Length of a Busy Period of an M/M/c queue with an initial length of k

Question

The expected length of a busy period in an M/M/1 queue is $\frac{1}{\mu - \lambda}$. I believe the time to empty a busy M/M/1 queue with a current line of k is $\frac{k}{\mu - \lambda}$. Does anyone know what the equivalent expression might be for a M/M/c queue with a current line of k? If I somewhat redefine the question as the first time there exists any open server as the renewal is it just $\frac{k}{c \mu - \lambda}$? As an additional question, does anyone know of an expression for the variance or moment generating function of the length of the busy period?