1 - 0.25) = 0.0833 jobs
• L = 0.25 * W
• Wq = 0.0833 / 0.25 = 0.333 hours
• W = 1 / (0.3125 - 0.25) = 4 hours

Answer 2

The arrival rate (λ) for the robotics manufacturing company's emergency repair jobs is 0.25 jobs per hour, and the service rate (μ) is 0.3125 jobs per hour. To compute the operating characteristics, more detailed information about the queuing system would be needed.

Step-by-step explanation:

The question provided is related to a robotics manufacturing company that also operates an emergency equipment repair business. The student is asked to determine the arrival rate (λ) and service rate (μ) in terms of jobs per hour, as well as the repair facility's operating characteristics, which include the average number of jobs in the queue (Lq), the average number of jobs in the system (L), the average time a job spends waiting in the queue (Wq), and the average time a job spends in the system (W).

To find the arrival rate (λ), we need to use the given rate of two jobs per 8-hour day. Since there are 8 hours in a working day, we divide 2 by 8 to convert the rate to jobs per hour:

λ = 2 jobs / 8 hours = 0.25 jobs/hour

The service rate (μ) is the inverse of the mean service time. Given the mean service time is 3.2 hours, the service rate is calculated as:

μ = 1 / mean service time = 1 / 3.2 hours = 0.3125 jobs/hour

However, we are not provided with enough information to calculate the operating characteristics (Lq, L, Wq, W) directly from the problem statement, as these values also depend on the specific distribution of service times and more detailed information about the queuing system. Generally, these are calculated using formulas for specific queuing models like M/M/1, M/D/1, etc