I have a queing system of type $M/M/n/\infty$.
The service time is exponential, and the arrival process is poisson.
I do understand that because of these two facts the future of the system in question does not depend on its past when its present is known and hence the system is markovian. (the property of exponential distribution)
However I dislike this kind of explanation because I don't find it strict enough.
Is there any book in which at least an idea of the strict proof is presented?
I'm not really sure how much stricter you can get.
By definition, a stochastic process is Markovian if its transition rates at any given point in time depend only on its state at the point in time.
In your queuing process, the state is the size of the queue, and the transitions consist of arrivals and departures. Since, by assumption, the arrival and departure events occur independently of each other, we only need to show that the arrival and departure rates each only depend on the state of the queue.
Since the arrival process in Poisson, arrivals occur at a constant rate (independently of the size of the queue, in fact), regardless of when then previous arrival occurred.
Since the service time is exponential, departures occur at a constant rate (which is zero if the queue is empty, and non-zero otherwise), regardless of when then previous departure occurred.
Together, these facts mean that the transition rates of the queueing process only depend on the state of the queue (and, in fact, only on whether it is empty or not), and thus the process is Markovian.