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On the rate of convergence and limiting characteristics for one quasi-birth–death process
I. A. Usova, Y. A. Satina, A. I. Zeifmanabcde a Vologda State University, 15 Lenin Str., Vologda 160000, Russian Federation
b Moscow Center for Fundamental and Applied Mathematics, M. V. Lomonosov Moscow State University,
1-52 Leninskie Gory, GSP-1, Moscow 119991, Russian Federation
c Federal Research Center “Computer Science and Control” of the Russian Academy of Sciences, 44-2 Vavilov Str., Moscow 119333, Russian Federation
d Vologda Research Center of the Russian Academy of Sciences, 56A Gorky Str., Vologda 160014, Russian Federation
e M. V. Lomonosov Moscow State University,
1-52 Leninskie Gory, GSP-1, Moscow 119991, Russian Federation
Abstract:
A queuing system with one server and different repair and failure options is considered, the number of requirements in which is described by a quasi-birth-death process. To reasonably find the limiting probabilistic characteristics of the system, the rate of convergence to them is studied (that is, the rate at which the initial conditions of the system are “forgotten”). To study the rate of convergence to the limiting regime, a recently developed version of the approach based on the concept of the logarithmic norm of the operator function corresponding to the estimate of the norm of the Cauchy matrix as well as a modernized special transformation of the forward Kolmogorov system was applied. A numerical example is considered for which the estimation of the rate of convergence is shown in detail as well as the construction of some limiting characteristics of the model based on these estimates.
Keywords:
quasi-birth–death processes, rate of convergence, ergodicity bounds, logarithmic norm, queuing systems.
Received: 14.04.2023
Citation:
I. A. Usov, Y. A. Satin, A. I. Zeifman, “On the rate of convergence and limiting characteristics for one quasi-birth–death process”, Inform. Primen., 17:3 (2023), 49–57
Linking options:
https://www.mathnet.ru/eng/ia858 https://www.mathnet.ru/eng/ia/v17/i3/p49
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Abstract page: | 59 | Full-text PDF : | 30 | References: | 11 |
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