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The inter-arrival times at a tool crib are exponential with an average time of 10 minutes and the length of the service time is assumed to be exponential with mean 6 minutes. The probability that a person arriving at the booth will have to wait is equal to
- a)0.15
- b)0.40
- c)0.42
- d)0.6
Correct answer is option 'D'. Can you explain this answer?
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Verified Answer
The inter-arrival times at a tool crib are exponential with an averag...
Probability that person has to wait = ρ =
= 10 persons/hour
Therefore probability that person has to wait
Most Upvoted Answer
The inter-arrival times at a tool crib are exponential with an averag...
Given Information:
- Inter-arrival times at a tool crib are exponential with an average time of 10 minutes.
- Service time is assumed to be exponential with a mean of 6 minutes.
To find:
The probability that a person arriving at the booth will have to wait.
Solution:
To find the probability that a person arriving at the booth will have to wait, we need to calculate the utilization factor (ρ) and then use Little's Law to find the probability of waiting.
Step 1: Calculate the Utilization Factor (ρ):
The utilization factor (ρ) is the ratio of the average service rate to the average arrival rate.
The average arrival rate (λ) is the reciprocal of the average inter-arrival time:
λ = 1 / Average Inter-arrival Time = 1 / 10 = 0.1 arrivals per minute.
The average service rate (μ) is the reciprocal of the average service time:
μ = 1 / Average Service Time = 1 / 6 = 0.1667 services per minute.
The utilization factor (ρ) is given by:
ρ = λ / μ = 0.1 / 0.1667 = 0.6
Step 2: Apply Little's Law:
Little's Law states that the average number of customers in a system (L) is equal to the average arrival rate (λ) multiplied by the average time a customer spends in the system (W):
L = λ * W
In this case, we are interested in the average number of customers waiting in the system (Lw), so we can modify Little's Law as follows:
Lw = λ * Ww
Where:
Lw = average number of customers waiting in the system
λ = average arrival rate
Ww = average time a customer spends waiting in the system
Since we want to find the probability of waiting, we can rewrite the equation as:
Probability of waiting = Lw / λ
Step 3: Calculate the Average Time a Customer Spends Waiting (Ww):
The average time a customer spends waiting in the system can be calculated using the following formula:
Ww = W - (1 / μ)
Where:
W = average time a customer spends in the system
μ = average service rate
The average time a customer spends in the system (W) can be calculated using the following formula:
W = 1 / (μ - λ)
Substituting the values of λ and μ, we get:
W = 1 / (0.1667 - 0.1) = 10 minutes
Substituting the value of W into the formula for Ww, we get:
Ww = 10 - (1 / 0.1667) = 3.9984 minutes
Step 4: Calculate the Probability of Waiting:
Using the formula:
Probability of waiting = Lw / λ
Substituting the values of λ and Ww, we get:
Probability of waiting = 0.1 * 3.9984 = 0.4
Therefore, the probability that a person arriving at the booth will have to wait is 0.4, which corresponds to option D.
- Inter-arrival times at a tool crib are exponential with an average time of 10 minutes.
- Service time is assumed to be exponential with a mean of 6 minutes.
To find:
The probability that a person arriving at the booth will have to wait.
Solution:
To find the probability that a person arriving at the booth will have to wait, we need to calculate the utilization factor (ρ) and then use Little's Law to find the probability of waiting.
Step 1: Calculate the Utilization Factor (ρ):
The utilization factor (ρ) is the ratio of the average service rate to the average arrival rate.
The average arrival rate (λ) is the reciprocal of the average inter-arrival time:
λ = 1 / Average Inter-arrival Time = 1 / 10 = 0.1 arrivals per minute.
The average service rate (μ) is the reciprocal of the average service time:
μ = 1 / Average Service Time = 1 / 6 = 0.1667 services per minute.
The utilization factor (ρ) is given by:
ρ = λ / μ = 0.1 / 0.1667 = 0.6
Step 2: Apply Little's Law:
Little's Law states that the average number of customers in a system (L) is equal to the average arrival rate (λ) multiplied by the average time a customer spends in the system (W):
L = λ * W
In this case, we are interested in the average number of customers waiting in the system (Lw), so we can modify Little's Law as follows:
Lw = λ * Ww
Where:
Lw = average number of customers waiting in the system
λ = average arrival rate
Ww = average time a customer spends waiting in the system
Since we want to find the probability of waiting, we can rewrite the equation as:
Probability of waiting = Lw / λ
Step 3: Calculate the Average Time a Customer Spends Waiting (Ww):
The average time a customer spends waiting in the system can be calculated using the following formula:
Ww = W - (1 / μ)
Where:
W = average time a customer spends in the system
μ = average service rate
The average time a customer spends in the system (W) can be calculated using the following formula:
W = 1 / (μ - λ)
Substituting the values of λ and μ, we get:
W = 1 / (0.1667 - 0.1) = 10 minutes
Substituting the value of W into the formula for Ww, we get:
Ww = 10 - (1 / 0.1667) = 3.9984 minutes
Step 4: Calculate the Probability of Waiting:
Using the formula:
Probability of waiting = Lw / λ
Substituting the values of λ and Ww, we get:
Probability of waiting = 0.1 * 3.9984 = 0.4
Therefore, the probability that a person arriving at the booth will have to wait is 0.4, which corresponds to option D.
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The inter-arrival times at a tool crib are exponential with an average time of 10 minutes and the length of the service time is assumed to be exponential with mean 6 minutes. The probability that a person arriving at the booth will have to wait is equal toa) 0.15b) 0.40c) 0.42d) 0.6Correct answer is option 'D'. Can you explain this answer?
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