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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?, 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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