Poisson Process Queueing System!
Consider a two-server parallel queueing system where customers arrive according to a Poisson process with rate λ, and where the service times are exponential with rate μ. Moreover, suppose that arrivals finding both servers busy immediately depart without receiving service (such a customer is said to be lost), whereas those finding at least one free server immediately enter service and then depart when their service is completed.
a)If both servers are presently busy, find the expected time until the next customer enters the system. [answer: 1/(2μ) + 1/λ]
b)Starting empty, find the expected time until both servers are busy. [answer: 2/λ + μ /λ^2]
I'm really really confused by this problem, to the point of being desperate. :(
First of all, I think I'm a little confused about the question itself (i.e. the setup). I believe the situation here is that we have one line and two parallel servers. But then what does it mean by "customers arrive according to a Poisson process with rate λ"? Arriving at where? Arriving at the line or the servers?
Next, for part a, I don't understand how we can rigorously justify the answer. Why is it not just 1/(2μ)?
I am seriously puzzled and really need your help.
Thanks in advance!