Renewal reward process problems

Question

Suppose that customers arrive at a single-server system in accordance with a Poisson process with rate λ. Upon arriving a customer must pass through a door that leads to the server. However, each time someone passes through, the door becomes locked for the next t units of time. An arrival finding a locked door is lost, and a cost c is incurred by the system. An arrival finding the door unlocked passes through to the server. If the server is free, the customer enters service; if the server is busy, the customer departs without service and a cost K is incurred. The service time of a customer is exponential with rate μ.

1. (A)Find the average cost per unit time incurred by the system.
2. (B)Find the long-run proportion of time that the server is busy.

The questions are referenced from here (https://www2.isye.gatech.edu/~sman/courses/6761/hw5f12solns.pdf)

I have read the reference, but some specific detail is missed. For example, I don't know how to derive E[C2] = K*e^(-ut)*(λ/λ+u) and E[B − Y] = -1/u*P(Y>B)=-1/u*(λ/λ+u)*e^(-ut)

I think both of these consider conditional probability, but I don't know what kind of condition we need to consider. I would like to know more about this detail or other solutions to the problems.

Complement for problems:

1. E[C2] = K*e^(-ut)*(λ/λ+u): I know the E[C2] is total fees from start t to end service time with customer who finding the door unlocked but the server busy. K is per customer fees without using service. So, E[C2] = K*Number of the customer without using service and leaving. However, I don't know what Number of the customer without using service and leaving equals to e^(-ut)*(λ/λ+u). What is meaning of the two items?
2. E[B−Y]: I think E[B−Y] is service time that a customer has already taken the time because E[Y] is the remaining service time of the person in service from the reference. The E[Y] equals to 1/u because of memoryless of exponential with rate μ, but I don't know why E[B − Y] = -1/u*P(Y>B)=-1/u*(λ/λ+u)*e^(-ut). Why -1/u multiplied by P(Y>B) equals to E[B − Y]? How to or any explanation to the above equation?