I am trying to model a queuing system, where the arrival process is deterministic and the service process is exponential, thus resulting in a D/M/1 queue. In that case the main factor for the rate at which messages come out of the system is the service process. I am interested in the variance of the time between messages leaving the system, a.k.a. the variance of the sojourn time. However I could not find any formulas for the variation of the sojourn time of a D/M/1 queue. What I found was the variance of the sojourn time of a M/M/1 queue:
$$ \sigma^2(t_s) = \frac{1}{(1-\rho)^2} \times \frac{1}{\mu^2} $$
However, I suspect it wont work in the case of D/M/1, since the arrival process is different. Can you point me to some open-source resources, where this has been discussed? Or provide the formula for a D/M/1 queue.