Consider an infinite server queue with channels $($servers$)$ numbered $1,2,....$ Customers arrive according to a Poisson process with rate $\backslash$lambda $.$ On arrival, a customer will choose the lowest numbered channel that is free. Thus, we can think of all arrivals as occurring at Channel $1,$ Those who find Channel $1$ busy overflow and become arrivals at Channel $2.$ Those finding both channels $1$ and $2$ busy overflow Channel $2$ and become arrivals at Channel $3,$ and so on$.$ The service times are indepenent exponentially distributed variables with mean m$.$

$$ What fraction of time is Channel $1$ busy? $$ What is the overflow rate from Channel k$?$