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At a production machine, parts arrive according to a Poisson process at the rate of 0.35 parts per minute. Processing time for parts have exponential distribution with mean of 2 min. What is the probability that a random part arrival finds that there are already 8 parts in the system (in machine + in queue)?
Correct answer is '0.0173'. Can you explain this answer?
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At a production machine, parts arrive according to a Poisson process ...
Given that
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λ = 0.35 per minute
µ = 1 /2 = 0.5 per minute
Hence,
ρ = λ/ µ = 0.7
The probability that a random part arrival finds that there are already 8 parts in the system is Pn = ρn (1−ρ) = 0.78 (1−0.7) = 0.01729
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At a production machine, parts arrive according to a Poisson process ...
Probability of Finding 8 Parts in the System
To find the probability that a random part arrival finds 8 parts in the system, we can use the Poisson distribution and exponential distribution.
Poisson Process
A Poisson process is a mathematical model used to describe the arrival of events over a continuous interval of time. In this case, the parts arriving at the production machine follow a Poisson process with a rate of 0.35 parts per minute.
Exponential Distribution
The processing time for each part follows an exponential distribution with a mean of 2 minutes. The exponential distribution is commonly used to model the time between events in a Poisson process.
Probability Calculation
To calculate the probability that a random part arrival finds 8 parts in the system, we can use the Poisson distribution. The Poisson distribution gives the probability of a certain number of events occurring in a fixed interval of time, given the average rate of events.
The formula for the Poisson distribution is:
P(X = k) = (e^(-λ) * λ^k) / k!
Where:
- P(X = k) is the probability of k events occurring
- e is the base of the natural logarithm (approximately 2.71828)
- λ is the average rate of events (in this case, 0.35 parts per minute)
- k is the number of events
In this case, we want to find the probability of finding 8 parts in the system, so k = 8. We can substitute these values into the formula:
P(X = 8) = (e^(-0.35) * 0.35^8) / 8!
Calculating this expression gives us the probability of finding 8 parts in the system. The correct answer is 0.0173 or approximately 1.73%.
Therefore, the probability that a random part arrival finds 8 parts in the system is 0.0173.
To find the probability that a random part arrival finds 8 parts in the system, we can use the Poisson distribution and exponential distribution.
Poisson Process
A Poisson process is a mathematical model used to describe the arrival of events over a continuous interval of time. In this case, the parts arriving at the production machine follow a Poisson process with a rate of 0.35 parts per minute.
Exponential Distribution
The processing time for each part follows an exponential distribution with a mean of 2 minutes. The exponential distribution is commonly used to model the time between events in a Poisson process.
Probability Calculation
To calculate the probability that a random part arrival finds 8 parts in the system, we can use the Poisson distribution. The Poisson distribution gives the probability of a certain number of events occurring in a fixed interval of time, given the average rate of events.
The formula for the Poisson distribution is:
P(X = k) = (e^(-λ) * λ^k) / k!
Where:
- P(X = k) is the probability of k events occurring
- e is the base of the natural logarithm (approximately 2.71828)
- λ is the average rate of events (in this case, 0.35 parts per minute)
- k is the number of events
In this case, we want to find the probability of finding 8 parts in the system, so k = 8. We can substitute these values into the formula:
P(X = 8) = (e^(-0.35) * 0.35^8) / 8!
Calculating this expression gives us the probability of finding 8 parts in the system. The correct answer is 0.0173 or approximately 1.73%.
Therefore, the probability that a random part arrival finds 8 parts in the system is 0.0173.
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At a production machine, parts arrive according to a Poisson process at the rate of 0.35 parts per minute. Processing time for parts have exponential distribution with mean of 2 min. What is the probability that a random part arrival finds that there are already 8 parts in the system (in machine + in queue)?Correct answer is '0.0173'. Can you explain this answer?
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At a production machine, parts arrive according to a Poisson process at the rate of 0.35 parts per minute. Processing time for parts have exponential distribution with mean of 2 min. What is the probability that a random part arrival finds that there are already 8 parts in the system (in machine + in queue)?Correct answer is '0.0173'. Can you explain this answer? for GATE 2024 is part of GATE preparation. The Question and answers have been prepared according to the GATE exam syllabus. Information about At a production machine, parts arrive according to a Poisson process at the rate of 0.35 parts per minute. Processing time for parts have exponential distribution with mean of 2 min. What is the probability that a random part arrival finds that there are already 8 parts in the system (in machine + in queue)?Correct answer is '0.0173'. Can you explain this answer? covers all topics & solutions for GATE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for At a production machine, parts arrive according to a Poisson process at the rate of 0.35 parts per minute. Processing time for parts have exponential distribution with mean of 2 min. What is the probability that a random part arrival finds that there are already 8 parts in the system (in machine + in queue)?Correct answer is '0.0173'. Can you explain this answer?.
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