Formulas for average transition times between states of the Markov birth-death process

• Autorzy: Zhernovyi Yuriy , Kopytko Bohdan

Identyfikatory

DOI

10.17512/jamcm.2021.4.09

• Języki publikacji: EN

Abstrakty

EN

In this paper, we consider Markov birth-death processes with constant intensities of transitions between neighboring states that have an ergodic property. Using the exponential distributions properties, we obtain formulas for the mean time of transition from the state i to the state j and transitions back, from the state j to the state i. We found expressions for the mean time spent outside the given state i, the mean time spent in the group of states (0,...,i-1) to the left from state i, and the mean time spent in the group of states (i+1,i+2,...) to the right. We derive the formulas for some special cases of the Markov birth-death processes, namely, for the Erlang loss system, the queueing systems with finite and with infinite waiting room and the reliability model for a recoverable system.

Słowa kluczowe

EN

birth-death process; Markov models; mean transition time; mean time spent in the group of states; queueing systems; reliability model

PL

proces narodzin i śmierci; modele Markova; średni czas przejścia; średni czas spędzony w grupie stanów; systemy kolejkowe; model niezawodności

• Wydawca: Wydawnictwo Politechniki Częstochowskiej
• Czasopismo: Journal of Applied Mathematics and Computational Mechanics
• Rocznik: 2021
• Tom: Vol. 20, nr 4
• Strony: 99--110
• Opis fizyczny: Bibliogr. 10 poz., rys.

Twórcy

autor

Zhernovyi Yuriy

• Ivan Franko National University of Lviv, Lviv, Ukraine

autor

Kopytko Bohdan

• Department of Mathematics, Czestochowa University of Technology Częstochowa, Poland

Bibliografia

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• [2] Novozhilov, A.S., Karev, G.P., & Koonin, E.V. (2006). Biological applications of the theory of birth-and-death processes. Briefings in Bioinformatics, 7(1), 70-85.
• [3] Crawford, F.W., & Suchard, M.A. (2012). Transition probabilities for general birth–death processes with applications in ecology, genetics, and evolution. Journal of Mathematical Biology, 65(3), 553-580.
• [4] Doss, C.R., Suchard, M.A., Holmes, I., Kato-Maeda, M., & Minin, V.N. (2013). Fitting birth-death processes to panel data with applications to bacterial DNA fingerprinting. The Annals of Applied Statistics, 7(4), 2315-2335.
• [5] Rabier, C.-E., Ta, T., & Ané С. (2014). Detecting and locating whole genome duplications on a phylogeny: a probabilistic approach. Molecular Biology and Evolution, 31(3), 750-762.
• [6] Crawford, F.W., Weiss, R.E., & Suchard, M.A. (2015). Sex, lies, and self-reported counts: Bayesian mixture models for longitudinal heaped count data via birth-death processes. Annals of Applied Statistics, 9, 572-596.
• [7] Crawford, F.W., Minin, V.N., & Suchard, M.A. (2014). Estimation for general birth-death processes. Journal of the American Statistical Association, 109(506), 730-747.
• [8] De Bruin, A.M., Bekker, R., van Zanten, L., & Koole, G.M. (2010). Dimensioning hospital wards using the Erlang loss model. Annals of Operations Research, 178(1), 23-43.
• [9] Zhernovyi, Yu. (2015). Creating Models of Queueing Systems Using GPSS World. Saarbrücken: LAP Lambert Academic Publishing.
• [10] Zhernovyi, Yu.V. (2020). Simulation Models of Reliability. Zhytomyr: PC ”Zhytomyr-Polygraph” (in Ukrainian).