Students arrive at the Administrative Services Office at an average of one every 15 minutes, and their requests take, on average, 10 minutes to be processed. The service counter is staffed by only one clerk, Judy Gumshoes, who works eight hours per day. Assume Poisson arrivals and exponential service times. a. What percentage of time is Judy idle? (Round your answer to 1 decimal place.)
Question
Students arrive at the Administrative Services Office at an average of one every 15 minutes, and their requests take, on average, 10 minutes to be processed. The service counter is staffed by only one clerk, Judy Gumshoes, who works eight hours per day. Assume Poisson arrivals and exponential service times. a. What percentage of time is Judy idle? (Round your answer to 1 decimal place.)
Solution
To solve this problem, we need to use the formula for the utilization rate in a single-server queue, which is λ/μ, where λ is the arrival rate and μ is the service rate.
Step 1: Convert the time to the same units. Judy works 8 hours a day, and there are 60 minutes in an hour, so she works 8*60 = 480 minutes per day.
Step 2: Calculate the arrival rate (λ). Students arrive every 15 minutes, so the arrival rate is 1/15 students per minute.
Step 3: Calculate the service rate (μ). Judy can process a request in 10 minutes, so her service rate is 1/10 requests per minute.
Step 4: Calculate the utilization rate (ρ), which is λ/μ. So, ρ = (1/15) / (1/10) = 2/3 = 0.67.
Step 5: To find the percentage of time Judy is idle, subtract the utilization rate from 1 and multiply by 100. So, the percentage of time Judy is idle is (1 - 0.67) * 100 = 33%.
Therefore, Judy is idle approximately 33.0% of the time.
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