Hogwarts University allows students and faculty to access its supercomputer by high$-$speed proxy servers. The university has $15$ proxy server connections that can be used. When all of the proxy server connections are in use, the system can keep up to $10$ users in a queue waiting for a connection to become available. If all $15$ connections are in use, and $10$ users are already holding, any new users are rejected. Requests to the proxy server pool follow a Poisson distribution and occur at an average rate of $60$ per hour. The length of each session with the supercomputer is an exponential random variable with a mean of $15$ minutes$-$therefore, each proxy server services an average of four users per hour $($use Q$.$xlsx$).$

a$)(20$ points$)$ On average, how many users are in the queue waiting for a connection?

b$)(20$ points$)$ On average, how long is a user kept in the queue before receiving a connection?

c$)(20$ points$)$ How many connections would the university need to add to its server pool in order for there to be no more than a $1\%$ chance of a user being rejected?