A note from Roberto: 'While preparing both recitations, I was under the impression that the triangular distribution is defined in terms of its minimum, mean and maximum values. This, however, is incorrect: it is defined in terms of its minimum, mode and maximum values. I am very sorry about this mistake: I should have known this. Unfortunately, I discovered this error too late to fix it in the videos. So$,$ every time in the recitation videos when I mention the value of a mean or average delay in a problem that I will then enter as part of a triangular distribution in the software, that value is actually a mode, not a mean. Apologies. In the problem below, and in the graded assignments, the correct term $($mode$,$ not mean$)$ is used."

The Windsor Bank in Eritrea is opening a new branch in the city of Keren, which will feature both ATMs and tellers to serve the bank's clients. The exact number of ATMs and tellers that will be available at any given time to serve the clients of this branch has not been decided yet, and is the subject of this problem.

Based on historical data from other branches in the city of Asmara, and some general information about the population and economic activity of Keren, experts with Windsor Bank estimates that the rate of arrivals of clients at the Keren branch will be around $40$ per hour. $($Assume that inter$-$arrival times follow an exponential distribution.$)$ Among these clients, the bank estimates that:

a third will only need to use an ATM, and will then leave,

another third will only need to see a teller, and will then leave, and

another third will need to first use the ATM and then see a teller, and only then leave.

Based on observations conducted in their Asmara branches $($which are considered applicable to the Keren branch, too$)$ over a period of one month, the Windsor Bank estimates that:

the time it takes a client to use the ATM, once it is available, follows a triangular distribution with a minimum of $2$ minutes, a mode of $10$ minutes, and a maximum of $20$ minutes, and

the time it takes a client to complete their business with a teller, once in front of the teller, follows a triangular distribution with a minimum of $3$ minutes, a mode of $10$ minutes, and a maximum of $30$ minutes.

The Windsor Bank prides itself in offering a good experience to its clients. To ensure prompt service, the bank wants to have enough ATMs and tellers to provide their clients with prompt service. Ideally, to ensure the clients are satisfied:

the average time spent waiting in line for an ATM to become available should be around $5$ minutes or so$,$ at most, and

the sum of the time spent waiting in line for an available teller plus the time spent being served by that teller, on average, should be around $15$ minutes or so$,$ at most.

While respecting these average times, the bank also wants to maintain costs reasonable: the bank does not want to install an unnecessarily large number of ATMs, which are very expensive, or to hire more tellers than are needed, which are also expensive $($although significantly less expensive than the ATMs$).$

Create a discrete$-$event simulation of the clients arriving at the bank, using the ATM and teller services as described above, and then leaving the bank. Model the ATMs and tellers as resources, since we will ask about utilization later. Do a simulation run from time $0$ to time $72,000$ minutes.