Main Article Content
Abstract
The probability generating function (PGF) of a discrete random variable is a concise way to describe the corresponding probability distribution and facilitate analysis. This paper will revisit and explore certain key properties of the PGFs, focusing on structural properties, moment characterization and stability under convolution. Through applications to distribution generation, branching processes, and queueing models, PGFs are shown to provide clear insight into extinction probabilities and steady-state behavior. The results confirm that probability generating functions provide a coherent and useful framework for both theory and applications in discrete stochastic modelling.
Keywords
Article Details
Copyright (c) 2026 Alfred Ayo Ayenigba, Olutunde Michael Ajao, Wisdom Chimaobi OKPECHI
This work is licensed under a Creative Commons Attribution 4.0 International License.
References
- Yechiali, U., Tandem stochastic systems: Jackson networks, asymmetric exclusion processes, asymmetric inclusion processes and Catalan numbers. Journal of Applied Probability, 56(3), 1-25, 2019.
- Conway, J., & Coombs, D., A stochastic model of latently infected cell reactivation and viral blip generation in treated HIV patients. PLoS Computational Biology, 7(4), e1002033, 2011.
- Saichev, A., & Sornette, D., Anomalous power law distribution of total lifetimes of branching processes: Application to earthquake aftershock sequences. Physical Review E, 70(4), 046123, 2004.
- Nikolaev, D. I., Beschastnyi, V., & Gaidamaka, Y. V., Two-queue polling system as a model of an integrated access and backhaul network node in half-duplex mode. Discrete and Continuous Models and Applied Computational Science, 32(4), 2024.
- Lanzoni, D., Pierre-Louis, O., & Montalenti, F., Accurate generation of stochastic dynamics based on multi-model generative adversarial networks. The Journal of Chemical Physics, 158(18), 2023.
- Hamze, F., & Andryash, E., Discrete equilibrium sampling with arbitrary nonequilibrium processes. Proceedings of the International Conference on Machine Learning, 15, 1-12, 2015.
- Feller, W., An introduction to probability theory and its applications (3rd ed., Vol. 1). John Wiley & Sons, 1968.
- Keating, Leah A., Gleeson, James P., and O’Sullivan, David J. P., A generating-function approach to modelling complex contagion on clustered networks with multi-type branching processes. Journal of Complex Networks, 11(6), cnad042, 2023.
- Eryilmaz, Serkan., Discrete Stochastic Models and Applications for Reliability Engineering and Statistical Quality Control. Boca Raton: CRC Press/Taylor & Francis Group, 2023.
- Eliwa, Mohamed S., Alghanem, Weed E., and Alenizi, Manahel A., A Novel Discrete Probability Distribution with Theoretical and Inferential Insights: Cutting-Edge Approaches to Sustainable Dispersion Data Modeling. European Journal of Pure and Applied Mathematics, 18(2), 6144–6160, 2025.
- Saini, Balveer, Singh, Dharamender, and Sharma, Kailash Chand., Application of Queueing Theory to Analyze the Performance Metrics of Manufacturing Systems. Asian Research Journal of Mathematics, 20(12), 2024.
- Abdulazeez, Sikiru Adeyinka., Branching Process Modelling: A Tool for Deciphering Complex Real-World Phenomena Using R Programming. FUDMA Journal of Sciences (FJS), 7(3), 2023.
- Wilf, H. S., generatingfunctionology (3rd ed.). Wellesley, MA: A K Peters/CRC Press, 2006.
- Harris, T. E., The Theory of Branching Processes. Berlin: Springer-Verlag, 1963.
- Grimmett, G.R., & Stirzaker, D. R., Probability and Random Processes (4th ed.). Oxford: Oxford University Press, 2020.
- Kleinrock, L., Queueing Systems, Volume 1: Theory. New York: Wiley-Interscience, 1975.
References
Yechiali, U., Tandem stochastic systems: Jackson networks, asymmetric exclusion processes, asymmetric inclusion processes and Catalan numbers. Journal of Applied Probability, 56(3), 1-25, 2019.
Conway, J., & Coombs, D., A stochastic model of latently infected cell reactivation and viral blip generation in treated HIV patients. PLoS Computational Biology, 7(4), e1002033, 2011.
Saichev, A., & Sornette, D., Anomalous power law distribution of total lifetimes of branching processes: Application to earthquake aftershock sequences. Physical Review E, 70(4), 046123, 2004.
Nikolaev, D. I., Beschastnyi, V., & Gaidamaka, Y. V., Two-queue polling system as a model of an integrated access and backhaul network node in half-duplex mode. Discrete and Continuous Models and Applied Computational Science, 32(4), 2024.
Lanzoni, D., Pierre-Louis, O., & Montalenti, F., Accurate generation of stochastic dynamics based on multi-model generative adversarial networks. The Journal of Chemical Physics, 158(18), 2023.
Hamze, F., & Andryash, E., Discrete equilibrium sampling with arbitrary nonequilibrium processes. Proceedings of the International Conference on Machine Learning, 15, 1-12, 2015.
Feller, W., An introduction to probability theory and its applications (3rd ed., Vol. 1). John Wiley & Sons, 1968.
Keating, Leah A., Gleeson, James P., and O’Sullivan, David J. P., A generating-function approach to modelling complex contagion on clustered networks with multi-type branching processes. Journal of Complex Networks, 11(6), cnad042, 2023.
Eryilmaz, Serkan., Discrete Stochastic Models and Applications for Reliability Engineering and Statistical Quality Control. Boca Raton: CRC Press/Taylor & Francis Group, 2023.
Eliwa, Mohamed S., Alghanem, Weed E., and Alenizi, Manahel A., A Novel Discrete Probability Distribution with Theoretical and Inferential Insights: Cutting-Edge Approaches to Sustainable Dispersion Data Modeling. European Journal of Pure and Applied Mathematics, 18(2), 6144–6160, 2025.
Saini, Balveer, Singh, Dharamender, and Sharma, Kailash Chand., Application of Queueing Theory to Analyze the Performance Metrics of Manufacturing Systems. Asian Research Journal of Mathematics, 20(12), 2024.
Abdulazeez, Sikiru Adeyinka., Branching Process Modelling: A Tool for Deciphering Complex Real-World Phenomena Using R Programming. FUDMA Journal of Sciences (FJS), 7(3), 2023.
Wilf, H. S., generatingfunctionology (3rd ed.). Wellesley, MA: A K Peters/CRC Press, 2006.
Harris, T. E., The Theory of Branching Processes. Berlin: Springer-Verlag, 1963.
Grimmett, G.R., & Stirzaker, D. R., Probability and Random Processes (4th ed.). Oxford: Oxford University Press, 2020.
Kleinrock, L., Queueing Systems, Volume 1: Theory. New York: Wiley-Interscience, 1975.