On the Superposition and Thinning of Generalized Counting Processes

TL;DR

This paper investigates how generalized counting processes can be merged and split, revealing their properties and applications in industrial and service systems, with new results on process composition and independence.

Contribution

It introduces new theoretical results on merging and splitting of GCPs, including conditions for independence and process characteristics, extending existing knowledge.

Findings

Merged GCPs have increased arrival rates and Poisson-distributed jump packets.

Split components are GCPs with decreased jump rates and are independent.

Applications demonstrated in fishing industry and hotel booking systems.

Abstract

In this paper, 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.