You are asked to perform a capacity analysis for a bank. The current resources of the bank are as follows:

Drive$-$through : Service providers: $3|$ Processing time $($mins$)$: $5$

Teller: Service providers: $5|$ Processing time $($mins$)$: $10$

ATM: Service providers: $3|$ Processing time $($mins$)$: $5$

Specialist: Service providers: $1|$ Processing time $($mins$)$: $30$

In addition, the following information is presented to you on the operations of the bank:

$$ On average, $50\%$ of the customers use the drive$-$through lanes without leaving their cars. Customers can perform tasks such as making a withdrawal, paying bills, or depositing a check via the drive$-$through lanes. There are $3$ identical drive$-$through lanes currently operating and the average processing time of a customer is $5$ minutes in each of these lanes.

$$ On average, $70\%$ of the customers who walk into the bank prefer talking to a bank teller and the remaining $30\%$ prefer the ATM, which is located inside the store. There are $5$ bank tellers and each of them services a customer in $10$ minutes on average. There are $3$ ATMs, and it takes $5$ minutes on average for a customer to complete service via ATM.

$$ After talking to a bank teller, $30\%$ of the customers are directed to a specialist because the bank tellers cannot perform operations such as opening a new account or applying for a credit card. There is $1$ specialist in the bank, and it takes an average of $30$ minutes for a specialist to complete the service for a customer.

a$.$ What is the capacity of each resource in the bank $($drive$-$through lanes, tellers, ATM, and specialist$)$ in number of customer$-$arrivals per hour?

b$.$ What is the system capacity of the bank in number of customer$-$arrivals per hour? Identify the bottleneck$($s$).$ What is the slack of each resource in total number of customer $-$arrivals per hour?

c$.$ If the bank can add $4$ additional unit of resource per bottleneck resource$($s$),$ what would be the new system capacity and bottleneck$($s$)?$