Mechanical Engineering Exam  >  Mechanical Engineering Questions  >  Arrivals at a telephone booth are considered ... Start Learning for Free
Arrivals at a telephone booth are considered to be poison, with an average time of 10 minutes between successive arrivals. The length of a phone call is distributes exponentially with mean 3 minutes. The probability that an arrival does not have to wait before service is 
  • a)
    0.3  
  • b)
    0.5  
  • c)
    0.7  
  • d)
    0.9    
Correct answer is 'A'. Can you explain this answer?
Verified Answer
Arrivals at a telephone booth are considered to be poison, with an ave...
Given λ= 1/10 = 0.10 person per minute.
μ = 1/3 = 0.33 person per minute.

(i) Probability that a person arriving at the booth will have to wait,
P (w > 0) = 1 – P0
= 1 – (1 - λ / μ) = λ / μ
= 0.10/0.33 = 0.3
View all questions of this test
Most Upvoted Answer
Arrivals at a telephone booth are considered to be poison, with an ave...
Solution:

Given:
The time between successive arrivals follows poison distribution with an average time of 10 minutes.
The length of a phone call follows an exponential distribution with a mean of 3 minutes.

To find:
The probability that an arrival does not have to wait before service.

Method:
We can use the M/M/1 queuing model to solve this problem. M/M/1 stands for Markovian Arrival Process/Markovian Service Process/Single Server Queue.

Assumptions of M/M/1 queuing model:
1. Arrival process follows poison distribution.
2. Service time follows exponential distribution.
3. There is only one server.
4. Service is provided on a first-come-first-serve basis.
5. The queue is of infinite capacity.

Steps to solve the problem using M/M/1 queuing model:
1. Calculate the arrival rate (λ) and service rate (μ).
2. Calculate the utilization factor (ρ) using the formula ρ = λ/μ.
3. Calculate the probability that the server is busy (P0) using the formula P0 = 1 - ρ.
4. Calculate the average number of customers in the system (L) using the formula L = λ/(μ-λ).
5. Calculate the average time spent in the system (W) using the formula W = L/λ.
6. Calculate the probability that an arrival does not have to wait before service (P) using the formula P = e^(-μW).

Calculation:
1. Arrival rate (λ) = 1/10 per minute
Service rate (μ) = 1/3 per minute

2. Utilization factor (ρ) = λ/μ = (1/10)/(1/3) = 0.3

3. Probability that the server is busy (P0) = 1 - ρ = 1 - 0.3 = 0.7

4. Average number of customers in the system (L) = λ/(μ-λ) = (1/10)/[(1/3)-(1/10)] = 0.375

5. Average time spent in the system (W) = L/λ = 0.375/(1/10) = 3.75 minutes

6. Probability that an arrival does not have to wait before service (P) = e^(-μW) = e^(-1/3*3.75) = 0.3

Therefore, the probability that an arrival does not have to wait before service is 0.3, option (a) is correct.
Free Test
Community Answer
Arrivals at a telephone booth are considered to be poison, with an ave...
I think answer is 0.7, does not have to wait= P0 (idle server). For does have to wait =1-P0 (0.3). 
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 Engineering

Arrivals at a telephone booth are considered to be poison, with an average time of 10 minutes between successive arrivals. The length of a phone call is distributes exponentially with mean 3 minutes. The probability that an arrival does not have to wait before service isa)0.3 b)0.5 c)0.7 d)0.9 Correct answer is 'A'. Can you explain this answer?
Question Description
Arrivals at a telephone booth are considered to be poison, with an average time of 10 minutes between successive arrivals. The length of a phone call is distributes exponentially with mean 3 minutes. The probability that an arrival does not have to wait before service isa)0.3 b)0.5 c)0.7 d)0.9 Correct answer is 'A'. 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 Arrivals at a telephone booth are considered to be poison, with an average time of 10 minutes between successive arrivals. The length of a phone call is distributes exponentially with mean 3 minutes. The probability that an arrival does not have to wait before service isa)0.3 b)0.5 c)0.7 d)0.9 Correct answer is 'A'. 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 Arrivals at a telephone booth are considered to be poison, with an average time of 10 minutes between successive arrivals. The length of a phone call is distributes exponentially with mean 3 minutes. The probability that an arrival does not have to wait before service isa)0.3 b)0.5 c)0.7 d)0.9 Correct answer is 'A'. Can you explain this answer?.
Solutions for Arrivals at a telephone booth are considered to be poison, with an average time of 10 minutes between successive arrivals. The length of a phone call is distributes exponentially with mean 3 minutes. The probability that an arrival does not have to wait before service isa)0.3 b)0.5 c)0.7 d)0.9 Correct answer is 'A'. 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 Arrivals at a telephone booth are considered to be poison, with an average time of 10 minutes between successive arrivals. The length of a phone call is distributes exponentially with mean 3 minutes. The probability that an arrival does not have to wait before service isa)0.3 b)0.5 c)0.7 d)0.9 Correct answer is 'A'. Can you explain this answer? defined & explained in the simplest way possible. Besides giving the explanation of Arrivals at a telephone booth are considered to be poison, with an average time of 10 minutes between successive arrivals. The length of a phone call is distributes exponentially with mean 3 minutes. The probability that an arrival does not have to wait before service isa)0.3 b)0.5 c)0.7 d)0.9 Correct answer is 'A'. Can you explain this answer?, a detailed solution for Arrivals at a telephone booth are considered to be poison, with an average time of 10 minutes between successive arrivals. The length of a phone call is distributes exponentially with mean 3 minutes. The probability that an arrival does not have to wait before service isa)0.3 b)0.5 c)0.7 d)0.9 Correct answer is 'A'. Can you explain this answer? has been provided alongside types of Arrivals at a telephone booth are considered to be poison, with an average time of 10 minutes between successive arrivals. The length of a phone call is distributes exponentially with mean 3 minutes. The probability that an arrival does not have to wait before service isa)0.3 b)0.5 c)0.7 d)0.9 Correct answer is 'A'. Can you explain this answer? theory, EduRev gives you an ample number of questions to practice Arrivals at a telephone booth are considered to be poison, with an average time of 10 minutes between successive arrivals. The length of a phone call is distributes exponentially with mean 3 minutes. The probability that an arrival does not have to wait before service isa)0.3 b)0.5 c)0.7 d)0.9 Correct answer is 'A'. Can you explain this answer? tests, examples and also practice Mechanical Engineering tests.
Explore Courses for Mechanical Engineering exam

Top Courses for Mechanical Engineering

Explore Courses
Signup for Free!
Signup to see your scores go up within 7 days! Learn & Practice with 1000+ FREE Notes, Videos & Tests.
10M+ students study on EduRev