Geodesic
Accuracy of estimation of the vector of queue lengths for open Jackson networks
Teoriâ veroâtnostej i ee primeneniâ, Tome 67 (2022) no. 4, pp. 792-801 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

The accuracy of approximation of the vector of queue lengths for an open Jackson network with regenerative input flow and unreliable servers is estimated. A theorem on the accuracy of approximation of the vector of queue lengths in open Jackson networks is put forward, i.e., an estimate for the probability of deviations of the norm of the difference between the process of queue lengths and the constructed reflected Brownian motion is obtained. As a corollary, an estimate of the Wasserstein distance is given.
Keywords: Jackson network, unreliable servers, reflected Brownian motion, heavy traffic conditions. 
@article{TVP_2022_67_4_a7,
     author = {E. O. Lenena},
     title = {Accuracy of estimation of the vector of queue lengths for open {Jackson} networks},
     journal = {Teori\^a vero\^atnostej i ee primeneni\^a},
     pages = {792--801},
     year = {2022},
     volume = {67},
     number = {4},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TVP_2022_67_4_a7/}
}
TY  - JOUR
AU  - E. O. Lenena
TI  - Accuracy of estimation of the vector of queue lengths for open Jackson networks
JO  - Teoriâ veroâtnostej i ee primeneniâ
PY  - 2022
SP  - 792
EP  - 801
VL  - 67
IS  - 4
UR  - http://geodesic.mathdoc.fr/item/TVP_2022_67_4_a7/
LA  - ru
ID  - TVP_2022_67_4_a7
ER  - 
%0 Journal Article
%A E. O. Lenena
%T Accuracy of estimation of the vector of queue lengths for open Jackson networks
%J Teoriâ veroâtnostej i ee primeneniâ
%D 2022
%P 792-801
%V 67
%N 4
%U http://geodesic.mathdoc.fr/item/TVP_2022_67_4_a7/
%G ru
%F TVP_2022_67_4_a7
E. O. Lenena. Accuracy of estimation of the vector of queue lengths for open Jackson networks. Teoriâ veroâtnostej i ee primeneniâ, Tome 67 (2022) no. 4, pp. 792-801. http://geodesic.mathdoc.fr/item/TVP_2022_67_4_a7/

[1] J. R. Jackson, “Networks of waiting lines”, Operations Res., 5:4 (1957), 518–521 | DOI | MR | Zbl

[2] M. I. Reiman, “Open queueing networks in heavy traffic”, Math. Oper. Res., 9:3 (1984), 441–458 | DOI | MR | Zbl

[3] F. Baccelli, S. Foss, J. Mairesse, “Stationary ergodic Jackson networks: results and counter-examples”, Stochastic networks. Theory and applications, Roy. Statist. Soc. Lecture Note Ser., 4, The Clarendon Press, Oxford Univ. Press, New York, 1996, 281–307 | Zbl

[4] A. N. Rybko, A. L. Stolyar, “Ergodicity of stochastic processes describing the operation of open queueing networks”, Problems Inform. Transmission, 28:3 (1992), 199–220 | MR | Zbl

[5] L. G. Afanas'eva, “On the ergodicity of an open queueing network”, Theory Probab. Appl., 32:4 (1987), 710–714 | DOI | MR | Zbl

[6] Hong Chen, D. D. Yao, Fundamentals of queueing networks. Performance, asymptotics, and optimization, Appl. Math. (N. Y.), 46, Springer-Verlag, New York, 2001, xviii+405 pp. | DOI | MR | Zbl

[7] E. Bashtova, E. Lenena, “Jackson network in a random environment: strong approximation”, Dalnevost. matem. zhurn., 20:2 (2020), 144–149 | DOI | MR | Zbl

[8] J. Komlós, P. Major, G. Tusnády, “An approximation of partial sums of independent $\mathbf{RV}$'s, and the sample $\mathbf{DF}$. I”, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete, 32 (1975), 111–131 | DOI | MR | Zbl

[9] A. Yu. Zaitsev, “The accuracy of strong Gaussian approximation for sums of independent random vectors”, Russian Math. Surveys, 68:4 (2013), 721–761 | DOI | DOI | MR | Zbl

[10] W. L. Smith, “Regenerative stochastic processes”, Proc. Royal Soc. London Ser. A, 232:1188 (1955), 6–31 | DOI | MR | Zbl

[11] J. B. Goodman, W. A. Massey, “The non-ergodic Jackson network”, J. Appl. Probab., 21:4 (1984), 860–869 | DOI | MR | Zbl

[12] J. M. Harrison, M. I. Reiman, “Reflected Brownian motion on an orthant”, Ann. Probab., 9:2 (1981), 302–308 | DOI | MR | Zbl

[13] E. Bashtova, A. Shashkin, “Strong Gaussian approximation for cumulative processes”, Stochastic Process. Appl., 150 (2022), 1–18 | DOI | MR | Zbl

[14] M. Csörgő, P. Deheuvels, L. Horváth, “An approximation of stopped sums with applications in queueing theory”, Adv. in Appl. Probab., 19:3 (1987), 674–690 | DOI | MR | Zbl

[15] M. Csörgő, L. Horváth, J. Steinebach, “Invariance principles for renewal processes”, Ann. Probab., 15:4 (1987), 1441–1460 | DOI | MR | Zbl

[16] M. Csörgő, P. Révész, “How big are the increments of a Wiener process?”, Ann. Probab., 7:4 (1979), 731–737 | DOI | MR | Zbl