Systems and Means of Informatics
2022, Volume 32, Issue 1, pp 34-45
TRUNCATION BOUNDS FOR INHOMOGENEOUS MARKOV CHAINS WITH CONTINUOUS TIME AND CATASTROPHES
- I. A. Usov
- I. A. Kovalev
- A. I. Zeifman
Abstract
The authors have obtained a new uniform estimate for the truncation bounds for a more general class of weakly ergodic Markov chains with continuous time and catastrophes. In contrast to the previously studied cases, for the corresponding direct Kolmogorov system, the matrix A has a more general form and less stringent restrictions on the intensity. The authors assume that the process is weakly ergodic in the li norm and in the "weighted" norm 1ù. The obtained estimate is valid for heterogeneous processes of birth and death as well as for queue with group admission and maintenance of requirements and for nonstationary service models with catastrophes and "heavy tails", i.e., when the intensities decrease at a power rate. The paper also describes an inhomogeneous queuing system Mt|Mt|S with catastrophes as a numerical example.
[+] References (17)
- Chen, A.Y., P. Pollett, J.P. Li, andH.J. Zhang. 2010. Markovian bulk-arrival and bulk-service queues with state-dependent control. Queueing Syst. 64:267-304.
- Zeifman, A. I., A. V. Korotysheva, T. L. Panfilova, and S. Y. Shorgin. 2011. Otsenki ustoychivosti dlya nekotorykh sistem obsluzhivaniya s katastrofami [Stability bounds for some queueing systems with catastrophes]. Sistemy i Sredstva Informatiki - Systems and Means of Informatics 5(3):27-33.
- Zeifman, A. I., A. V. Korotysheva, Ya. A. Satin, V. Yu. Korolev, S. Ya. Shorgin, and R. V. Razumchik. 2015. Ergodicity and perturbation bounds for inhomogeneous birth and death processes with additional transitions from and to origin. Int. J. Appl. Math. Comp. 25:787-802.
- Zeifman, A. I., A. V. Korotysheva, V. Yu. Korolev, and Ya. A. Satin. 2016. Truncation bounds for approximations of inhomogeneous continuous-time Markov chains. Theor. Probab. Appl. 61 (3):513-520.
- Zeifman, A.I., V. Yu. Korolev, A.V. Korotysheva, and Ya. A. Satin. 2016. Otsenki dlya neodnorodnykh markovskikh sistem obsluzhivaniya s osobennostyami v nule [Bounds for inhomogeneous Markov service systems with singularities at zero]. Moscow: FRC CSCRAS. 54 p.
- Zeifman, A.I., A.V. Korotysheva, Ya. A. Satin, K. M. Kiseleva, R.V. Razumchik, V. Yu. Korolev, and S. Ya. Shorgin. 2017. Otsenki pogreshnosti approksimatsii dlya markovskikh sistem obsluzhivaniya, opisyvaemykh protsessami rozhdeniya i gibeli s dopolnitel'nymi perekhodami [Truncation bounds for a class of inhomogeneous birth and death queueing models with additional transitions]. Sistemy i Sredstva Informatiki - Systems and Means of Informatics 29(3):37-51.
- Zeifman, A.I., A.V. Korotysheva, Y. A. Satin, V.Yu. Korolev, and K. M. Kiseleva. 2017. Bounds for Markovian queues with possible catastrophes. 31st European Conference on Modelling and Simulation Proceedings. Budapest, Hungary: ECMS. 628-634.
- Satin, Ya.A., A.I. Zeifman, and A.V. Korotysheva. 2012. Convergence rate and truncations for one class of Markov queueing systems. Theor. Probab. Appl. 57(3): 529-539.
- Zeifman, A.I., V.Yu. Korolev, Y. A. Satin, A.V. Korotysheva, and V. E. Bening. 2014. Perturbation bounds and truncations for a class of Markovian queues. Queueing Syst. 76(2):205-221.
- Zeifman, A.I., Ya.A. Satin, G. N. Shilova, V.Yu. Korolev, V.E. Bening, and
S. Ya. Shorgin. 2014. On truncations for SZK model. 28th European Conference on Modelling and Simulation Proceedings. Brescia, Italy: ECMS. 577-582.
- Zeifman, A.I., Ya.A. Satin, and I. A. Kovalev. 2021. Ob odnoy nestatsionarnoy modeli obsluzhivaniya s katastrofami i tyazhelymi khvostami [On one nonstationary service model with catastrophes and heavy tails]. Informatika i ee Primeneniya - Inform. Appl. 15(2): 19-24.
- Zeifman, A. I. 1995. Upper and lower bounds on the rate of convergence for nonhomo- geneous birth and death processes. Stoch. Proc. Appl. 59:157-173.
- Granovsky, B. L., and A. I. Zeifman. 2004. Nonstationary queues: Estimation of the rate of convergence. Queueing Syst. 46:363-388.
- Zeifman, A.I., S. Leorato, E. Orsingher, Ya. Satin, and G. Shilova. 2006. Some universal limits for nonhomogeneous birth and death processes. Queueing Syst. 52:139151.
- Zeifman, A.I., V.E. Bening, and I. A. Sokolov. 2008. Markovskie tsepi i modeli s nepreryvnym vremenem [Markov chains and models with continuous time]. Moscow: ELEKS-KM Publs. 168 p.
- Zeifman, A.I., and A. V. Korotysheva. 2012. Perturbation bounds for Mt|Mt |N queue with catastrophes. Stoch. Models 28(1):49-62.
- Zeifman, A.I., A. V. Korotysheva, Ya. A. Satin, R. V. Razumchik, V. Yu. Korolev, and S. Ya. Shorgin. 2017. Ergodicity and truncation bounds for inhomogeneous birth and death processes with additional transitions from and to origin. Stoch. Models 33:598-616
[+] About this article
Title
TRUNCATION BOUNDS FOR INHOMOGENEOUS MARKOV CHAINS WITH CONTINUOUS TIME AND CATASTROPHES
Journal
Systems and Means of Informatics
Volume 32, Issue 1, pp 34-45
Cover Date
2022-05-10
DOI
10.14357/08696527220103
Print ISSN
0869-6527
Publisher
Institute of Informatics Problems, Russian Academy of Sciences
Additional Links
Key words
approximations; truncations; catastrophes; queuing systems; weak ergodicity
Authors
I. A. Usov  , I. A. Kovalev , and A. I. Zeifman ,  , 
Author Affiliations
Department of Applied Mathematics, Vologda State University, 15 Lenin Str., Vologda 160000, Russian Federation
Federal Research Center "Computer Science and Control", Russian Academy of Sciences, 44-2 Vavilov Str., Moscow 119333, Russian Federation
Vologda Research Center of the Russian Academy of Sciences, 56A Gorky Str., Vologda 160014, Russian Federation
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