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Trains arrive at a yard every 15 minutes and the service time is 5 minutes. The probability that the yard has 3 trains in the system is __________
- a)0.020
- b)0.030
Correct answer is between ' 0.020, 0.030'. Can you explain this answer?
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Verified Answer
Trains arrive at a yard every 15 minutes and the service time is 5 mi...
Given data,
Probability that there are ‘n’ trains in a system is
ρn (1 − ρ)
for 3 train in a system = 
= 0.024
Most Upvoted Answer
Trains arrive at a yard every 15 minutes and the service time is 5 mi...
Solution:
Given data:
Inter-arrival time = 15 min
Service time = 5 min
To find: Probability of having 3 trains in the system
Assumptions made:
- The arrival of trains follows a Poisson distribution
- The service times follow an exponential distribution
Calculation:
1. Finding average number of arrivals in a minute
Average number of arrivals in a minute (λ) = 1/15
2. Finding average number of departures in a minute
Average number of departures in a minute (μ) = 1/5
3. Finding utilization factor (ρ)
Utilization factor (ρ) = λ/μ
ρ = (1/15)/(1/5)
ρ = 1/3
4. Finding probability of having 0, 1, 2, and 3 trains in the system
Probability of having 0 trains in the system:
P0 = 1 - ρ = 1 - 1/3 = 2/3
Probability of having 1 train in the system:
P1 = ρ(1 - ρ) = (1/3)(2/3) = 2/9
Probability of having 2 trains in the system:
P2 = ρ^2(1 - ρ/2) = (1/9)(2/3) = 2/27
Probability of having 3 trains in the system:
P3 = ρ^3/3!(1 - ρ/3) = (1/27)(3/4) = 1/36
5. Finding the required probability
The required probability is the probability of having 3 trains in the system, which is P3 = 1/36
Therefore, the probability that the yard has 3 trains in the system is 0.028 or 0.03 (approx.)
Answer: b) 0.030
Given data:
Inter-arrival time = 15 min
Service time = 5 min
To find: Probability of having 3 trains in the system
Assumptions made:
- The arrival of trains follows a Poisson distribution
- The service times follow an exponential distribution
Calculation:
1. Finding average number of arrivals in a minute
Average number of arrivals in a minute (λ) = 1/15
2. Finding average number of departures in a minute
Average number of departures in a minute (μ) = 1/5
3. Finding utilization factor (ρ)
Utilization factor (ρ) = λ/μ
ρ = (1/15)/(1/5)
ρ = 1/3
4. Finding probability of having 0, 1, 2, and 3 trains in the system
Probability of having 0 trains in the system:
P0 = 1 - ρ = 1 - 1/3 = 2/3
Probability of having 1 train in the system:
P1 = ρ(1 - ρ) = (1/3)(2/3) = 2/9
Probability of having 2 trains in the system:
P2 = ρ^2(1 - ρ/2) = (1/9)(2/3) = 2/27
Probability of having 3 trains in the system:
P3 = ρ^3/3!(1 - ρ/3) = (1/27)(3/4) = 1/36
5. Finding the required probability
The required probability is the probability of having 3 trains in the system, which is P3 = 1/36
Therefore, the probability that the yard has 3 trains in the system is 0.028 or 0.03 (approx.)
Answer: b) 0.030
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Trains arrive at a yard every 15 minutes and the service time is 5 minutes. The probability that the yard has 3 trains in the system is __________a) 0.020b) 0.030Correct answer is between ' 0.020, 0.030'. Can you explain this answer?
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Trains arrive at a yard every 15 minutes and the service time is 5 minutes. The probability that the yard has 3 trains in the system is __________a) 0.020b) 0.030Correct answer is between ' 0.020, 0.030'. Can you explain this answer? for Mechanical Engineering 2024 is part of Mechanical Engineering preparation. The Question and answers have been prepared according to the Mechanical Engineering exam syllabus. Information about Trains arrive at a yard every 15 minutes and the service time is 5 minutes. The probability that the yard has 3 trains in the system is __________a) 0.020b) 0.030Correct answer is between ' 0.020, 0.030'. Can you explain this answer? covers all topics & solutions for Mechanical Engineering 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Trains arrive at a yard every 15 minutes and the service time is 5 minutes. The probability that the yard has 3 trains in the system is __________a) 0.020b) 0.030Correct answer is between ' 0.020, 0.030'. Can you explain this answer?.
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