Uniqueness, recurrence and decay properties of collision branching processes with immigration

We consider a kind of Collision Branching Processes with Immigration (CBIP). Some important properties of the generating functions for the CBI q-matrix were first investigated in detail. Then for any given CBI q-matrix, the existence and uniqueness of CBIP is proved, and sufficient and easily checked conditions for the CBIP to be recurrent are given. Moreover, the exact value of the decay parameter λZ is obtained and expressed explicitly for the communicating class Z+ in the case that the immigration is independent of states. It is shown that this λZ can be directly obtained from the generating functions of the corresponding q-matrix. Finally, the invariant vectors and invariant measures are considered.