[Paper Review] A Mathematical Model of Population Growth as a Queuing System

Mariia Nosova|arXiv (Cornell University)|May 21, 2020

Insurance, Mortality, Demography, Risk Management22 references4 citations

TL;DR

This paper proposes a novel stochastic queuing model of human population growth as an autonomous non-Markovian system with unlimited servers and two application types (e.g., births and deaths). Using the virtual phase method and asymptotic analysis of age-specific density, it proves that the number of served applications asymptotically follows a Gaussian distribution, deriving explicit expressions for mean, variance, and covariance of population size over time.

ABSTRACT

In this article, a new mathematical model of human population growth as an autonomous non-Markov queuing system with an unlimited number of servers and two types of applications is proposed. The research of this system was carried out a virtual phase method and a modified method of asymptotic analysis of a stochastic age-specific density for a number of applications served in the system at time t and was proofed that the asymptotic distribution is Gaussian. The main probabilistic characteristics of this distribution are found. The mathematical model and methods of its research can be applied to the analysis of the population growth both in a single country and around the world.

Motivation & Objective

• To develop a new mathematical framework for modeling human population growth using queuing theory.
• To analyze demographic dynamics as a stochastic process with age-specific intensities of fertility and mortality.
• To establish the asymptotic distribution of the number of individuals served in the system over time.
• To derive explicit analytical expressions for the first and second moments of the population density.
• To enable forecasting of population size, age structure, and migration using a generalized, analyzable model.

Proposed method

• Modeling population growth as an autonomous non-Markov queuing system with two types of applications: births and deaths.
• Applying the virtual phase method to approximate service time distributions via sums of exponentially distributed phases.
• Using a modified asymptotic analysis technique to study the stochastic age-specific density of individuals served at time t.
• Deriving explicit analytical solutions for the mean, variance, and covariance of the number of applications served.
• Formulating integral equations for the second moments of the population density using survival and fertility functions.
• Proving that the asymptotic distribution of the number of served applications converges to a Gaussian distribution.

Experimental results

Research questions

• RQ1 How can human population growth be modeled as a non-Markovian queuing system with unlimited servers and two application types?
• RQ2 What are the asymptotic statistical properties of the number of individuals served in the system over time?
• RQ3 Does the distribution of the population size converge to a Gaussian distribution under the proposed model?
• RQ4 What are the explicit expressions for the mean and second-order moments of the population density in this stochastic system?
• RQ5 How do fertility and mortality intensities influence the long-term behavior of the population distribution?

Key findings

• The asymptotic distribution of the number of applications served in the system is proven to be Gaussian, providing a foundation for probabilistic forecasting.
• Explicit analytical expressions are derived for the mean and variance of the population size at time t, depending on fertility and survival functions.
• The variance of the number of individuals in the system is expressed through integrals involving the fertility function, survival function, and their derivatives.
• The covariance between different population components (e.g., births and deaths) is derived using joint moment equations.
• The model allows for the incorporation of time-varying fertility and mortality intensities through the use of time-series functions.
• The derived solutions are general and applicable to both national and global demographic forecasting, including age structure and social status considerations.