The Annals of Probability

Stationary distributions of multi-type totally asymmetric exclusion processes

Pablo A. Ferrari and James B. Martin

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Abstract

We consider totally asymmetric simple exclusion processes with n types of particle and holes (n-TASEPs) on ℤ and on the cycle ℤN. Angel recently gave an elegant construction of the stationary measures for the 2-TASEP, based on a pair of independent product measures. We show that Angel’s construction can be interpreted in terms of the operation of a discrete-time M/M/1 queueing server; the two product measures correspond to the arrival and service processes of the queue. We extend this construction to represent the stationary measures of an n-TASEP in terms of a system of queues in tandem. The proof of stationarity involves a system of n 1-TASEPs, whose evolutions are coupled but whose distributions at any fixed time are independent. Using the queueing representation, we give quantitative results for stationary probabilities of states of the n-TASEP on ℤN, and simple proofs of various independence and regeneration properties for systems on ℤ.

Article information

Source
Ann. Probab., Volume 35, Number 3 (2007), 807-832.

Dates
First available in Project Euclid: 10 May 2007

Permanent link to this document
https://projecteuclid.org/euclid.aop/1178804315

Digital Object Identifier
doi:10.1214/009117906000000944

Mathematical Reviews number (MathSciNet)
MR2319708

Zentralblatt MATH identifier
1117.60089

Subjects
Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]
Secondary: 82C22: Interacting particle systems [See also 60K35] 90B22: Queues and service [See also 60K25, 68M20]

Keywords
Totally asymmetric simple exclusion process multi-type process multiclass queue

Citation

Ferrari, Pablo A.; Martin, James B. Stationary distributions of multi-type totally asymmetric exclusion processes. Ann. Probab. 35 (2007), no. 3, 807--832. doi:10.1214/009117906000000944. https://projecteuclid.org/euclid.aop/1178804315


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References

  • Angel, O. (2006). The stationary measure of a 2-type totally asymmetric exclusion process. J. Combin. Theory Ser. A 113 625--635.
  • Derrida, B., Janowsky, S. A., Lebowitz, J. L. and Speer, E. R. (1993). Exact solution of the totally asymmetric simple exclusion process: Shock profiles. J. Statist. Phys. 73 813--842.
  • Duchi, E. and Schaeffer, G. (2005). A combinatorial approach to jumping particles. J. Combin. Theory Ser. A 110 1--29.
  • Ferrari, P. A., Fontes, L. R. G. and Kohayakawa, Y. (1994). Invariant measures for a two-species asymmetric process. J. Statist. Phys. 76 1153--1177.
  • Ferrari, P. A., Kipnis, C. and Saada, E. (1991). Microscopic structure of travelling waves in the asymmetric simple exclusion process. Ann. Probab. 19 226--244.
  • Ferrari, P. A. and Martin, J. B. (2006). Multi-class processes, dual points and $M/M/1$ queues. Markov Process. Related Fields 12 175--201.
  • Hsu, J. and Burke, P. J. (1976). Behavior of tandem buffers with geometric input and Markovian output. IEEE Trans. Comm. COM-24 358--361.
  • Liggett, T. M. (1976). Coupling the simple exclusion process. Ann. Probab. 4 339--356.
  • Liggett, T. M. (1985). Interacting Particle Systems. Springer, New York.
  • Liggett, T. M. (1999). Stochastic Interacting Systems: Contact, Voter and Exclusion Processes. Springer, Berlin.
  • Mallick, K., Mallick, S. and Rajewsky, N. (1999). Exact solution of an exclusion process with three classes of particles and vacancies. J. Phys. A 32 8399--8410.
  • Speer, E. R. (1994). The two species asymmetric simple exclusion process. In On Three Levels: Micro, Meso and Macroscopic Approaches in Physics (C. M. M. Fannes and A. Verbuere, eds.) 91--102. Plenum, New York.