Skip to main content
Disciplines
Automotive Business IT + Informatics Construction + Real Estate Electrical Engineering + Electronics Energy + Sustainability Insurance + Risk Finance + Banking Management + Leadership Marketing + Sales Mechanical Engineering + Materials
Events
DE
EN
Books
Journals
Topic Page
Marketing
Start single access now
Access for companies
EXTENDED SEARCH
Log in
Top
Queueing Systems
Published in: Queueing Systems 1-2/2019

27-03-2019

Attractiveness of Brownian queues in tandem

Authors: Eric A. Cator, Sergio I. López, Leandro P. R. Pimentel

Published in: Queueing Systems | Issue 1-2/2019

Log in

Activate our intelligent search to find suitable subject content or patents.

search-config
loading …

Abstract

Consider a sequence of n bi-infinite and stationary Brownian queues in tandem. Assume that the arrival process entering the first queue is a zero mean ergodic process. We prove that the departure process from the n-th queue converges in distribution to a Brownian motion as n goes to infinity. In particular this implies that the Brownian motion is an attractive invariant measure for the Brownian queueing operator. Our proof exploits the relationship between Brownian queues in tandem and the last-passage Brownian percolation model, developing a coupling technique in the second setting. The result is also interpreted in the related context of Brownian particles acting under one-sided reflection.
previous article Replicate to the shortest queues
next article Marked point processes in discrete time

Dont have a licence yet? Then find out more about our products and how to get one now:

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

  • über 102.000 Bücher
  • über 537 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Maschinenbau + Werkstoffe
  • Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

inform now

Springer Professional "Technik"

Online-Abonnement

Mit Springer Professional "Technik" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 390 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Maschinenbau + Werkstoffe




 

Jetzt Wissensvorsprung sichern!

inform now
Literature
1.
Baryshnikov, Y.: GUEs and queues. Probab. Theory Relat. Fields 119, 256–274 (2001)CrossRef
2.
Cator, E.A., Groeneboom, P.: Second class particles and cube root asymptotics for Hammersley’s process. Ann. Probab. 34, 1273–1295 (2006)CrossRef
3.
Cator, E.A., Pimentel, L.P.R.: Busemman functions and equilibrium measures in last-passage percolation models. Prob. Theory Relat. Fields. 154, 89–125 (2012)CrossRef
4.
Ferrari, P.A.: Shocks in the Burgers equation and the asymmetric simple exclusion process. In: Goles, E., Martínez, S. (eds.) Statistical Physics, Automata Networks and Dynamical Systems. Mathematics and its Applications, vol. 75, pp. 25–64. Springer, Dordrecht (1992)
5.
Ferrari, P.L., Spohn, H., Weiss, T.: Scaling limit for Brownian with one-sided collisions. Ann. Appl. Probab. 25, 1349–1382 (2015)CrossRef
6.
Ferrari, P.L., Spohn, H., Weiss, T.: Brownian motions with one-sided collisions: the stationary case. Eletron. J. Probab. 69, 1–41 (2015)
7.
Georgiou, N., Rassoul-Agha, F., Seppäläinen, T.: Geodesics and the competition interface for the corner growth model. Probab. Theory Relat. Fields 169, 223–255 (2015)CrossRef
8.
Georgiou, N., Rassoul-Agha, F., Seppäläinen, T.: Stationary cocycles and Busemann functions for the corner growth model. Probab. Theory Relat. Fields 169, 177–222 (2015)CrossRef
9.
Glynn, P.W., Whitt, W.: Departures from many queues in series. Ann. Appl. Probab. 1, 546–572 (1991)CrossRef
10.
Gravner, W.J., Tracy, C.A., Widom, H.: Limit theorems for height fluctuations in a class of discrete space and time growth models. J. Stat. Phys. 102(5–6), 1085–1132 (2001)CrossRef
11.
Hambly, B.M., Martin, J.B., O’Connell, N.: Concentration results for a Brownian directed percolation problem. Stoch. Process. Appl. 102, 207–220 (2002)CrossRef
12.
Harrison, M., Williams, R.: On the quasireversibility of a multiclass Brownian service station. Ann. Probab. 18, 1249–1268 (1990)CrossRef
13.
Ichiba, T., Karatzas, I.: On collisions of Brownian particles. Ann. Appl. Probab 20, 951–977 (2012)CrossRef
14.
Karatzas, I., Pal, S., Shkolnikov, M.: Systems of Brownian particles with asymmetric collisions. Ann. Inst. H. Poincaré Probab. Stat. 52(1), 323–354 (2016)CrossRef
15.
López, S.I.: Convergence of tandem Brownian queues. J. Appl. Probab. 53(2), 585–592 (2016)CrossRef
16.
López, S.I., Pimentel, L.P.R.: On the location of the maximum of a process: lévy, Gaussian and multidimensional cases. Stochastics 90(8), 1221–1237 (2018)CrossRef
17.
Loynes, R.M.: The stability of a queue with non-independent interarrival and service times. Proc. Camb. Philos. Soc. 58, 497–520 (1962)CrossRef
18.
Mairesse, J., Prabhakar, B.: The existence of fixed points for the \(\cdot /GI/1\) queue. Ann. Probab. 31, 2216–2236 (2003)CrossRef
19.
Martin, J.B.: Last passage percolation with general weight distribution. Markov Proc. Relat. Fields. 12, 273–299 (2006)
20.
Mountford, T., Prabhakar, B.: On the weak convergence of departures from an infinite series of \(\cdot /M/ 1\) queues. Ann. Appl. Probab. 5(1), 121–127 (1995)CrossRef
21.
O’Connell, N., Yor, M.: Brownian analogues of Burke’s theorem. Stoch. Process. Appl. 2, 285–304 (2001)CrossRef
22.
Pal, S., Pitman, J.: One-dimensional Brownian particle systems with rank-dependent drifts. Ann. Appl. Prob. 18, 2179–2207 (2008)CrossRef
23.
Prabhakar, B.: The attractiveness of the fixed points of a \(\cdot /GI/1\) queue. Ann. Probab 31, 2237–2269 (2003)CrossRef
24.
Seppäläinen, T.: A scaling limit for queues in series. Ann. Appl. Probab. 7, 855–872 (1997)CrossRef
Metadata
Title
Attractiveness of Brownian queues in tandem
Authors
Eric A. Cator
Sergio I. López
Leandro P. R. Pimentel
Publication date
27-03-2019
Publisher
Springer US
Published in
Queueing Systems / Issue 1-2/2019
Print ISSN: 0257-0130
Electronic ISSN: 1572-9443
DOI
https://doi.org/10.1007/s11134-019-09609-y

Other articles of this Issue 1-2/2019

Queueing Systems 1-2/2019 Go to the issue

OriginalPaper

Strategic arrivals to queues offering priority service

OriginalPaper

Marked point processes in discrete time

OriginalPaper

Replicate to the shortest queues

OriginalPaper

Exponential ergodicity and convergence for generalized reflected Brownian motion

OriginalPaper

A sequential update algorithm for computing the stationary distribution vector in upper block-Hessenberg Markov chains

Correction

Correction to: A sequential update algorithm for computing the stationary distribution vector in upper block-Hessenberg Markov chains

Premium Partner

    Image Credits