A residential construction company has opened a registration office. Customers arrive at the rate of $200$ per hour $($Poisson Distribution$)$ and the cost of their waiting in the queue is estimated $$100$ per person per hour. The local convention bureau provides servers to register customers at the fee of $$15$ per person per hour. Registration process takes $1$ minute $($Exponential Distribution$)$ and a single waiting line, with multiple servers is set up$.$

a$)$ Compute the minimum number of servers for this system $(3$ marks$).$

b$)$ Compute the optimal number of servers for this system $(8$ marks$).$

c$)$ What is the cost of system $($per hour$)$ at the optimum number of servers? $(2$ marks$)$

d$)$ Compute the server utilization rate with the minimum number of servers. $(3$ marks$)$

e$)$ A new registration manager is hired who initiates a program to entertain the people in line with juggler whom she pays $$15/$hour$.$ This reduces the waiting cost to $$50/$hour$.$ What is the optimal number of servers and the cost of that in this case? $(9$ marks$)$