Executive Summary : | The Poisson process is an important counting process mainly used in modeling the count data. Its jump intensity is constant, that is, it does not depend on the state of the process. It’s a L\'evy as well as a renewal process with exponentially distributed waiting times. It has certain limitations in modeling the phenomena with long memory due to its light-tailed distributed waiting times. To overcome these limitations, many researchers have introduced and studied various generalizations of the Poisson process in the past two decades. Some of these variants are generally characterized by their heavy-tailed distributed waiting times. Recently, a generalization of the Poisson process, namely, the generalized counting process (GCP) which performs k kinds of jumps of amplitude 1,2,...,k with positive rates \lambda_1, \lambda_2,...,\lambda_k has been introduced and studied. It is important to note that the jump intensities in GCP do not depend on its state. For k = 1, the GCP reduces to the Poisson process. We propose to study the superposition of independent GCPs, where jumps in the merged process are registered with some positive probability only when one of the merging processes is performing jumps and other merging components perform zero jump in an infinitesimal interval of length h. The merged process turns out to be a GCP with some increased rates. We obtain the differential equation that governs the state probabilities of merged process. First, we study the merging of two independent GCPs. Then extend it to the superposition of countable number of independent GCPs. We will also discuss the conditional probabilities of jump originating from particular GCP conditioning on jumps registered in the merged process. Also, we propose to study the splitting of GCP where jumps are sent to two different processes either with a independent flip of a coin or mutliple independent flips of a coin. In case of single flip of coin, all jumps registered in the original GCPs are sent to one of the splitted process, and in the case of multiple flips of a coin the flip number equals the number of jumps registered in the original GCP and the jumps are sent to the splitted processes according to results on the independent flips of coin. We expect that the resulting splitted processes are independent GCPs with reduced rates of jumps. Some real life applications of the merged and splitting GCPs will be studied. |