You have discrete event simulation in Kendall’s notation a M/G/3 model. Since you have a memory-less arrival process, 3 servers and the service time is not exponentially distributed. In R you “need” a priority queue for the events, rpois and runif for the events and service times. You sample an event and put it into the queue.
I don’t see what it has to do with Stan? There is a way with Stochastic Differential Equations. Thus if
you transform the problem, it might be put into Stan ODE solver.