Informatics and Applications
2019, Volume 13, Issue 3, pp 1419
ON THE BOUNDS OF THE RATE OF CONVERGENCE FOR SOME QUEUEING MODELS WITH INCOMPLETELY DEFINED INTENSITIES
 A. I. Zeifman
 Y. A. Satin
 K. M. Kiseleva
Abstract
The authors consider some queuing systems with incompletely defined 1periodical intensities under corresponding conditions. The authors deal with Mt/Mt/S queue for any number of servers S and Mt/Mt/S/S (the Erlang model). Estimates of the rate of convergence in weakly ergodic situation are obtained by applying the method of the logarithmic norm of the operator of a linear function. The examples with exact given values of intensities and different variations of amplitude and frequency are considered, ergodicity conditions and estimates of the rate of convergence are obtained for each model, and plots of the effect of intensities' amplitude and frequency of incoming requirements on the limiting characteristics of the process are constructed. The authors use the general algorithm to build graphs, it is associated with solving the Cauchy problem for the forward Kolmogorov system on the corresponding interval, which has already been used by the authors in previous papers.
[+] References (17)
 Guo, Y., and Z. Wang. 2013. Stability of Markovian jump systems with generally uncertain transition rates. J. Frankl. Inst. 350(9):28262836.
 Crawford, F.W., V. N. Minin, and M. A. Suchard. 2014. Estimation for birthdeath processes. J. Am. Stat. Assoc. 109(506):730747.
 Dong J., and W. Whitt. 2015. Stochastic greybox modeling of queueing systems: Fitting birthanddeath processes to data. Queueing Syst. 79:391426.
 Zhu, D. M., W K. Ching, and S. M. Guu. 2016. Sufficient conditions for the ergodicity of fuzzy Markov chains. Fuzzy Set. Syst. 304:8293.
 Cruz, FR.B., R. C. Quinino, and L. L. Ho. 2017. Bayesian estimation of traffic intensity based on queue length in a multiserver M/M/s queue. Commun. Stat. Simulat. 46:73197331.
 Ho, L.S.T., J. Xu, FW. Crawford, V. N. Minin, and M.A. Suchard. 2018. Birth/birthdeath processes and their computable transition probabilities with biological applications. J. Math. Biol. 76:911944.
 Zeifman, A. I. 1989. Some properties of a system with losses in the case of variable rates. Automat. Rem. Contr. 50(1):8287.
 Kijima, M. 1990. On the largest negative eigenvalue of the infinitesimal generator associated with M/M/n/n queues. Oper. Res. Lett. 9:5964.
 Zeifman, A. I. 1995. Upper and lower bounds on the rate of convergence for nonhomogeneous birth and death processes. Stoch. Proc. Appl. 59:157173.
 Fricker C., P. Robert, and D. Tibi. 1999. On the rate of convergence of Erlang's model. J. Appl. Probab. 36:1167 1184.
 Voit, M. 2000. A note of the rate of convergence to equilibrium for Erlang's model in the subcritical case. J. Appl. Probab. 37:918923.
 Zeifman, A., 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 modelis nepreryvnym vremenem [Markov
chains and models with continuous time]. Moscow: ELEKSKM Publs. 168 p.
 Van Doorn, E. A., and A. I. Zeifman. 2009. On the speed of convergence to stationarity of the Erlang loss system. Queueing Syst. 63:241252.
 Zeifman, A. I. 2009. On the nonstationary Erlang loss model. Automat. Rem. Contr. 70(12):20032012.
 Van Doorn, E.A., A.I. Zeifman, and T. L. Panfilova. 2010. Bounds and asymptotics forthe rate of convergence ofbirthdeath processes. Theor. Probab. Appl. 54:97113.
 Zeifman, A., Ya. Satin, V. Korolev, and S. Shorgin. 2014. On truncations for weakly ergodic inhomogeneous birth and death processes. Int. J. Appl. Math. Comp. 24:503518.
[+] About this article
Title
ON THE BOUNDS OF THE RATE OF CONVERGENCE FOR SOME QUEUEING MODELS WITH INCOMPLETELY DEFINED INTENSITIES
Journal
Informatics and Applications
2019, Volume 13, Issue 3, pp 1419
Cover Date
20190930
DOI
10.14357/19922264190303
Print ISSN
19922264
Publisher
Institute of Informatics Problems, Russian Academy of Sciences
Additional Links
Key words
queuing systems; incompletely defined intensities; rate of convergence; ergodicity; logarithmic norm; Mt/Mt/S queue; Mt/Mt/S/S queue
Authors
A. I. Zeifman , , , Y. A. Satin , and K. M. Kiseleva
Author Affiliations
Vologda State University, 15 Lenin Str., Vologda 160000, Russian Federation
Institute of Applied Mathematical Research, Karelian Research Centre of the Russian Academy of Sciences,
11 Pushkinskaya Str., Petrozavodsk 185910, Karelia, Russian Federation
Vologda Research Center of the Russian Academy of Sciences, 56A Gorky Str., Vologda 160014, Russian Federation
