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Article

On First-Passage Times and Sojourn Times in Finite QBD Processes and Their Applications in Epidemics

1
Department of Statistics and Operations Research, School of Mathematical Sciences, Plaza de Ciencias 3, Complutense University of Madrid, 28040 Madrid, Spain
2
Department of Applied Mathematics, School of Mathematics, University of Leeds, Leeds LS2 9JT, UK
3
Department of Statistics and Data Science, School of Statistical Studies, Avda. Puerta de Hierro s/n, Complutense University of Madrid, 28040 Madrid, Spain
*
Author to whom correspondence should be addressed.
Mathematics 2020, 8(10), 1718; https://doi.org/10.3390/math8101718
Received: 8 September 2020 / Revised: 26 September 2020 / Accepted: 28 September 2020 / Published: 7 October 2020
(This article belongs to the Special Issue Queue and Stochastic Models for Operations Research)
In this paper, we revisit level-dependent quasi-birth-death processes with finitely many possible values of the level and phase variables by complementing the work of Gaver, Jacobs, and Latouche (Adv. Appl. Probab. 1984), where the emphasis is upon obtaining numerical methods for evaluating stationary probabilities and moments of first-passage times to higher and lower levels. We provide a matrix-analytic scheme for numerically computing hitting probabilities, the number of upcrossings, sojourn time analysis, and the random area under the level trajectory. Our algorithmic solution is inspired from Gaussian elimination, which is applicable in all our descriptors since the underlying rate matrices have a block-structured form. Using the results obtained, numerical examples are given in the context of varicella-zoster virus infections. View Full-Text
Keywords: epidemic modeling; first-passage times; hitting probabilities; quasi-birth-death processes; sojourn times epidemic modeling; first-passage times; hitting probabilities; quasi-birth-death processes; sojourn times
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MDPI and ACS Style

Gómez-Corral, A.; López-García, M.; Lopez-Herrero, M.J.; Taipe, D. On First-Passage Times and Sojourn Times in Finite QBD Processes and Their Applications in Epidemics. Mathematics 2020, 8, 1718. https://doi.org/10.3390/math8101718

AMA Style

Gómez-Corral A, López-García M, Lopez-Herrero MJ, Taipe D. On First-Passage Times and Sojourn Times in Finite QBD Processes and Their Applications in Epidemics. Mathematics. 2020; 8(10):1718. https://doi.org/10.3390/math8101718

Chicago/Turabian Style

Gómez-Corral, Antonio, Martín López-García, Maria J. Lopez-Herrero, and Diana Taipe. 2020. "On First-Passage Times and Sojourn Times in Finite QBD Processes and Their Applications in Epidemics" Mathematics 8, no. 10: 1718. https://doi.org/10.3390/math8101718

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