A queueing system with Erlang incoming flow with relative priority

A one-channel queueing system with waiting delay and relative priority is considered. Incoming claims are separated into $r$ classes, numbered by $1,\dots,r$, each claim getting to the $i$-th class with probability $p_i$ ($i=1,\dots,r$), $\displaystyle\sum_i p_i=1$. Claims of the $i$-th class have priority with respect to those of the $j$-th class for $j>i$. Claims arrive at the input of the system according to the Erlang law. Service times are jointly independent absolutely continuous random variables. To each priority class, there corresponds a distribution function of the service time.
The behaviour of the queue size (in non-stationary regime) and busy period is studied.