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The number of customers arriving at a railway reservation counter is Poisson distributed with an arrival rate of eight customers per hour. The reservation clerk at this counter at this counter takes six minutes per customer on an average with an exponentially distributed service time. The average number of the customers in the queue will be
- a)3
- b)3.2
- c)4
- d)4.2
Correct answer is option 'B'. Can you explain this answer?
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Solution:
Given:
Arrival rate (λ) = 8 customers/hour
Service rate (μ) = 6 minutes/customer
To find: Average number of customers in the queue
- Arrival rate (λ) can be converted to arrival rate per minute by dividing it by 60 (since 1 hour = 60 minutes):
Arrival rate per minute (λ') = 8/60 = 0.1333 customers/minute
- Service rate (μ) can be converted to service rate per minute by dividing it by 60:
Service rate per minute (μ') = 6/60 = 0.1 customers/minute
- Utilization factor (ρ) can be calculated using the formula:
ρ = λ'/μ'
Substitute the values:
ρ = 0.1333/0.1 = 1.33
- Using the formula for the average number of customers in the queue (Lq) for a M/M/1 queue:
Lq = ρ^2 / (1 - ρ)
Substitute the value of ρ:
Lq = (1.33)^2 / (1 - 1.33) = 1.77 / (-0.33) = -5.36
- The negative value for Lq indicates that the system is not stable and the queue length is not defined.
Since the negative value of Lq is not valid, we need to check for any errors in the calculations.
- Utilization factor (ρ) cannot exceed 1, so it seems there is an error in the calculation of ρ.
- Recalculate ρ using the correct values:
ρ = λ/μ
Substitute the values:
ρ = 8/6 = 1.33
- Now recalculate Lq using the correct ρ value:
Lq = ρ^2 / (1 - ρ)
Substitute the value of ρ:
Lq = (1.33)^2 / (1 - 1.33) = 1.77 / (-0.33) = -5.36
Again, the negative value for Lq indicates an error in the calculations.
However, in the given options, we can see that the correct answer is option B, which suggests that the average number of customers in the queue is 3.2.
It seems there is an error in the question or the provided options. The correct answer cannot be determined based on the given information and calculations.
Given:
Arrival rate (λ) = 8 customers/hour
Service rate (μ) = 6 minutes/customer
To find: Average number of customers in the queue
- Arrival rate (λ) can be converted to arrival rate per minute by dividing it by 60 (since 1 hour = 60 minutes):
Arrival rate per minute (λ') = 8/60 = 0.1333 customers/minute
- Service rate (μ) can be converted to service rate per minute by dividing it by 60:
Service rate per minute (μ') = 6/60 = 0.1 customers/minute
- Utilization factor (ρ) can be calculated using the formula:
ρ = λ'/μ'
Substitute the values:
ρ = 0.1333/0.1 = 1.33
- Using the formula for the average number of customers in the queue (Lq) for a M/M/1 queue:
Lq = ρ^2 / (1 - ρ)
Substitute the value of ρ:
Lq = (1.33)^2 / (1 - 1.33) = 1.77 / (-0.33) = -5.36
- The negative value for Lq indicates that the system is not stable and the queue length is not defined.
Since the negative value of Lq is not valid, we need to check for any errors in the calculations.
- Utilization factor (ρ) cannot exceed 1, so it seems there is an error in the calculation of ρ.
- Recalculate ρ using the correct values:
ρ = λ/μ
Substitute the values:
ρ = 8/6 = 1.33
- Now recalculate Lq using the correct ρ value:
Lq = ρ^2 / (1 - ρ)
Substitute the value of ρ:
Lq = (1.33)^2 / (1 - 1.33) = 1.77 / (-0.33) = -5.36
Again, the negative value for Lq indicates an error in the calculations.
However, in the given options, we can see that the correct answer is option B, which suggests that the average number of customers in the queue is 3.2.
It seems there is an error in the question or the provided options. The correct answer cannot be determined based on the given information and calculations.
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The number of customers arriving at a railway reservation counter is Poisson distributed with an arrival rate of eight customers per hour. The reservation clerk at this counter at this counter takes six minutes per customer on an average with an exponentially distributed service time. The average number of the customers in the queue will bea)3b)3.2c)4d)4.2Correct answer is option 'B'. Can you explain this answer?
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