In the general framework of $M/M/1$ queue we have rate $\lambda$ and an exponential service time $\mu$, we can set up the transition rate matrix intuitively. However, if the service times satisfy shifted exponential distribution, i.e. they will always take some constant time plus some exponentially distributed random variable the case becomes less intuitive, since the memoryless property disappears when we shift the exponential distribution.
I would like to know the expected value and variance of number of customers in my queue.
Is there any simple model I´m unaware of that describes this behavior to find answers to my simple questions or should I just simulate?