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In a single-channel queuing model, the customer arrival rate is 12 per hour and the waiting time in the queue is 2.5 minute. The proportion of time that a server actually spends with customers is
- a)33.33%
- b)50%
- c)66.67%
- d)75%
Correct answer is option 'B'. Can you explain this answer?
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Verified Answer
In a single-channel queuing model, the customer arrival rate is 12 pe...
Given, arrival rate, λ = 12 per hr
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Waiting time in queue, Wq = 2.5min = 1/24hr
Wq = p2(1−ρ)λ = 1/24
ρ = traffic intensity or the proportion of time that a server actually spends with customers
2p2 + p − 1 = 0
2p2 + 2p − p − 1 = 0
ρ = −1(not possible) ρ = 1/2 = 0.5 = 50%
Most Upvoted Answer
In a single-channel queuing model, the customer arrival rate is 12 pe...
Given information:
- Arrival rate of customers: 12 per hour
- Waiting time in the queue: 2.5 minutes
To find:
The proportion of time that a server actually spends with customers.
Solution:
To find the proportion of time that a server spends with customers, we need to consider the service rate and the arrival rate.
Service rate:
The service rate can be determined by taking the reciprocal of the waiting time in the queue. In this case, the waiting time is given as 2.5 minutes. Therefore, the service rate is 1/2.5 = 0.4 customers per minute.
Utilization factor:
The utilization factor is the ratio of the arrival rate to the service rate. It represents the proportion of time the server is busy.
Utilization factor = Arrival rate / Service rate
Utilization factor = 12 (customers per hour) / 0.4 (customers per minute)
Utilization factor = 30
Proportion of time spent with customers:
The proportion of time spent with customers is the complement of the utilization factor. It represents the proportion of time the server is idle or not serving customers.
Proportion of time spent with customers = 1 - Utilization factor
Proportion of time spent with customers = 1 - 30
Proportion of time spent with customers = 0.0333 (or 33.33%)
Correct answer:
Option B: 50%
Explanation:
The correct answer is not option B. The correct answer is option D: 75%.
The proportion of time that a server actually spends with customers is given by the utilization factor, which is the ratio of the arrival rate to the service rate. In this case, the utilization factor is 30, which means that the server is busy 30 out of 31 minutes.
Therefore, the proportion of time that the server spends with customers is 1 - 30/31 = 1/31 = 0.0323 (or 3.23%). This is not equal to any of the given options.
Hence, the correct answer is not among the options provided.
- Arrival rate of customers: 12 per hour
- Waiting time in the queue: 2.5 minutes
To find:
The proportion of time that a server actually spends with customers.
Solution:
To find the proportion of time that a server spends with customers, we need to consider the service rate and the arrival rate.
Service rate:
The service rate can be determined by taking the reciprocal of the waiting time in the queue. In this case, the waiting time is given as 2.5 minutes. Therefore, the service rate is 1/2.5 = 0.4 customers per minute.
Utilization factor:
The utilization factor is the ratio of the arrival rate to the service rate. It represents the proportion of time the server is busy.
Utilization factor = Arrival rate / Service rate
Utilization factor = 12 (customers per hour) / 0.4 (customers per minute)
Utilization factor = 30
Proportion of time spent with customers:
The proportion of time spent with customers is the complement of the utilization factor. It represents the proportion of time the server is idle or not serving customers.
Proportion of time spent with customers = 1 - Utilization factor
Proportion of time spent with customers = 1 - 30
Proportion of time spent with customers = 0.0333 (or 33.33%)
Correct answer:
Option B: 50%
Explanation:
The correct answer is not option B. The correct answer is option D: 75%.
The proportion of time that a server actually spends with customers is given by the utilization factor, which is the ratio of the arrival rate to the service rate. In this case, the utilization factor is 30, which means that the server is busy 30 out of 31 minutes.
Therefore, the proportion of time that the server spends with customers is 1 - 30/31 = 1/31 = 0.0323 (or 3.23%). This is not equal to any of the given options.
Hence, the correct answer is not among the options provided.
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In a single-channel queuing model, the customer arrival rate is 12 per hour and the waiting time in the queue is 2.5 minute. The proportion of time that a server actually spends with customers isa)33.33%b)50%c)66.67%d)75%Correct answer is option 'B'. Can you explain this answer?
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