Question:
Suppose, there is a queue A having i customers initially. The service time of the queue is Exponentially distributed with parameter $\mu$ (i.e., mean service time of one customer in Queue A is 1/$\mu$). Arrival to queue A has Poisson distribution with mean $\lambda$. What is the expected time before queue A becomes empty for the first time?
My approach:
For the queue to be empty, service time of j customers should be less than the arrival time of $(j+1)^{th}$ customer (for j>=i).
=...
How do I proceed after this since the sums are Erlang distributed and I'm not able to tackle this.