1. Collect data on the call arrival rates throughout different time periods of the day. (350 calls daily)
2. Record the service times for each call, including the time taken to resolve the customer's inquiry. (4 minutes, 22 seconds per call)
3. Measure the number of servers available at the call center during different shifts. (AM Shift: 12 servers, PM Shift: 4 servers)
4. Assume the call center is open 24/7, 365 days a year. AM shift is defined as 6am-6pm and PM shift is defined as 6pm-6am.

Case Study Questions:

1. Modeling Call Arrivals:

1. Explain the Poisson process and its relevance to modeling call arrivals at the call center.
2. Using the collected data, determine the average call arrival rate () for different time periods of the day.
3. Construct a Poisson distribution model for call arrivals based on the calculated arrival rates.

2. Discrete Random Variables:

1. Define a discrete random variable and discuss its application in the context of call center operations.
2. Explain how the number of calls arriving within a specific time interval can be modeled as a discrete random variable.
3. Calculate the probability mass function (PMF) for the number of calls arriving within a given time interval using the Poisson distribution model.

3. Random Number Generation:

1. Describe the concept of random number generation and its significance in simulating call center operations.
2. Implement a random number generator in a computer program of your choice to simulate call arrivals based on the Poisson distribution model.
3. Generate a series of random numbers to simulate call arrivals during a specific time period and analyze the distribution of the generated numbers.

4. Multi-Server Queue Analysis:

1. Define the concept of a multi-server queue and explain its applicability to call center scenarios.
2. Derive the formulas for calculating the average number of customers in the system, the average number of customers in the queue, and the average waiting time for a multi-server queue.
3. Apply the derived formulas to the call center's scenario, considering the number of servers available and the average service time per call.

5. Optimizing Call Center Operations:

1. Propose strategies for optimizing call center operations to minimize customer wait times.
2. Discuss the advantages and limitations of increasing the number of servers at the call center.
3. Evaluate the impact of varying the number of servers on the average waiting time and customer satisfaction using the multi-server queue model.

6. Sensitivity Analysis:

1. Explain the concept of sensitivity analysis and its relevance to the call center's optimization.
2. Identify key factors or parameters that could influence the performance of the call center's operations.
3. Conduct a sensitivity analysis by varying the call arrival rates, number of servers, or service times and analyze their impact on customer wait times.

7. Real-world Implementation:

1. Develop a plan for implementing the recommended improvements based on the analysis conducted.
2. Discuss the challenges and potential risks associated with implementing changes to the call center's operations.
3. Propose a monitoring and evaluation framework to assess the effectiveness of the implemented changes and ensure ongoing optimization.