Problem 3: Queue Configurations

A store has two employees in charge of customer checkout and no separate self-checkout kiosks. Customers finish shopping and arrive at the checkout area following a Poisson process with a mean arrival rate of 54 customers/hour. The time required by an employee to finish the checkout process for a customer is exponentially distributed with a mean of 2 minutes. To make the checking out process smooth for customers, the store is considering three different queue configurations.

I. Independent Queues: An independent queue is formed in front of each employee. Customers arriving to the checkout area randomly join each queue, and they do not have the option of jumping from one queue to another. Hence, each employee serves half of the arriving customers.

II. Pooled Queue: The customers form a single queue from which they are routed to the first available employee on a first-come-first-serve basis.

III. Quick Service: The two employees together serve each customer; while one employee scans items, the other bags them. Since the workload is shared by the two employees, the average service times are now exponentially distributed with a mean of 1 minute. Customers form a single queue and are served on a first-come-first-serve basis.

1. On average, how many customers are waiting in line (to receive service) under the Independent Queues configuration (configuration I)? If your answer is not an integer, provide at least three decimal places, e.g., 7.500.

2. How long does the entire checkout process (waiting in line and then checking out) take for an average customer (in minutes) under the Independent Queue configuration (configuration I)? If your answer is not an integer, provide at least three decimal places, e.g., 7.500.

3. On average, how many customers are waiting in line (to receive service) under the Pooled Queue configuration (configuration II)? If your answer is not an integer, provide at least three decimal places, e.g., 7.500.

4. How long does the entire checkout process (waiting in line and then checking out) take for an average customer (in minutes) under the Pooled Queue configuration (configuration II)? If your answer is not an integer, provide at least three decimal places, e.g., 7.500.

5. On average, how many customers are waiting in line (to receive service) under the Quick Service configuration (configuration III)? If your answer is not an integer, provide at least three decimal places, e.g., 7.500.

6. How long does the entire checkout process (waiting in line and then checking out) take for an average customer (in minutes) under the Quick Service configuration (configuration III)? If your answer is not an integer, provide at least three decimal places, e.g., 7.500.