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A self-service store employs one cashier at its counter. Nine customers arrive on an average every 5 minutes, while the cashier can serve 10 customers in 5 minutes. Assuming poison distribution for arrival rate and exponential distribution for service rate, find the average number of customers in the queue.
- a)6.7
- b)7.8
- c)9.6
- d)8.1
Correct answer is option 'D'. Can you explain this answer?
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A self-service store employs one cashier at its counter. Nine custome...
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A self-service store employs one cashier at its counter. Nine custome...
Given data:
- Arrival rate follows Poison distribution with an average of 9 customers every 5 minutes.
- Service rate follows Exponential distribution with an average of 10 customers in 5 minutes.
- There is only one cashier available at the counter.
We need to find the average number of customers in the queue.
Solution:
1. Calculation of arrival rate:
- The arrival rate is given as 9 customers every 5 minutes.
- We can calculate the arrival rate per minute by dividing it by 5: 9/5 = 1.8 customers per minute.
2. Calculation of service rate:
- The service rate is given as 10 customers in 5 minutes.
- We can calculate the service rate per minute by dividing it by 5: 10/5 = 2 customers per minute.
3. Calculation of utilization factor:
- Utilization factor (ρ) is the ratio of service rate to arrival rate: ρ = 2/1.8 = 1.11.
4. Calculation of average number of customers in the queue:
- Average number of customers in the queue (Lq) can be calculated using the formula: Lq = ρ^2 / (1-ρ) * (1 - p0), where p0 is the probability of zero customers in the system.
- Probability of zero customers in the system can be calculated as: p0 = 1 - ρ = 1 - 1.11 = -0.11 (which is not possible, as probability cannot be negative).
- We can assume p0 to be 0, as it is very unlikely that there would be no customers in the system.
- So, Lq = ρ^2 / (1-ρ) = 1.11^2 / (1-1.11) = 8.1.
Therefore, the average number of customers in the queue is 8.1, which is option D.
- Arrival rate follows Poison distribution with an average of 9 customers every 5 minutes.
- Service rate follows Exponential distribution with an average of 10 customers in 5 minutes.
- There is only one cashier available at the counter.
We need to find the average number of customers in the queue.
Solution:
1. Calculation of arrival rate:
- The arrival rate is given as 9 customers every 5 minutes.
- We can calculate the arrival rate per minute by dividing it by 5: 9/5 = 1.8 customers per minute.
2. Calculation of service rate:
- The service rate is given as 10 customers in 5 minutes.
- We can calculate the service rate per minute by dividing it by 5: 10/5 = 2 customers per minute.
3. Calculation of utilization factor:
- Utilization factor (ρ) is the ratio of service rate to arrival rate: ρ = 2/1.8 = 1.11.
4. Calculation of average number of customers in the queue:
- Average number of customers in the queue (Lq) can be calculated using the formula: Lq = ρ^2 / (1-ρ) * (1 - p0), where p0 is the probability of zero customers in the system.
- Probability of zero customers in the system can be calculated as: p0 = 1 - ρ = 1 - 1.11 = -0.11 (which is not possible, as probability cannot be negative).
- We can assume p0 to be 0, as it is very unlikely that there would be no customers in the system.
- So, Lq = ρ^2 / (1-ρ) = 1.11^2 / (1-1.11) = 8.1.
Therefore, the average number of customers in the queue is 8.1, which is option D.
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A self-service store employs one cashier at its counter. Nine customers arrive on an average every 5 minutes, while the cashier can serve 10 customers in 5 minutes. Assuming poison distribution for arrival rate and exponential distribution for service rate, find the average number of customers in the queue.a)6.7b)7.8c)9.6d)8.1Correct answer is option 'D'. Can you explain this answer?
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