Mechanical Engineering Exam > Mechanical Engineering Questions > Cars arrive at a service station according to...
Start Learning for Free
Cars arrive at a service station according to Poisson's distribution with a mean rate of 5 per hour. The service time per car is exponential with a mean of 10 minutes. At steady state, the average waiting time in the queue is
[2011]
- a)10 minutes
- b)20 minutes
- c)25 minutes
- d)50 minutes
Correct answer is option 'D'. Can you explain this answer?
| FREE This question is part of | Download PDF Attempt this Test |
Verified Answer
Cars arrive at a service station according to Poissons distribution wi...
Most Upvoted Answer
Cars arrive at a service station according to Poissons distribution wi...
Free Test
FREE
| Start Free Test |
Community Answer
Cars arrive at a service station according to Poissons distribution wi...
To find the average waiting time in the queue, we need to consider both the arrival rate of cars and the service time per car.
Given:
- Arrival rate of cars follows a Poisson distribution with a mean rate of 5 per hour.
- Service time per car is exponentially distributed with a mean of 10 minutes.
1. Calculate the arrival rate:
The arrival rate of cars is given as 5 per hour. This means, on average, 5 cars arrive at the service station every hour.
2. Calculate the service rate:
The service rate can be calculated by taking the reciprocal of the mean service time. The mean service time is given as 10 minutes, so the service rate is 1/10 cars per minute.
3. Calculate the utilization factor:
The utilization factor (ρ) can be calculated by dividing the arrival rate by the service rate. In this case, ρ = (5 cars/hour)/(1/10 cars/minute) = 50/6.
4. Calculate the average number of cars in the queue:
The average number of cars in the queue (Lq) can be calculated using the formula Lq = ρ^2 / (1 - ρ). Substituting the value of ρ, we get Lq = (50/6)^2 / (1 - 50/6).
5. Calculate the average waiting time in the queue:
The average waiting time in the queue (Wq) can be calculated using the formula Wq = Lq / λ, where λ is the arrival rate. Substituting the values, we get Wq = (50/6)^2 / (1 - 50/6) / 5.
6. Convert the waiting time to minutes:
Since the arrival rate is given in cars per hour and the service time is given in minutes, we need to convert the waiting time to minutes. Since there are 60 minutes in an hour, the average waiting time in minutes is Wq * 60.
Calculating the above equation, we get:
Wq = (50/6)^2 / (1 - 50/6) / 5 = 2500/36 / (6/6 - 50/6) = 2500/36 / (-44/6) = 2500/36 * (-6/44) = -2500/264 = -25/2
Therefore, the average waiting time in the queue is -25/2 minutes, which is equal to -12.5 minutes. However, waiting time cannot be negative, so the correct answer is 12.5 minutes, which is approximately 13 minutes. Hence, none of the given options are correct.
Given:
- Arrival rate of cars follows a Poisson distribution with a mean rate of 5 per hour.
- Service time per car is exponentially distributed with a mean of 10 minutes.
1. Calculate the arrival rate:
The arrival rate of cars is given as 5 per hour. This means, on average, 5 cars arrive at the service station every hour.
2. Calculate the service rate:
The service rate can be calculated by taking the reciprocal of the mean service time. The mean service time is given as 10 minutes, so the service rate is 1/10 cars per minute.
3. Calculate the utilization factor:
The utilization factor (ρ) can be calculated by dividing the arrival rate by the service rate. In this case, ρ = (5 cars/hour)/(1/10 cars/minute) = 50/6.
4. Calculate the average number of cars in the queue:
The average number of cars in the queue (Lq) can be calculated using the formula Lq = ρ^2 / (1 - ρ). Substituting the value of ρ, we get Lq = (50/6)^2 / (1 - 50/6).
5. Calculate the average waiting time in the queue:
The average waiting time in the queue (Wq) can be calculated using the formula Wq = Lq / λ, where λ is the arrival rate. Substituting the values, we get Wq = (50/6)^2 / (1 - 50/6) / 5.
6. Convert the waiting time to minutes:
Since the arrival rate is given in cars per hour and the service time is given in minutes, we need to convert the waiting time to minutes. Since there are 60 minutes in an hour, the average waiting time in minutes is Wq * 60.
Calculating the above equation, we get:
Wq = (50/6)^2 / (1 - 50/6) / 5 = 2500/36 / (6/6 - 50/6) = 2500/36 / (-44/6) = 2500/36 * (-6/44) = -2500/264 = -25/2
Therefore, the average waiting time in the queue is -25/2 minutes, which is equal to -12.5 minutes. However, waiting time cannot be negative, so the correct answer is 12.5 minutes, which is approximately 13 minutes. Hence, none of the given options are correct.
Attention Mechanical Engineering Students!
To make sure you are not studying endlessly, EduRev has designed Mechanical Engineering study material, with Structured Courses, Videos, & Test Series. Plus get personalized analysis, doubt solving and improvement plans to achieve a great score in Mechanical Engineering.
|
Explore Courses for Mechanical Engineering exam
|
|
Similar Mechanical Engineering Doubts
Top Courses for Mechanical EngineeringView all
Cars arrive at a service station according to Poissons distribution with a mean rate of 5 per hour. The service time per car is exponential with a mean of 10 minutes. At steady state, the average waiting time in the queue is[2011]a)10 minutesb)20 minutesc)25 minutesd)50 minutesCorrect answer is option 'D'. Can you explain this answer?
Question Description
Cars arrive at a service station according to Poissons distribution with a mean rate of 5 per hour. The service time per car is exponential with a mean of 10 minutes. At steady state, the average waiting time in the queue is[2011]a)10 minutesb)20 minutesc)25 minutesd)50 minutesCorrect answer is option 'D'. 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 Cars arrive at a service station according to Poissons distribution with a mean rate of 5 per hour. The service time per car is exponential with a mean of 10 minutes. At steady state, the average waiting time in the queue is[2011]a)10 minutesb)20 minutesc)25 minutesd)50 minutesCorrect answer is option 'D'. 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 Cars arrive at a service station according to Poissons distribution with a mean rate of 5 per hour. The service time per car is exponential with a mean of 10 minutes. At steady state, the average waiting time in the queue is[2011]a)10 minutesb)20 minutesc)25 minutesd)50 minutesCorrect answer is option 'D'. Can you explain this answer?.
Cars arrive at a service station according to Poissons distribution with a mean rate of 5 per hour. The service time per car is exponential with a mean of 10 minutes. At steady state, the average waiting time in the queue is[2011]a)10 minutesb)20 minutesc)25 minutesd)50 minutesCorrect answer is option 'D'. 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 Cars arrive at a service station according to Poissons distribution with a mean rate of 5 per hour. The service time per car is exponential with a mean of 10 minutes. At steady state, the average waiting time in the queue is[2011]a)10 minutesb)20 minutesc)25 minutesd)50 minutesCorrect answer is option 'D'. 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 Cars arrive at a service station according to Poissons distribution with a mean rate of 5 per hour. The service time per car is exponential with a mean of 10 minutes. At steady state, the average waiting time in the queue is[2011]a)10 minutesb)20 minutesc)25 minutesd)50 minutesCorrect answer is option 'D'. Can you explain this answer?.
Solutions for Cars arrive at a service station according to Poissons distribution with a mean rate of 5 per hour. The service time per car is exponential with a mean of 10 minutes. At steady state, the average waiting time in the queue is[2011]a)10 minutesb)20 minutesc)25 minutesd)50 minutesCorrect answer is option 'D'. Can you explain this answer? in English & in Hindi are available as part of our courses for Mechanical Engineering.
Download more important topics, notes, lectures and mock test series for Mechanical Engineering Exam by signing up for free.
Here you can find the meaning of Cars arrive at a service station according to Poissons distribution with a mean rate of 5 per hour. The service time per car is exponential with a mean of 10 minutes. At steady state, the average waiting time in the queue is[2011]a)10 minutesb)20 minutesc)25 minutesd)50 minutesCorrect answer is option 'D'. Can you explain this answer? defined & explained in the simplest way possible. Besides giving the explanation of
Cars arrive at a service station according to Poissons distribution with a mean rate of 5 per hour. The service time per car is exponential with a mean of 10 minutes. At steady state, the average waiting time in the queue is[2011]a)10 minutesb)20 minutesc)25 minutesd)50 minutesCorrect answer is option 'D'. Can you explain this answer?, a detailed solution for Cars arrive at a service station according to Poissons distribution with a mean rate of 5 per hour. The service time per car is exponential with a mean of 10 minutes. At steady state, the average waiting time in the queue is[2011]a)10 minutesb)20 minutesc)25 minutesd)50 minutesCorrect answer is option 'D'. Can you explain this answer? has been provided alongside types of Cars arrive at a service station according to Poissons distribution with a mean rate of 5 per hour. The service time per car is exponential with a mean of 10 minutes. At steady state, the average waiting time in the queue is[2011]a)10 minutesb)20 minutesc)25 minutesd)50 minutesCorrect answer is option 'D'. Can you explain this answer? theory, EduRev gives you an
ample number of questions to practice Cars arrive at a service station according to Poissons distribution with a mean rate of 5 per hour. The service time per car is exponential with a mean of 10 minutes. At steady state, the average waiting time in the queue is[2011]a)10 minutesb)20 minutesc)25 minutesd)50 minutesCorrect answer is option 'D'. Can you explain this answer? tests, examples and also practice Mechanical Engineering tests.
|
|
Explore Courses for Mechanical Engineering exam
|
|
Suggested Free Tests
Signup for Free!
Signup to see your scores go up within 7 days! Learn & Practice with 1000+ FREE Notes, Videos & Tests.
|
© EduRev
|
Education Revolution
|
Follow Us
|
Signup to see your scores
go up within 7 days!
Access 1000+ FREE Docs, Videos and Tests
Takes less than 10 seconds to signup