Problem 2

Emma and Ethan are contemplating opening a drive-thru convenience store in their town. The new store format has gain popularity as people look for contactless ways to shop during the pandemic. This convenience grocery store is open 24 hours and features thousands of items, offering almost every category you would expect to find in a traditional grocery store, and doesnt require you to order ahead. Customers would pull up to a drive-thru window to place their order, the attendance would process their payment. They would be given an assigned spot to park and an employee/server would then walk out to their car and deliver the items. They want to provide a level of convenience and customer service that is lacking in the area. Based upon their own experience, they believe the number of customers waiting in the drive-thru lane as well as time in the system are important considerations with respect to customer service. Consequently, they are debating how many servers are needed and the right queuing system to achieve their desired level of customer service. Their biggest concern is scheduling for Monday-Friday 3:00 -7:00pm when they expect to experience their largest number of customers.

• The calling population is assumed to be large.
• Customers are assumed to arrive according to a Poisson distribution at an average of 18 customers per hour.
• Emma and Ethan are concerned that if too many customers are waiting in the drive through line, it may potentially cause people to balk or renege. They would like to limit the number of total customers waiting in the queue to no more than 8 customers on average. They would also like to limit the amount of time a customer spends waiting in the queue to place their order a maximum of 6 minutes on average.
• Based upon the proposed size and layout of the drive-thru and store parking lot (their system) can accommodate a maximum of 20 customers at any given point in time. Having more than 20 customers waiting will create traffic and parking issues, and create a poor customer experience.

They are considering 2 plans for their stores queueing and delivery configuration:

Scenario 1:

• The drive-thru will be configured such that customers arriving will wait in a single common drive-thru lane to be served on a first-in-first-out (FIFO) basis.
• Their system (drive-thru lane and parking lot) can accommodate up to 20 customers.
• Service (place order, process payment and delivery of items to car) is assumed to be exponentially distributed at a rate capable of serving an average of 7 customers per hour.

1. Calculate the following statistics for five separate systems consisting of one, two, three, four, and five servers using one drive-thru lane, respectively, based upon customers arriving at the drive-thru an average of 18 customers per hour, and each server serving an average of 7 customers per hour.
   • Server utilization
   • Mean number of customers in system
   • Mean time in system
   • Mean time in queue
   • Mean number of customers in queue
   • Probability of an empty system
   • Probability system is full

2. Based upon the preceding statistics, what is the minimum number of servers that will be required to satisfy the specified system performance criteria (i.e., maximum of 8 customers waiting in line on average, and maximum of 6 minutes spent waiting in line to place order on average)?

Scenario 2:

• Emma and Ethan want to evaluate system performance using two drive-thru lanes configured such that each drive-thru lane will have its own waiting line, or independent queues. Customers choose one of two drive-thru lanes to be served (two single-channel queuing system).
• The system can accommodate up to 20 customers total (those in queues and waiting for delivery of items).
• There will be two servers per drive-thru lane (each independent queues) and service is assumed to be exponentially distributed at a rate capable of serving an average of 7 customers per hour.

3. Based on the above scenario 2 conditions, calculate the following statistics based upon customers arriving at an average of 18 customers per hour, each customer choosing one of two drive-thru lanes and with each server handling an average of 7 customers per hour.
   1. Server utilization
   2. Mean number of customers in system
   3. Mean time in system
   4. Mean time in queue
   5. Mean number of customers in queue
   6. Probability of an empty system
   7. Probability system is full

4. Compare the system configurations chosen from (2b) and (2c) above for the following statistics, and what option you would recommend and why.
   1. Server utilization
   2. Mean number of customers in system
   3. Mean time in system
   4. Mean time in queue
   5. Mean number of customers in queue
   6. Probability of an empty system

5. What other recommendations can you suggest to the store owners to reduce waiting time and time in the system and improve customer service in their new drive-thru convenience store?