On the Generalized Birth-Death Process and its Linear Versions

TL;DR

This paper introduces a generalized birth-death process and its linear variants, deriving their probabilistic properties, differential equations, and applications, including an analysis of immigration effects and a parking management system example.

Contribution

It provides new analytical results for the generalized birth-death process and its linear version, including explicit formulas, differential equations, and an application to real-world systems.

Findings

Derived differential equations for state probabilities

Obtained explicit forms of distributions under constant rates

Analyzed the impact of immigration in the process

Abstract

In this paper, we consider a generalized birth-death process (GBDP) and examined its linear versions. Using its transition probabilities, we obtain the system of differential equations that governs its state probabilities. The distribution function of its waiting-time in state $s$ given that it starts in state $s$ is obtained. For a linear version of it, namely, the generalized linear birth-death process (GLBDP), we obtain the probability generating function, mean, variance and the probability of ultimate extinction of population. Also, we obtain the maximum likelihood estimate of one of its parameter. The differential equations that govern the joint cumulant generating functions of the population size with cumulative births and cumulative deaths are derived. In the case of constant birth and death rates in GBDP, the explicit forms of the state probabilities, joint probability mass functions of population size with cumulative births and cumulative deaths, and their marginal probability mass functions are obtained. It is shown that the Laplace transform of a stochastic integral of GBDP satisfies its Kolmogorov backward equation with certain scaled parameters. Also, the first two moments of the stochastic path integral of GLBDP are obtained. Later, we consider the immigration effect in GLBDP for two different cases. An application of a linear version of GBDP and its stochastic path integral to vehicles parking management system is discussed.