I would like to know if there is a name for a type of process that works as follows: Suppose we have a Poisson process with rate paramter $\lambda$. At each new arrival at time $t$ of this process, a new sub-process begins, which is also Poisson with some new parameter $\lambda_t$. This parameter can be deterministic or generated via some distribution. So this gives us many Possion processes that are running in parallel.
Is anything know about this type of process? Can it be described via one large Poisson process?