Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

On the Generalized Birth-Death Process and its Linear Versions

P. Vishwakarma, K. K. Kataria

TL;DR

The paper extends classical birth-death processes to a generalized framework (GBDP) that allows multiple births or deaths in an infinitesimal interval and analyzes its linear specialization (GLBDP) with state-dependent linear rates. It develops a comprehensive analytic toolkit including systems of differential equations for state probabilities, waiting-time distributions, probability generating functions, and cumulant generating functions, as well as stochastic path integrals and their moments. Key contributions include explicit results for GLBDP under linear and constant-rate regimes, parameter estimation for the GLBDP, and detailed treatment of immigration effects, both at zero and all states, plus an application to a parking-system with capacity constraints. The work provides exact closed-form expressions and generating-function approaches that facilitate extinction analysis, joint distributions of cumulative births/deaths, and path-functionals relevant to population dynamics and logistics-like parking systems, with potential applicability to engineering and operations research problems.

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.

On the Generalized Birth-Death Process and its Linear Versions

TL;DR

The paper extends classical birth-death processes to a generalized framework (GBDP) that allows multiple births or deaths in an infinitesimal interval and analyzes its linear specialization (GLBDP) with state-dependent linear rates. It develops a comprehensive analytic toolkit including systems of differential equations for state probabilities, waiting-time distributions, probability generating functions, and cumulant generating functions, as well as stochastic path integrals and their moments. Key contributions include explicit results for GLBDP under linear and constant-rate regimes, parameter estimation for the GLBDP, and detailed treatment of immigration effects, both at zero and all states, plus an application to a parking-system with capacity constraints. The work provides exact closed-form expressions and generating-function approaches that facilitate extinction analysis, joint distributions of cumulative births/deaths, and path-functionals relevant to population dynamics and logistics-like parking systems, with potential applicability to engineering and operations research problems.

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 given that it starts in state 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.
Paper Structure (15 sections, 12 theorems, 173 equations, 7 figures)

This paper contains 15 sections, 12 theorems, 173 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Generalized birth-death process
  3. A linear case of GBDP
  4. Parameter estimation in GLBDP
  5. GBDP with constant birth and death rates
  6. Cumulative births and deaths in GLBDP
  7. GBDP with constant birth and death rates
  8. Stochastic integral of GBDP
  9. Stochastic path integral of GBDP
  10. A linear case of GBDP
  11. Stochastic path integral of GBDP in the case of constant birth and death rates
  12. Immigration effect in GLBDP
  13. Immigration effect at state $n=0$ only
  14. Immigration effect at any state
  15. Application to vehicles parking management system

Key Result

Proposition 2.1

Let $W_s(t)=\mathrm{Pr}\{T_s\ge t\}.$ Then, $W_s(t)$ solves the following linear differential equation: with initial condition $W_s(0)=1.$

Figures (7)

  • Figure 1: Cumulative births in GLBDP for $k_1=k_2=2$ at $t=1$ .
  • Figure 2: Correlation between $N(t)$ and $B(t)$ for $k_1=k_2=2$
  • Figure 3: Cumulative deaths in GLBDP for $k_1=k_2=2$ at time $t=1$.
  • Figure 4: Correlation between $N(t)$ and $D(t)$ for $k_1=k_2=2$
  • Figure 5: Values of stochastic path integral of GLBDP at $t=1$.
  • ...and 2 more figures

Theorems & Definitions (26)

  • Proposition 2.1
  • proof
  • Proposition 2.2
  • proof
  • proof
  • proof
  • proof
  • Proposition 3.1
  • proof
  • Proposition 3.2
  • ...and 16 more