The First American bank is trying to determine whether it should install one or two drive-through windows. The following probability distributions for arrivals and service times have been developed from historical data

Time between
Automobile arrivals in (min) Probability
1 .20
2 .60
3 .10
4 .10
1.00

Service Time (min) Probability
2 .10
3 .40
4 .20
5 .20
6 .10
1.00

Assume that in the two-server system an arriving car will join the shorter queue. When queues are of equal length there is a 50-50 chance the driver will enter the queue for either window.

a. Simulate both the one- and the two-teller systems. Compute the average queue length, waiting time, and percentage of utilization for each system.

To simulate both the one- and two-teller systems and compute the average queue length, waiting time, and percentage of utilization for each system, we can use a simulation model. Here's how you can approach the problem step-by-step:

1. Start by setting up the simulation environment. Define the variables and parameters you will need, such as the number of customers, arrival and service time distributions, queue lengths, and waiting times.

2. Generate a random number for each customer to determine their inter-arrival time. You can do this by using a random number generator and the given probability distribution for arrival times.

3. Determine the arrival times for each customer by accumulating the inter-arrival times.

4. Simulate the arrival of each customer by adding them to the appropriate queue based on the current state of the system (e.g., one-teller or two-teller).

5. Generate a random number for each customer to determine their service time. Use the given probability distribution for service times and a random number generator.

6. Start serving customers from the queues. If there is only one teller, serve customers in a first-come-first-serve manner. If there are two tellers, serve customers from the shorter queue and alternate between the queues when they have the same length.

7. Keep track of the queue lengths, waiting times, and service times for each customer.

8. Repeat steps 4-7 until all customers have been served.

9. Calculate the average queue length by summing up the queue lengths at each time observation and dividing by the total number of observations.

10. Calculate the average waiting time by summing up the waiting times for all customers and dividing by the total number of customers.

11. Calculate the percentage of utilization for each teller by dividing the total service time by the total simulation time (number of customers multiplied by the average service time).

By following these steps and running the simulation multiple times, you can obtain the average queue length, waiting time, and percentage of utilization for both the one-teller and two-teller systems.