For an M/M/c/c queueing system (when it's at equilibrium) the mean sojourn time of each state can be calculated using the diagonal entries of its transition rate matrix (or the infinitesimal generator matrix). Because the transition rates remain constant for exponential distributions.
However, my question is when the service time follows a general distribution (say, a Gamma distribution with mean E), then the departure rate is now depending on how much time the system has been spending on the current state. And when the system is at equilibrium, the departure rate for each customer is not necessarily equal to 1/E. So my question is: is there a way to calculate (or approximate) the (long run) mean sojourn time of this M/G/c/c queueing system? Or more generally, for any non-Markovian chain?
I would really appreciate your suggestions/comments! Thanks in advance!
M/G/c/c :the last 2 letters indicate that there are c servers and max c customers in the queue.
So your queueing system is of a specific (degenerate) type (there is no queue, only servers). Either a packet gets in (when not all c servers are occupied) and has sojourn time equal to the service time (f.i. Gamma in your example) or it is dropped (when all servers are busy) and has an undefined sojourn time.