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Customers arrive at a ticket counter with an arrival rate of 12 customers per hour. A clerk serves the customers at a rate of 24 per hour. The probability that the clerk is busy is __________

  • a)
    0.5

  • b)
    0.9

  • c)
    1.0

  • d)
    2.0

Correct answer is option 'A'. Can you explain this answer?
Verified Answer
Customers arrive at a ticket counter with an arrival rate of 12 custom...
Arrival rate λ = 12/hour
Service rate μ = 24/hour
Probability that the clerk is busy is when there is at least one customer in the system
= λ/μ
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Most Upvoted Answer
Customers arrive at a ticket counter with an arrival rate of 12 custom...
The probability that the clerk is busy can be determined using the concept of the Poisson process. In a Poisson process, the arrival of customers follows a Poisson distribution, and the service time of customers also follows a Poisson distribution.

Given that the arrival rate is 12 customers per hour, we can calculate the arrival rate parameter (λ) as λ = 12 customers/hour. Similarly, the service rate is 24 customers per hour, so the service rate parameter (μ) is μ = 24 customers/hour.

The utilization factor (ρ) is the ratio of the arrival rate to the service rate, which gives an indication of how busy the clerk is. It is calculated as ρ = λ/μ.

Let's calculate ρ:

ρ = 12 customers/hour / 24 customers/hour = 0.5

Now, we can interpret the value of ρ:

- If ρ < 1,="" it="" means="" that="" the="" arrival="" rate="" is="" less="" than="" the="" service="" rate,="" and="" the="" clerk="" is="" not="" busy="" most="" of="" the="" />
- If ρ = 1, it means that the arrival rate is equal to the service rate, and the clerk is busy 100% of the time.
- If ρ > 1, it means that the arrival rate is greater than the service rate, and the clerk cannot keep up with the demand.

In this case, ρ = 0.5, which means that the arrival rate is less than the service rate. Therefore, the probability that the clerk is busy is less than 1, indicating that the clerk is not busy most of the time.

Hence, the correct answer is option A) 0.5.
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Customers arrive at a ticket counter with an arrival rate of 12 customers per hour. A clerk serves the customers at a rate of 24 per hour. The probability that the clerk is busy is __________a)0.5b)0.9c)1.0d)2.0Correct answer is option 'A'. Can you explain this answer?
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