On the superposition and thinning of generalized counting processes

In this article, we study the merging and splitting of generalized counting processes (GCPs). First, we study the merging of a finite number of independent GCPs and then extend it to the case of countably infinite. The merged process is observed to be a GCP with increased arrival rates. It is shown that a packet of jumps arrives in the merged process according to the Poisson process. Also, we study two different types of splitting of a GCP. In the first type, we study the splitting of jumps of a GCP where the probability of simultaneous jumps in the split components is negligible. In the second type, we consider the splitting of jumps in which there is a possibility of simultaneous jumps in the split components. It is shown that the split components are GCPs with certain decreased jump rates. Moreover, the independence of split components is established. Later, we discuss applications of the obtained results to industrial fishing problem and hotel booking management system.