Queueing Systems

, Volume 78, Issue 2, pp 155–187 | Cite as

On the departure process of the linear loss network



This paper considers a \(k\)-node linear loss network consisting of bufferless nodes. In particular, the asymptotic behavior of the departure process is investigated, as the size of the network grows. Our result provides a complete characterization of a properly scaled limiting departure process, i.e., the joint probability density function of any finite number of consecutive inter-departure times, as the size of the network increases.


Linear loss networks Departure process Asymptotic behavior Fixed-point process 

Mathematics Subject Classification




The research was supported by the National Science Foundation under Grant CNS-0643213.


  1. 1.
    Anantharam, V.: Uniqueness of stationary ergodic fixed point for a \(\cdot \)/M/k node. Ann. Appl. Probab. 3(1), 154–172 (1993)CrossRefGoogle Scholar
  2. 2.
    Arratia, R.A.: Coalescing Brownian motions on the line. Ph.D. thesis, University of Wisconsin, Madison (1979)Google Scholar
  3. 3.
    Arratia, R.A.: Limiting point processes for rescalings of coalescing and annihilating random walks on \(Z^d\). Ann. Probab. 9(6), 909–936 (1981)CrossRefGoogle Scholar
  4. 4.
    Baccelli, F., Brémaud, P.: Elements of Queueing Theory, 2nd edn. Springer, Berlin (2003)CrossRefGoogle Scholar
  5. 5.
    Ben-Avraham, D.: Complete exact solution of diffusion-limited coalescence, \(A+A\rightarrow A\). Phys. Rev. Lett. 81(21), 4756–4759 (1998)CrossRefGoogle Scholar
  6. 6.
    Ben-Avraham, D., Brunet, É.: On the relation between one-species diffusion-limited coalescence and annihilation in one dimension. J. Phys. A Math. Gen. 38(15), 3247–3252 (2005)CrossRefGoogle Scholar
  7. 7.
    Ben-Avraham, D., Havlin, S.: Diffusion and Reactions in Fractals and Disordered Systems. Cambridge University Press, Cambridge (2000)CrossRefGoogle Scholar
  8. 8.
    Ben-Avraham, D., Burschka, M.A., Doering, C.R.: Statics and dynamics of a diffusion-limited reaction: anomalous kinetics, nonequilibrium self-ordering, and a dynamic transition. J. Stat. Phys. 60(5–6), 695–728 (1990)CrossRefGoogle Scholar
  9. 9.
    Billingsley, P.: Probability and Measure, 3rd edn. Wiley, New York (1995)Google Scholar
  10. 10.
    Billingsley, P.: Convergence of Probability Measures, 2nd edn. Wiley, New York (1999)CrossRefGoogle Scholar
  11. 11.
    Bramson, M., Griffeath, D.: Clustering and dispersion rates for some interacting particle systems on \({\mathbb{Z}}\). Ann. Probab. 8(2), 183–213 (1980)CrossRefGoogle Scholar
  12. 12.
    Burke, P.J.: The output of a queuing system. Oper. Res. 4(6), 699–704 (1956)CrossRefGoogle Scholar
  13. 13.
    Chang, C.-S.: On the input–output map of a G/G/1 queue. J. Appl. Probab. 31(4), 1128–1133 (1994)CrossRefGoogle Scholar
  14. 14.
    Choi, Y., Momčilović, P.: On a critical regime for linear finite-buffer networks. In: Proceedings of the 50th Annual Allerton Conference on Communication, Control, and Computing, October (2012)Google Scholar
  15. 15.
    Daley, D.J.: Queueing output processes. Adv. Appl. Probab. 8(2), 395–415 (1976)CrossRefGoogle Scholar
  16. 16.
    Doering, C.R.: Microscopic spatial correlations induced by external noise in a reaction–diffusion system. Phys. A 188(1–3), 386–403 (1992)CrossRefGoogle Scholar
  17. 17.
    Doob, J.L.: Stochastic Processes. Wiley, New York (1953)Google Scholar
  18. 18.
    Gelenbe, E., Pujolle, G.: Introduction to Queueing Networks, 2nd edn. Wiley, New York (1998)Google Scholar
  19. 19.
    Karatzas, I., Shreve, S.E.: Brownian Motion and Stochastic Calculus, 2nd edn. Springer-Verlag, New York (1991)Google Scholar
  20. 20.
    Lawler, G.F.: Introduction to Stochastic Processes, 2nd edn. Chapman & Hall/CRC, New York (2006)Google Scholar
  21. 21.
    Loynes, R.M.: The stability of a queue with non-independent inter-arrival and service times. Proc. Camb. Philos. Soc. 58, 497–520 (1962)CrossRefGoogle Scholar
  22. 22.
    Mairesse, J., Prabhakar, B.: The existence of fixed points for the \(\cdot \)/GI/1 queue. Ann. Probab. 31(4), 2216–2236 (2003)CrossRefGoogle Scholar
  23. 23.
    Momčilović, P., Squillante, M.S.: Linear loss networks. Queueing Syst. 68(2), 111–131 (2011)CrossRefGoogle Scholar
  24. 24.
    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)CrossRefGoogle Scholar
  25. 25.
    Prabhakar, B.: The attractiveness of the fixed points of a \(\cdot \)/GI/1 queue. Ann. Probab. 31(4), 2237–2269 (2003)CrossRefGoogle Scholar
  26. 26.
    Pyke, R.: On renewal processes related to type I and type II counter models. Ann. Math. Stat. 29(3), 737–754 (1958)CrossRefGoogle Scholar
  27. 27.
    Takács, L.: On a probability problem arising in the theory of counters. Proc. Camb. Philos. Soc. 52, 488–498 (1956)CrossRefGoogle Scholar
  28. 28.
    Whitt, W.: Some useful functions for functional limit theorems. Math. Oper. Res. 5(1), 67–85 (1980)CrossRefGoogle Scholar
  29. 29.
    Whitt, W.: Stochastic-Process Limits. Springer, New York (2002)Google Scholar

Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  1. 1.Mobile Solutions LabSamsung Electronics US R&D CenterSan DiegoUSA
  2. 2.Department of Industrial and Systems EngineeringUniversity of FloridaGainesvilleUSA

Personalised recommendations