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Answer :
a) The average wait time for customers in line at UTea is 0.693 minutes.
b) The new demand rate is 30 customers per hour, with a 33.33% decrease in sales.
c) There are approximately 22.50 offline customers served per hour, and the total number of customers per hour is 62.50.
d) Based on the Paramount Diner case, the demand mix between online and offline is expected to shift towards online orders, impacting UTea's operations and resource allocation.
a) To calculate the average wait time for customers in line at UTea, we need to use the queuing formula for an M/M/1 queue. In this case, the arrival rate (λ) is 1 customer every 45 seconds, the service rate (μ) is 1 customer every 2 minutes, and the coefficient of variation for service time (CVP) is 0.9.
Using the queuing formula for an M/M/1 queue, the average wait time (Wq) is given by:
Wq = (ρ / (μ - λ)) * (1 + (CVP^2 / 2))
where ρ is the traffic intensity, calculated as ρ = λ / μ.
First, calculate the traffic intensity,
ρ = λ / μ = (1 customer/45 seconds) / (1 customer/2 minutes) = (1/45) / (1/2) = 2/45
Then, substitute the values into the formula,
Wq = (2/45 / (1/2 - 1/45)) * (1 + (0.9^2 / 2))
Wq = (2/45 / (23/90)) * (1 + (0.81 / 2))
Wq = (4/23) * (1 + 0.405)
Wq ≈ 0.693 minutes (rounded to three decimal places)
Therefore, the average wait time for customers in line at UTea is approximately 0.693 minutes.
b) If the average line length is 3 customers, we can calculate the new demand rate (λ) in customers per hour using Little's Law. Little's Law states that the average number of customers in a system (L) is equal to the arrival rate (λ) multiplied by the average time a customer spends in the system (W).
Given that the average line length is 3 customers, and the service rate (μ) is still 1 customer every 2 minutes,
L = 3 customers
W = (L / λ) = (3 / (1/2)) = 6 minutes
Now, we convert the average time spent in the system to hours,
W (in hours) = 6 minutes / 60 = 0.1 hours
Using Little's Law: L = λ * W
3 = λ * 0.1
λ = 3 / 0.1 = 30 customers per hour
The new demand rate is 30 customers per hour.
To calculate the percentage decrease in sales (i.e., throughput), we compare the new demand rate with the initial demand rate:
Percentage decrease in sales = ((Initial demand rate - New demand rate) / Initial demand rate) * 100
Percentage decrease in sales = ((45 customers per hour - 30 customers per hour) / 45 customers per hour) * 100
Percentage decrease in sales ≈ 33.33%
c) With the launch of the mobile app, UTea receives an average of 40 orders per hour from the app. These orders are not affected by wait time (CVa = 1). Since the processing time for online orders remains the same and there are three visible customers in line, we can calculate the number of offline (in-store) customers served per hour by subtracting the number of online orders from the total number of customers served per hour.
Number of offline customers served per hour = Total customers served per hour - Online orders per hour
Number of offline customers served per hour = (Initial demand rate - Percentage decrease in sales) - Online orders per hour
Number of offline customers served per hour = (45 customers per hour - (33.33% of
45 customers per hour)) - 40 orders per hour
Number of offline customers served per hour ≈ 22.50 customers per hour
To calculate the total number of customers per hour, we add the number of offline customers served per hour and the number of online orders per hour,
Total number of customers per hour = Number of offline customers served per hour + Online orders per hour
Total number of customers per hour = 22.50 customers per hour + 40 orders per hour
Total number of customers per hour ≈ 62.50 customers per hour
Therefore, UTea serves approximately 22.50 offline customers and receives 40 online orders per hour, resulting in a total of 62.50 customers per hour.
d) Based on the Paramount Diner case, where online orders increased significantly, we can expect a similar trend at UTea. The introduction of the mobile app and the convenience it offers to customers for ordering and pick-up are likely to attract more online orders in the future. As the online ordering channel becomes more popular, the demand mix between online and offline (in-store) is expected to shift further towards online orders.
The implications for UTea include the need to allocate resources efficiently to handle the increasing volume of online orders. It may require additional staff or specific workflows to manage the online order fulfillment process effectively. UTea should also focus on providing a seamless and satisfactory experience for online customers to maintain customer satisfaction and loyalty. Additionally, UTea should continue monitoring and adapting its operations to meet the changing demand mix and leverage the benefits of the mobile app to improve profitability and customer engagement.
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