Previous |  Up |  Next


Title: Infinite queueing systems with tree structure (English)
Author: Fajfrová, Lucie
Language: English
Journal: Kybernetika
ISSN: 0023-5954
Volume: 42
Issue: 5
Year: 2006
Pages: 585-604
Summary lang: English
Category: math
Summary: We focus on invariant measures of an interacting particle system in the case when the set of sites, on which the particles move, has a structure different from the usually considered set $\mathbb{Z}^d$. We have chosen the tree structure with the dynamics that leads to one of the classical particle systems, called the zero range process. The zero range process with the constant speed function corresponds to an infinite system of queues and the arrangement of servers in the tree structure is natural in a number of situations. The main result of this work is a characterisation of invariant measures for some important cases of site-disordered zero range processes on a binary tree. We consider the single particle law to be a random walk on the binary tree. We distinguish four cases according to the trend of this random walk for which the sets of extremal invariant measures are completely different. Finally, we shall discuss the model with an external source of customers and, in this context, the case of totally asymmetric single particle law on a binary tree. (English)
Keyword: invariant measures
Keyword: zero range process
Keyword: binary tree
Keyword: queues
MSC: 37L40
MSC: 60K25
MSC: 60K35
MSC: 82B44
idZBL: Zbl 1249.60194
idMR: MR2283507
Date available: 2009-09-24T20:19:01Z
Last updated: 2015-03-29
Stable URL:
Reference: [1] Andjel E. D.: Invariant measures for the zero range process.Ann. Probab. 10 (1982), 525–547 Zbl 0492.60096, MR 0659526, 10.1214/aop/1176993765
Reference: [3] Harris T. E.: Nearest-neighbor Markov interaction processes on multidimensional lattice.Adv. in Math. 9 (1972), 66–89 MR 0307392, 10.1016/0001-8708(72)90030-8
Reference: [4] Liggett T. M.: Interacting Particle Systems.Springer–Verlag, New York 1985 MR 0776231
Reference: [5] Saada E.: Processus de zero-range avec particule marquee.Ann. Inst. H. Poincaré 26 (1990), 1, 5–17 Zbl 0703.60101, MR 1075436
Reference: [6] Sethuraman S.: On extremal measures for conservative particle systems.Ann. Inst. H. Poincaré 37 (2001), 2, 139–154 Zbl 0981.60098, MR 1819121, 10.1016/S0246-0203(00)01062-1
Reference: [7] Spitzer F.: Interaction of Markov processes.Adv. in Math. 5 (1970), 246–290 Zbl 0312.60060, MR 0268959, 10.1016/0001-8708(70)90034-4
Reference: [8] Štěpán J.: A noncompact Choquet theorem.Comment. Math. Univ. Carolin. 25 (1984), 1, 73–89 Zbl 0562.60006, MR 0749117
Reference: [9] Waymire E.: Zero-range interaction at Bose-Einstein speeds under a positive recurrent single particle law.Ann. Probab. 8 (1980), 3, 441–450 Zbl 0442.60095, MR 0573285, 10.1214/aop/1176994719


Files Size Format View
Kybernetika_42-2006-5_5.pdf 975.2Kb application/pdf View/Open
Back to standard record
Partner of
EuDML logo