From Burke's theorem, we know that for the M/M/$\infty$ queue in the steady-state with arrivals a Poisson process with rate parameter $\lambda$ and service time exponentially distributed with parameter $\delta$ (the mean of the exponential distribution is $EX = \frac{1}{\delta}$), the departure process from the server is a Poisson process with parameter $\lambda$.
I want to, however, determine what is the departure process in the following case: a server receives $x$ packets at the same time and the time a packet is processed by the server is described by a random variable from an exponential distribution with parameter $\delta$ (i.e., each packet has a different random service time). How can I model the departure process? Is it the case that the departure process is a Poisson process with parameter $\delta$? Or am I completely wrong?
Also, how would that change in case I have 3 chained servers, the first server receives $x$ packets at the same time, each packet is serviced by the server for a random amount of time (selected from the exponential distribution), and then forwarded to the next server, where it is again serviced for a random amount of time, and then forwarded to the last server, where the situation repeats. Is the departure process from the last server a Poisson process with parameter $\delta$?