Is there a known closed-form distribution of the number of customers in a $M/Erlang-2/1$ system with service rate $2/\mu$ and $\lambda$ as the arrival rate to the system? I wonder if such a general distribution formula exists for this specific setting. Is working through the recursive procedure yield from the Pollaczek-Khintchine formula the only way about it, or did someone develop a closed-form distribution in the past?
Thanks so much if anyone is familiar with such a formula.