Estimating the Rate of Convergence of the PH/M/1 Model by Reducing to Quasi-Birth-Death Processes

Ilya Usov, Yacov Satin and Alexander Zeifman ()
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Ilya Usov: Department of Applied Mathematics, Vologda State University, 15 Lenina Str., 160000 Vologda, Russia
Yacov Satin: Department of Applied Mathematics, Vologda State University, 15 Lenina Str., 160000 Vologda, Russia
Alexander Zeifman: Department of Applied Mathematics, Vologda State University, 15 Lenina Str., 160000 Vologda, Russia

Mathematics, 2023, vol. 11, issue 6, 1-11

Abstract: We are studying the quasi-birth-death process and the property of weak ergodicity. Using the C-matrix method, we derive estimates for the rate of convergence to the limiting regime for the general case of the P H / M / 1 model, as well as the particular case when m = 3 . We provide a numerical example for the case m = 3 , and construct graphs showing the probability of an empty queue and the probability of p 1 ( t ) .

Keywords: bounds on the rate of convergence; C-matrix method; limiting regime; logarithmic norm method; queuing system (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2023
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