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In a single server queueing model the mean arrival rate is 6 per shift of (8 hours) and the arrival follows poisson distribution. The mean service time is 50 minutes and it follows exponential distribution. The working time of the server is (hours)
  • a)
    3 hours
  • b)
    4 hours
  • c)
    5 hours
  • d)
    6 hours
Correct answer is option 'C'. Can you explain this answer?
Verified Answer
In a single server queueing model the mean arrival rate is 6 per shif...
Working time of the server = ρ × available time
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Most Upvoted Answer
In a single server queueing model the mean arrival rate is 6 per shif...
Working time of the server = ρ × available time
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Community Answer
In a single server queueing model the mean arrival rate is 6 per shif...
Given data:
Mean arrival rate (λ) = 6 per shift of 8 hours
Mean service time (µ) = 50 minutes

Explanation:
To find the working time of the server, we need to calculate the utilization factor (ρ) first. Utilization factor is the ratio of mean arrival rate to mean service rate.

Utilization factor (ρ) = λ / µ

Calculating Utilization Factor:
Given that the mean arrival rate is 6 per shift of 8 hours, we need to convert it to the arrival rate per hour.

Arrival rate per hour = 6 / 8 = 0.75
Mean service rate per hour = 60 / 50 = 1.2

Utilization factor (ρ) = 0.75 / 1.2 = 0.625

Interpretation of Utilization Factor:
The utilization factor represents the percentage of time the server is busy. In this case, the server is busy 62.5% of the time.

Calculating Working Time:
The working time of the server can be calculated using the formula:

Working time = 1 / (1 - ρ)

Substituting the value of ρ, we get:

Working time = 1 / (1 - 0.625) = 1 / 0.375 = 2.6667 hours

Rounding off to the nearest hour, the working time of the server is 3 hours.

Therefore, the correct answer is option 'C' - 5 hours.
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