Systems and Means of Informatics
2022, Volume 32, Issue 4, pp 2131
CONVERGENCE RATE AND STABILITY ESTIMATES FOR A CLASS OF NONSTATIONARY MARKOV MODELS OF QUEUES WITH IMPATIENT CUSTOMERS
 I. A. Kovalev
 Y. A. Satin
 A. I. Zeifman
Abstract
A nonstationary queuing system with S servers and impatient customers is considered, i. e., the arrival intensities decrease with the growth of the queue. The process X (t) describing the number of customers in such a system is considered, the existence of a limiting mode of the probability distribution of states and a limiting mean for X (t) is proved, and the estimates of the rate of convergence to the limiting mode and the limiting mean are obtained. Also, the
perturbation estimates are obtained. The authors apply an approach based on the concept of the logarithmic norm of the operator function. As an example, a simple model of a nonstationary system is considered in which potential customers are discouraged by queue length.
[+] References (11)
 Reynolds, J.E. 1968. The stationary solution of a multiserver queueing model with discouragement. Oper. Res. 16:6471.
 Natvig, B. 1974. On the transient state probabilities for a queueing model where potential customers are discouraged by queue length. J. Appl. Probab. 11:345354.
 Van Doorn, E. 1981. The transient state probabilities for a queueing model where potential customers are discouraged by queue length. J. Appl. Probab. 18:499506.
 Srivastava, H. M., and B. R. K. Kashyap. 1984. Special functions in queueing theory and related stochastic processes. B. Am. Math. Soc. 10(1): 139141.
 Parthasarathy, P.R., and N. Selvaraju. 2001. Transient analysis of a queue where potential customers are discouraged by queue length. Math. Probl. Eng. 7:433454.
 Granovsky, B. L., and A. I. Zeifman. 2004. Nonstationary queues: Estimation of the rate of convergence. Queueing Syst. 46:363388.
 Jain M., and M. Singh. 2020. Transient analysis of a Markov queueing model with feedback, discouragement and disaster. Int. J. Applied Computational Mathematics 6(2): 114.
 Zeifman, A. 2020. On the study of forward Kolmogorov system and the corresponding problems for inhomogeneous continuoustime Markov chains. Differential and difference equations with applications. Eds. S. Pinelas, J. R. Graef, S. Hilger, P. Kloeden, and C. Schinas. Springer proceedings in mathematics and statistics ser. Springer. 333:2139.
 Zeifman, A., V. Korolev, and Y. Satin. 2020. Two approaches to the construction of perturbation bounds for continuoustime Markov chains. Mathematics 8(2):253. 25 p.
 Zeifman, A. I., Y. A. Satin, V. Yu. Korolev, and S. Ya. Shorgin. 2014. On truncations for weakly ergodic inhomogeneous birth and death processes. Int. J. Appl. Math. Comp. 24:503518.
 Zeifman, A. I., A. V. Korotysheva, V. Yu. Korolev, and Ya. A. Satin. 2016. Truncation bounds for approximations of inhomogeneous continuoustime Markov chains. Theor. Probab. Appl. 61 (3):513520.
[+] About this article
Title
CONVERGENCE RATE AND STABILITY ESTIMATES FOR A CLASS OF NONSTATIONARY MARKOV MODELS OF QUEUES WITH IMPATIENT CUSTOMERS
Journal
Systems and Means of Informatics
Volume 32, Issue 4, pp 2131
Cover Date
20223011
DOI
10.14357/08696527220403
Print ISSN
08696527
Publisher
Institute of Informatics Problems, Russian Academy of Sciences
Additional Links
Key words
rate of convergence; ergodicity bounds; logarithmic norm; perturbation; queuing systems
Authors
I. A. Kovalev , , Y. A. Satin , and A. I. Zeifman , , ,
Author Affiliations
Department of Applied Mathematics, Vologda State University, 15 Lenin Str., Vologda 160000, Russian Federation
Moscow Center for Fundamental and Applied Mathematics, M.V. Lomonosov Moscow State University, 152 Leninskie Gory, GSP1, Moscow 119991, Russian Federation
Federal Research Center "Computer Science and Control", Russian Academy of Sciences, 442 Vavilov Str., Moscow 119333, Russian Federation
Vologda Research Center of the Russian Academy of Sciences, 56A Gorky Str., Vologda 160014, Russian Federation
