Infinite queueing systems with tree structure

Lucie Fajfrová

Kybernetika (2006)

  • Volume: 42, Issue: 5, page 585-604
  • ISSN: 0023-5954

Abstract

top
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 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.

How to cite

top

Fajfrová, Lucie. "Infinite queueing systems with tree structure." Kybernetika 42.5 (2006): 585-604. <http://eudml.org/doc/33826>.

@article{Fajfrová2006,
abstract = {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.},
author = {Fajfrová, Lucie},
journal = {Kybernetika},
keywords = {invariant measures; zero range process; binary tree; queues; invariant measures; zero range process; binary tree; queues},
language = {eng},
number = {5},
pages = {585-604},
publisher = {Institute of Information Theory and Automation AS CR},
title = {Infinite queueing systems with tree structure},
url = {http://eudml.org/doc/33826},
volume = {42},
year = {2006},
}

TY - JOUR
AU - Fajfrová, Lucie
TI - Infinite queueing systems with tree structure
JO - Kybernetika
PY - 2006
PB - Institute of Information Theory and Automation AS CR
VL - 42
IS - 5
SP - 585
EP - 604
AB - 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.
LA - eng
KW - invariant measures; zero range process; binary tree; queues; invariant measures; zero range process; binary tree; queues
UR - http://eudml.org/doc/33826
ER -

References

top
  1. Andjel E. D., 10.1214/aop/1176993765, Ann. Probab. 10 (1982), 525–547 (1982) Zbl0492.60096MR0659526DOI10.1214/aop/1176993765
  2. Harris T. E., 10.1016/0001-8708(72)90030-8, Adv. in Math. 9 (1972), 66–89 (1972) MR0307392DOI10.1016/0001-8708(72)90030-8
  3. Liggett T. M., Interacting Particle Systems, Springer–Verlag, New York 1985 MR0776231
  4. Saada E., Processus de zero-range avec particule marquee, Ann. Inst. H. Poincaré 26 (1990), 1, 5–17 (1990) Zbl0703.60101MR1075436
  5. Sethuraman S., 10.1016/S0246-0203(00)01062-1, Ann. Inst. H. Poincaré 37 (2001), 2, 139–154 Zbl0981.60098MR1819121DOI10.1016/S0246-0203(00)01062-1
  6. Spitzer F., 10.1016/0001-8708(70)90034-4, Adv. in Math. 5 (1970), 246–290 (1970) Zbl0312.60060MR0268959DOI10.1016/0001-8708(70)90034-4
  7. Štěpán J., A noncompact Choquet theorem, Comment. Math. Univ. Carolin. 25 (1984), 1, 73–89 (1984) Zbl0562.60006MR0749117
  8. Waymire E., 10.1214/aop/1176994719, Ann. Probab. 8 (1980), 3, 441–450 (1980) Zbl0442.60095MR0573285DOI10.1214/aop/1176994719

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.