Bank Drive$-$In Teller Analysis:

Question $6$: Evaluate the queuing system for the bank's drive$-$in teller, where customers arrive randomly at an average of $30$ per hour.

Calculate average arrivals per minute and per $10-$minute intervals.Use given equations to find probabilities of specific arrival counts and more than three arrivals in a $10-$minute interval.

Service Time Analysis:

Question $7$: Given the service rate of $40$ customers per hour at the bank, and an exponential distribution:

Calculate the expected service time.Determine probabilities for different service durations using provided equations.

Comprehensive Drive$-$In Teller Analysis:

Question $8$: Based on your answers to Questions $6$ and $7$:

Find probabilities related to the system being empty or customers needing to wait.Calculate average wait times, number of cars in the queue, and total time spent in the system.Propose a new service rate that could reduce the average total time in the system to $2$ minutes.