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Jobs arrive at a facility at an average rate of 5 in an 8 hour shift. The arrival of the jobs follows Poisson distribution. The average service time of a job on the facility is 40 minutes. The service time follows exponential distribution. Idle time (in hours) at the facility per shift will be
[2014]
- a)5/7
- b)14/3
- c)7/3
- d)10/3
Correct answer is option 'B'. Can you explain this answer?
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Verified Answer
Jobs arrive at a facility at an average rate of 5 in an 8 hour shift. ...
Arrival rate = 5 jobs in 8 hr service time = 40 min/job
Total service time = 40 × 5 = 200 min
Total service time = 40 × 5 = 200 min
Most Upvoted Answer
Jobs arrive at a facility at an average rate of 5 in an 8 hour shift. ...
Given:
- Average arrival rate of jobs = 5 in an 8-hour shift
- Arrival of jobs follows a Poisson distribution
- Average service time of a job = 40 minutes
- Service time follows an exponential distribution
To find:
Idle time at the facility per shift
Approach:
1. Calculate the arrival rate per hour
2. Calculate the service rate per hour
3. Calculate the utilization factor
4. Calculate the average number of jobs in the system
5. Calculate the average time a job spends in the system
6. Calculate the idle time at the facility per shift
Solution:
Step 1: Calculate the arrival rate per hour
- Given average arrival rate of jobs = 5 in an 8-hour shift
- Arrival rate per hour = (5 jobs / 8 hours) = 0.625 jobs per hour
Step 2: Calculate the service rate per hour
- Given average service time of a job = 40 minutes
- Service rate = 1 / average service time = 1 / 40 minutes = 0.025 jobs per minute
- Service rate per hour = (0.025 jobs per minute) * (60 minutes per hour) = 1.5 jobs per hour
Step 3: Calculate the utilization factor
- Utilization factor = Arrival rate per hour / Service rate per hour
- Utilization factor = 0.625 jobs per hour / 1.5 jobs per hour = 0.4167
Step 4: Calculate the average number of jobs in the system
- Average number of jobs in the system = Utilization factor / (1 - Utilization factor)
- Average number of jobs in the system = 0.4167 / (1 - 0.4167) = 0.7143
Step 5: Calculate the average time a job spends in the system
- Average time a job spends in the system = Average number of jobs in the system / Arrival rate per hour
- Average time a job spends in the system = 0.7143 jobs / (0.625 jobs per hour) = 1.1429 hours
Step 6: Calculate the idle time at the facility per shift
- Idle time at the facility per shift = Total shift time - Average time a job spends in the system
- Idle time at the facility per shift = 8 hours - 1.1429 hours = 6.8571 hours
Therefore, the idle time at the facility per shift is approximately 6.8571 hours, which is equivalent to 6 hours and 52 minutes.
Hence, the correct answer is option B) 14/3.
- Average arrival rate of jobs = 5 in an 8-hour shift
- Arrival of jobs follows a Poisson distribution
- Average service time of a job = 40 minutes
- Service time follows an exponential distribution
To find:
Idle time at the facility per shift
Approach:
1. Calculate the arrival rate per hour
2. Calculate the service rate per hour
3. Calculate the utilization factor
4. Calculate the average number of jobs in the system
5. Calculate the average time a job spends in the system
6. Calculate the idle time at the facility per shift
Solution:
Step 1: Calculate the arrival rate per hour
- Given average arrival rate of jobs = 5 in an 8-hour shift
- Arrival rate per hour = (5 jobs / 8 hours) = 0.625 jobs per hour
Step 2: Calculate the service rate per hour
- Given average service time of a job = 40 minutes
- Service rate = 1 / average service time = 1 / 40 minutes = 0.025 jobs per minute
- Service rate per hour = (0.025 jobs per minute) * (60 minutes per hour) = 1.5 jobs per hour
Step 3: Calculate the utilization factor
- Utilization factor = Arrival rate per hour / Service rate per hour
- Utilization factor = 0.625 jobs per hour / 1.5 jobs per hour = 0.4167
Step 4: Calculate the average number of jobs in the system
- Average number of jobs in the system = Utilization factor / (1 - Utilization factor)
- Average number of jobs in the system = 0.4167 / (1 - 0.4167) = 0.7143
Step 5: Calculate the average time a job spends in the system
- Average time a job spends in the system = Average number of jobs in the system / Arrival rate per hour
- Average time a job spends in the system = 0.7143 jobs / (0.625 jobs per hour) = 1.1429 hours
Step 6: Calculate the idle time at the facility per shift
- Idle time at the facility per shift = Total shift time - Average time a job spends in the system
- Idle time at the facility per shift = 8 hours - 1.1429 hours = 6.8571 hours
Therefore, the idle time at the facility per shift is approximately 6.8571 hours, which is equivalent to 6 hours and 52 minutes.
Hence, the correct answer is option B) 14/3.
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Jobs arrive at a facility at an average rate of 5 in an 8 hour shift. The arrival of the jobs follows Poisson distribution. The average service time of a job on the facility is 40 minutes. The service time follows exponential distribution. Idle time (in hours) at the facility per shift will be[2014]a)5/7b)14/3c)7/3d)10/3Correct answer is option 'B'. Can you explain this answer?
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