# Queuing Theory with Poisson Distribution

Suppose customers arrive in a one-server queue according to a Poisson distribution with rate lambda=1 (in hours). Suppose that the service times equal 1/4 hour, 1/2 hour, or one hour each with probability 1/3.

(a) Assume that the queue is empty and a customer arrives. What is the expected amount of time until that customer leaves?

(b) Assume that the queue is empty and a customer arrives. What is the expected amount of time until the queue is empty again?

(c) At a large time t what is the probability that there are no customers in the queue?

I'm trying to do couple of practice problems involving queuing before my exam and I am really confused, I would really appreciate it if someone can show me how to do this problem.

Thanks

• We usually use $\mu$ to refer to the service rate, not the mean service duration. – Brian Tung Jul 17 '18 at 20:18