# How to model this queueing process

I need help with the following problem. In my eyes, the description of it is a bit sloppy/unclear, so hopefully someone can help me figure out how the related questions can be answered satisfactorily.

You have been hired by an international airport to simulate the check-in process during the forenoon. It can be assumed, that the time with the highest arrival rate is approximately normally distributed with a mean at 11 pm and a standard deviation of 45 minutes. You, as a manager are interested in the simulation of a scenario, in which 275 people arrive during this highest arrival rate. The check-in time (waiting time not included) at the counter depends on the experience of the employees. Assume a triangular distribution for the service duration with minimum = 1 minute, maximum = 12 minutes, and mode = 4 minutes.

I am then asked the following questions:

ai) Model the arrival time and service duration of 275 passengers.
aii) Simulate the arrival time and service time for one of the 275 passengers 5000 times and plot the results in two separate histograms to verify your model. Do the histograms represent the given distributions?
b) How can the mean of the triangular distribution be calculated in theory? Show that your simulation yields a comparable value.
c) Why is it important to draw separate random numbers for the arrival and the service process?
d) Simulate how many passengers (average and maximum) will be at the counter at the same time. Assume an infinite number of employees, i.e., the check-in time of each passenger can begin immediately after their arrival. Based on the average and maximum amount of passengers report the summary statistics min, max, mean, and standard deviation after 2000 Monte Carlo Replications.

What is confusing to me in this task description is that only the distribution of the time with the highest arrival rate is given. Nowhere is anything said about the arrival rate (or its distribution) in general. I only know that the highest arrival rate should be 275 people who arrive at the airport exactly at time $$T\sim N(11,0.75)$$. Then, every passenger $$i$$ is checked in with check-in time $$C_i\sim Triangular(1,12,4)$$

Thus, the arrival time of all 275 is the same and the service duration of all 275 passengers is the wait time and the check in time of the last passenger. So for ai) I got the following answer: Arrival Time $$T\sim N(11,0.75)$$ and Service Time $$S=\sum_{i=1}^{150}C_i$$

But aii) then gives me a hard time as I have no idea how I can verify my model of the arrival time with a simulation of one single passenger since I only have information of 275 passengers arriving at the same time. Same problem goes for d).

Maybe I am just stupid, but I spent quite some time now thinking about this and couldn't come up with an answer. Hopefully someone can help me solve these questions.