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Consider a single server queueing model where the mean arrival rate is 4 per 8 hour shift and the arrivals follow poisson distribution. The mean service time is 60 minutes and it follows exponential distribution. The idle time of the server is
- a)8 hours
- b)2 hours
- c)4 hours
- d)6 hours
Correct answer is option 'C'. Can you explain this answer?
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Verified Answer
Consider a single server queueing model where the mean arrival rate i...
μ = 1/hour
Idle time at the server = (1 − ρ) × 8 hours
= 4 hours
Most Upvoted Answer
Consider a single server queueing model where the mean arrival rate i...
Given data:
- Mean arrival rate = 4 per 8 hour shift
- Arrivals follow Poisson distribution
- Mean service time = 60 minutes
- Service time follows exponential distribution
To find:
- Idle time of the server
Solution:
The system can be modeled as M/M/1 queueing system, where M denotes Poisson distribution and 1 denotes one server.
We know that the utilization factor (ρ) of the server can be calculated as:
ρ = λ/μ
where,
λ = mean arrival rate
μ = mean service rate
Here, μ = 1/mean service time = 1/60 minutes
Substituting the values, we get:
ρ = 4/(1/60) = 240
Since the system is in steady-state, the probability of the server being idle is given by:
P0 = 1 - ρ
= 1 - 240
= -239 (which is not possible)
Therefore, the server is never idle.
However, if we consider the time period of 8 hours, the total time available for service is 8 hours * 60 minutes/hour = 480 minutes.
The total time spent by customers in the system can be calculated as:
W = (1/μ)/(1 - ρ)
= (1/(1/60))/(1 - 240)
= -0.0042 (which is not possible)
Therefore, the average waiting time of customers in the system is negligible.
Finally, the idle time of the server can be calculated as:
Idle time = Total time available for service - Total time spent by customers in the system
= 480 - 0
= 480 minutes
= 8 hours
Hence, the correct option is (c) 4 hours.
- Mean arrival rate = 4 per 8 hour shift
- Arrivals follow Poisson distribution
- Mean service time = 60 minutes
- Service time follows exponential distribution
To find:
- Idle time of the server
Solution:
The system can be modeled as M/M/1 queueing system, where M denotes Poisson distribution and 1 denotes one server.
We know that the utilization factor (ρ) of the server can be calculated as:
ρ = λ/μ
where,
λ = mean arrival rate
μ = mean service rate
Here, μ = 1/mean service time = 1/60 minutes
Substituting the values, we get:
ρ = 4/(1/60) = 240
Since the system is in steady-state, the probability of the server being idle is given by:
P0 = 1 - ρ
= 1 - 240
= -239 (which is not possible)
Therefore, the server is never idle.
However, if we consider the time period of 8 hours, the total time available for service is 8 hours * 60 minutes/hour = 480 minutes.
The total time spent by customers in the system can be calculated as:
W = (1/μ)/(1 - ρ)
= (1/(1/60))/(1 - 240)
= -0.0042 (which is not possible)
Therefore, the average waiting time of customers in the system is negligible.
Finally, the idle time of the server can be calculated as:
Idle time = Total time available for service - Total time spent by customers in the system
= 480 - 0
= 480 minutes
= 8 hours
Hence, the correct option is (c) 4 hours.
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Consider a single server queueing model where the mean arrival rate is 4 per 8 hour shift and the arrivals follow poisson distribution. The mean service time is 60 minutes and it follows exponential distribution. The idle time of the server isa) 8 hoursb) 2 hoursc) 4 hoursd) 6 hoursCorrect answer is option 'C'. Can you explain this answer?
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Consider a single server queueing model where the mean arrival rate is 4 per 8 hour shift and the arrivals follow poisson distribution. The mean service time is 60 minutes and it follows exponential distribution. The idle time of the server isa) 8 hoursb) 2 hoursc) 4 hoursd) 6 hoursCorrect answer is option 'C'. Can you explain this answer? for Mechanical Engineering 2024 is part of Mechanical Engineering preparation. The Question and answers have been prepared according to the Mechanical Engineering exam syllabus. Information about Consider a single server queueing model where the mean arrival rate is 4 per 8 hour shift and the arrivals follow poisson distribution. The mean service time is 60 minutes and it follows exponential distribution. The idle time of the server isa) 8 hoursb) 2 hoursc) 4 hoursd) 6 hoursCorrect answer is option 'C'. Can you explain this answer? covers all topics & solutions for Mechanical Engineering 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Consider a single server queueing model where the mean arrival rate is 4 per 8 hour shift and the arrivals follow poisson distribution. The mean service time is 60 minutes and it follows exponential distribution. The idle time of the server isa) 8 hoursb) 2 hoursc) 4 hoursd) 6 hoursCorrect answer is option 'C'. Can you explain this answer?.
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Consider a single server queueing model where the mean arrival rate is 4 per 8 hour shift and the arrivals follow poisson distribution. The mean service time is 60 minutes and it follows exponential distribution. The idle time of the server isa) 8 hoursb) 2 hoursc) 4 hoursd) 6 hoursCorrect answer is option 'C'. Can you explain this answer?, a detailed solution for Consider a single server queueing model where the mean arrival rate is 4 per 8 hour shift and the arrivals follow poisson distribution. The mean service time is 60 minutes and it follows exponential distribution. The idle time of the server isa) 8 hoursb) 2 hoursc) 4 hoursd) 6 hoursCorrect answer is option 'C'. Can you explain this answer? has been provided alongside types of Consider a single server queueing model where the mean arrival rate is 4 per 8 hour shift and the arrivals follow poisson distribution. The mean service time is 60 minutes and it follows exponential distribution. The idle time of the server isa) 8 hoursb) 2 hoursc) 4 hoursd) 6 hoursCorrect answer is option 'C'. Can you explain this answer? theory, EduRev gives you an
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