The Annals of Applied Probability

A scaling limit for queues in series

Timo Seppäläinen

Full-text: Open access

Abstract

We derive a law of large numbers for a tagged particle in the one-dimensional totally asymmetric simple exclusion process under a scaling different from the usual Euler scaling. By interpreting the particles as the servers of a series of queues we use this result to verify an open conjecture about the scaling behavior of the departure times from a long series of queues.

Article information

Source
Ann. Appl. Probab., Volume 7, Number 4 (1997), 855-872.

Dates
First available in Project Euclid: 29 January 2003

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1043862414

Digital Object Identifier
doi:10.1214/aoap/1043862414

Mathematical Reviews number (MathSciNet)
MR1484787

Zentralblatt MATH identifier
0897.60094

Subjects
Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]
Secondary: 60K25: Queueing theory [See also 68M20, 90B22]

Keywords
Hydrodynamic limit exclusion process queues in series

Citation

Seppäläinen, Timo. A scaling limit for queues in series. Ann. Appl. Probab. 7 (1997), no. 4, 855--872. doi:10.1214/aoap/1043862414. https://projecteuclid.org/euclid.aoap/1043862414


Export citation

References

  • Andjel, E. D. (1982). Invariant measures for the zero-range process. Ann. Probab. 10 525-547.
  • Andjel, E. D. and Kipnis, C. (1984). Derivation of the hy drody namical equation for the zero-range interaction process. Ann. Probab. 12 325-334.
  • Andjel, E. D. and Vares, M. E. (1987). Hy drody namic equations for attractive particle sy stems on Z. J. Statist. Phy s. 47 265-288.
  • Bardi, M. and Evans, L. C. (1984). On Hopf's formulas for solutions of Hamilton-Jacobi equations. Nonlinear Anal. 8 1373-1381.
  • Benassi, A. and Fouque, J-P. (1987). Hy drody namical limit for the asy mmetric simple exclusion process. Ann. Probab. 15 546-560.
  • Benassi, A. and Fouque, J-P. (1988). Hy drody namical limit for the asy mmetric zero-range process. Ann. Inst. H. Poincar´e Probab. Statist. 24 189-200.
  • Ferrari, P. A. (1994). Shocks in one-dimensional processes with drift. In Probability and Phase Transition (G. Grimmett, ed.) 35-48. Kluwer, Dordrecht.
  • Ferrari, P. A. and Fontes, L. R. G. (1994). The net output process of a sy stem with infinitely many queues. Ann. Appl. Probab. 4 1129-1144.
  • Gly nn, P. W. and Whitt, W. (1991). Departures from many queues in series. Ann. Appl. Probab. 1 546-572.
  • Greenberg, A. G., Schlunk, O. and Whitt, W. (1993). Using distributed-event parallel simulation to study departures from many queues in series. Probab. Engrg. Inform. Sci. 7 159-186.
  • Griffeath, D. (1979). Additive and Cancellative Interacting Particle Sy stems. Lecture Notes in Math. 724. Springer, Berlin.
  • Kelly, F. (1979). Reversibility and Stochastic Networks. Wiley, New York.
  • Kipnis, C. (1986). Central limit theorems for infinite series of queues and applications to simple exclusion. Ann. Probab. 14 397-408. Landim, C. (1991a). Hy drody namical equation for attractive particle sy stems on Zd. Ann. Probab. 19 1537-1558. Landim, C. (1991b). Hy drody namical limit for asy mmetric attractive particle sy stems on Zd. Ann. Inst. H. Poincar´e Probab. Statist. 27 559-581.
  • Liggett, T. M. (1985). Interacting Particle Sy stems. Springer, New York.
  • Lions, P. L. (1982). Generalized Solutions of Hamilton-Jacobi Equations. Pitman, London.
  • Rezakhanlou, F. (1991). Hy drody namic limit for attractive particle sy stems on Zd. Comm. Math. Phy s. 140 417-448.
  • Rezakhanlou, F. (1994). Evolution of tagged particles in nonreversible particle sy stems. Comm. Math. Phy s. 165 1-32.
  • Rost, H. (1981). Non-equilibrium behaviour of a many particle process: density profile and local equilibrium. Z. Wahrsch. Verw. Gebiete 58 41-53.
  • Saada, E. (1987). A limit theorem for the position of a tagged particle in a simple exclusion process. Ann. Probab. 15 375-381.
  • Sepp¨al¨ainen, T. (1996). Hy drody namic scaling, convex duality, and asy mptotic shapes of growth models. Preprint.
  • Spitzer, F. (1970). Interaction of Markov processes. Adv. Math. 5 246-290.
  • Srinivasan, R. (1991). Stochastic comparisons of density profiles for the road-hog process. J. Appl. Probab. 28 852-863.
  • Srinivasan, R. (1993). Queues in series via interacting particle sy stems. Math. Oper. Res. 18 39-50.