On the Generalized Birth–Death Process and Its Linear Versions

P. Vishwakarma () and K. K. Kataria ()
Additional contact information
P. Vishwakarma: Indian Institute of Technology Bhilai
K. K. Kataria: Indian Institute of Technology Bhilai

Journal of Theoretical Probability, 2024, vol. 37, issue 4, 3540-3580

Abstract: Abstract In this paper, we consider a generalized birth–death process (GBDP) and examine 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 its parameters. 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 an integral of GBDP satisfies its Kolmogorov backward equation with certain scaled parameters. The first two moments of the path integral of GLBDP are obtained. Also, we consider the immigration effect in GLBDP for two different cases. An application of a linear version of GBDP and its path integral to a vehicles parking management system is discussed. Later, we introduce a time-changed version of the GBDP where time is changed via an inverse stable subordinator. We show that its state probabilities are governed by a system of fractional differential equations.

Keywords: Extinction probability; Birth–death process; Generalized birth–death process; Generalized linear birth–death process; Generalized linear birth–death process with immigration; 60J27; 60J80 (search for similar items in EconPapers)
Date: 2024
References: View complete reference list from CitEc
Citations: View citations in EconPapers (1)

Downloads: (external link)
http://link.springer.com/10.1007/s10959-024-01355-z Abstract (text/html)
Access to the full text of the articles in this series is restricted.

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:spr:jotpro:v:37:y:2024:i:4:d:10.1007_s10959-024-01355-z

Ordering information: This journal article can be ordered from
https://www.springer.com/journal/10959

DOI: 10.1007/s10959-024-01355-z

Access Statistics for this article

Journal of Theoretical Probability is currently edited by Andrea Monica

More articles in Journal of Theoretical Probability from Springer
Bibliographic data for series maintained by Sonal Shukla () and Springer Nature Abstracting and Indexing ().