The infinite server queue with arrivals generated by a non-homogeneous compound Poisson process and heterogeneous customers

We consider the infinite server queue where customers arrive in groups of various sizes according to a non-homogeneous Poisson process. The customers in each group may be of different types and also may have dependent service times. Assuming that the system is initially empty, we obtain the probability generating function of the transient distribution of the numbers of customers of various types present in the system, thus extending recent relevant results. At the same time the probability generating function of the corresponding departure counting process is derived as well.