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If the arrivals at a service facility are distributed as per the poisson distribution with a mean rate of 10 per hour and the services are exponentially distributed with a mean service time of 4 minutes, what is the probability that a customer may have to wait to be served?
- a)0.40
- b)0.50
- c)0.67
- d)1.00
Correct answer is option 'C'. Can you explain this answer?
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Verified Answer
If the arrivals at a service facility are distributed as per the pois...
Arrivals at a rate of 10/hour (λ = 10)
Service is at the rate of 4 minutes interval (μ = 15)
ρ = Probability that the customer has to wait
Most Upvoted Answer
If the arrivals at a service facility are distributed as per the pois...
To calculate the probability that a customer may have to wait to be served, we need to consider the arrival rate and the service rate at the facility.
1. Calculation of Arrival Rate:
The arrival rate is given as 10 customers per hour. The Poisson distribution is used to model the arrival of customers in a given time period. The Poisson distribution is defined by its mean, which is equal to the arrival rate. Therefore, the mean arrival rate is 10 customers per hour.
2. Calculation of Service Rate:
The service time is exponentially distributed with a mean of 4 minutes. The exponential distribution is defined by its mean, which is equal to the reciprocal of the service rate. Therefore, the mean service rate is 1/4 customers per minute.
3. Utilization Factor:
The utilization factor (ρ) is the ratio of the arrival rate to the service rate. It represents the level of utilization of the facility. In this case, the utilization factor can be calculated as follows:
ρ = Arrival Rate / Service Rate
ρ = 10 customers per hour / (1/4) customers per minute
ρ = 10 * 60 / 1/4
ρ = 600 / 1/4
ρ = 600 * 4
ρ = 2400
4. Probability of Wait Time:
The probability that a customer may have to wait to be served can be calculated using the following formula:
P(wait) = ρ^2 / (1 - ρ)
Substituting the calculated value of ρ into the formula:
P(wait) = 2400^2 / (1 - 2400)
P(wait) = 5760000 / (-2399)
P(wait) ≈ 0.67
Therefore, the probability that a customer may have to wait to be served is approximately 0.67, which corresponds to option 'C'.
1. Calculation of Arrival Rate:
The arrival rate is given as 10 customers per hour. The Poisson distribution is used to model the arrival of customers in a given time period. The Poisson distribution is defined by its mean, which is equal to the arrival rate. Therefore, the mean arrival rate is 10 customers per hour.
2. Calculation of Service Rate:
The service time is exponentially distributed with a mean of 4 minutes. The exponential distribution is defined by its mean, which is equal to the reciprocal of the service rate. Therefore, the mean service rate is 1/4 customers per minute.
3. Utilization Factor:
The utilization factor (ρ) is the ratio of the arrival rate to the service rate. It represents the level of utilization of the facility. In this case, the utilization factor can be calculated as follows:
ρ = Arrival Rate / Service Rate
ρ = 10 customers per hour / (1/4) customers per minute
ρ = 10 * 60 / 1/4
ρ = 600 / 1/4
ρ = 600 * 4
ρ = 2400
4. Probability of Wait Time:
The probability that a customer may have to wait to be served can be calculated using the following formula:
P(wait) = ρ^2 / (1 - ρ)
Substituting the calculated value of ρ into the formula:
P(wait) = 2400^2 / (1 - 2400)
P(wait) = 5760000 / (-2399)
P(wait) ≈ 0.67
Therefore, the probability that a customer may have to wait to be served is approximately 0.67, which corresponds to option 'C'.
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If the arrivals at a service facility are distributed as per the poisson distribution with a mean rate of 10 per hour and the services are exponentially distributed with a mean service time of 4 minutes, what is the probability that a customer may have to wait to be served?a) 0.40b) 0.50c) 0.67d) 1.00Correct answer is option 'C'. Can you explain this answer?
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