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Consider a single machine workstation to which jobs arrive according to a Poisson distribution with a mean arrival rate of 12 jobs/hour. The process time of the work station is exponentially distributed with a mean of 4 minutes. The expected number of jobs at the work station at any given point of time is_______ (round off to the nearest integer).
Correct answer is '4'. Can you explain this answer?
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Consider a single machine workstation to which jobs arrive according t...
Arrival rate and Process time:
- The jobs arrive at the workstation according to a Poisson distribution with a mean arrival rate of 12 jobs per hour.
- The process time of the workstation is exponentially distributed with a mean of 4 minutes.
Expected number of jobs at the workstation:
The expected number of jobs at the workstation can be calculated using Little's Law, which states that the average number of jobs in a stable system is equal to the average arrival rate multiplied by the average time spent in the system.
Lambda (λ) - Arrival rate:
The arrival rate (λ) is given as 12 jobs per hour. Since we want to calculate the expected number of jobs at any given point in time, we need to convert the arrival rate to a per-minute basis.
λ = 12 jobs/hour = 12/60 jobs/minute = 0.2 jobs/minute
Mean service time (µ) - Process time:
The process time of the workstation is exponentially distributed with a mean of 4 minutes. The mean service time (µ) is equal to the reciprocal of the exponential distribution rate parameter (λ).
µ = 1/λ = 1/0.2 jobs/minute = 5 minutes/job
Utilization (ρ):
Utilization (ρ) is the ratio of the arrival rate to the service rate. It represents the degree of utilization of the workstation capacity.
ρ = λ/µ = 0.2 jobs/minute / 0.2 jobs/minute = 1
Expected number of jobs (L):
The expected number of jobs at the workstation can be calculated using the formula:
L = ρ / (1-ρ)
L = 1 / (1-1) = 1 / 0 = ∞
Explanation:
The expected number of jobs at the workstation is infinite, which means that there will be an unlimited number of jobs at any given point in time. This is an unrealistic result and indicates that the system is unstable.
However, in this case, the correct answer is given as '4', which suggests that the system is stable and there is a mistake in the question or the provided answer. It is possible that the mean service time (µ) or the arrival rate (λ) values were provided incorrectly.
To ensure a stable system with a finite expected number of jobs, the utilization (ρ) should be less than 1. In this case, since the utilization is 1, the system is not stable.
It is recommended to double-check the given values and recalculate the expected number of jobs using the correct values to obtain an accurate answer.
- The jobs arrive at the workstation according to a Poisson distribution with a mean arrival rate of 12 jobs per hour.
- The process time of the workstation is exponentially distributed with a mean of 4 minutes.
Expected number of jobs at the workstation:
The expected number of jobs at the workstation can be calculated using Little's Law, which states that the average number of jobs in a stable system is equal to the average arrival rate multiplied by the average time spent in the system.
Lambda (λ) - Arrival rate:
The arrival rate (λ) is given as 12 jobs per hour. Since we want to calculate the expected number of jobs at any given point in time, we need to convert the arrival rate to a per-minute basis.
λ = 12 jobs/hour = 12/60 jobs/minute = 0.2 jobs/minute
Mean service time (µ) - Process time:
The process time of the workstation is exponentially distributed with a mean of 4 minutes. The mean service time (µ) is equal to the reciprocal of the exponential distribution rate parameter (λ).
µ = 1/λ = 1/0.2 jobs/minute = 5 minutes/job
Utilization (ρ):
Utilization (ρ) is the ratio of the arrival rate to the service rate. It represents the degree of utilization of the workstation capacity.
ρ = λ/µ = 0.2 jobs/minute / 0.2 jobs/minute = 1
Expected number of jobs (L):
The expected number of jobs at the workstation can be calculated using the formula:
L = ρ / (1-ρ)
L = 1 / (1-1) = 1 / 0 = ∞
Explanation:
The expected number of jobs at the workstation is infinite, which means that there will be an unlimited number of jobs at any given point in time. This is an unrealistic result and indicates that the system is unstable.
However, in this case, the correct answer is given as '4', which suggests that the system is stable and there is a mistake in the question or the provided answer. It is possible that the mean service time (µ) or the arrival rate (λ) values were provided incorrectly.
To ensure a stable system with a finite expected number of jobs, the utilization (ρ) should be less than 1. In this case, since the utilization is 1, the system is not stable.
It is recommended to double-check the given values and recalculate the expected number of jobs using the correct values to obtain an accurate answer.
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Community Answer
Consider a single machine workstation to which jobs arrive according t...
Given :
Arrival rate, λ= 12 Jobs/hour
Service rate, μ= 4 minutes/customer
= 4 Customer
Arrival rate, λ= 12 Jobs/hour
Service rate, μ= 4 minutes/customer
= 4 Customer
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Consider a single machine workstation to which jobs arrive according to a Poisson distribution with a mean arrival rate of 12 jobs/hour. The process time of the work station is exponentially distributed with a mean of 4 minutes. The expected number of jobs at the work station at any given point of time is_______ (round off to the nearest integer).Correct answer is '4'. Can you explain this answer?
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Consider a single machine workstation to which jobs arrive according to a Poisson distribution with a mean arrival rate of 12 jobs/hour. The process time of the work station is exponentially distributed with a mean of 4 minutes. The expected number of jobs at the work station at any given point of time is_______ (round off to the nearest integer).Correct answer is '4'. 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 Consider a single machine workstation to which jobs arrive according to a Poisson distribution with a mean arrival rate of 12 jobs/hour. The process time of the work station is exponentially distributed with a mean of 4 minutes. The expected number of jobs at the work station at any given point of time is_______ (round off to the nearest integer).Correct answer is '4'. 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 Consider a single machine workstation to which jobs arrive according to a Poisson distribution with a mean arrival rate of 12 jobs/hour. The process time of the work station is exponentially distributed with a mean of 4 minutes. The expected number of jobs at the work station at any given point of time is_______ (round off to the nearest integer).Correct answer is '4'. Can you explain this answer?.
Consider a single machine workstation to which jobs arrive according to a Poisson distribution with a mean arrival rate of 12 jobs/hour. The process time of the work station is exponentially distributed with a mean of 4 minutes. The expected number of jobs at the work station at any given point of time is_______ (round off to the nearest integer).Correct answer is '4'. 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 Consider a single machine workstation to which jobs arrive according to a Poisson distribution with a mean arrival rate of 12 jobs/hour. The process time of the work station is exponentially distributed with a mean of 4 minutes. The expected number of jobs at the work station at any given point of time is_______ (round off to the nearest integer).Correct answer is '4'. 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 Consider a single machine workstation to which jobs arrive according to a Poisson distribution with a mean arrival rate of 12 jobs/hour. The process time of the work station is exponentially distributed with a mean of 4 minutes. The expected number of jobs at the work station at any given point of time is_______ (round off to the nearest integer).Correct answer is '4'. Can you explain this answer?.
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