Skip to main content
SpringerLink
Log in

Coupling in the queue with impatience: case of several servers

  • Published:
Discrete Event Dynamic Systems Aims and scope Submit manuscript

Abstract

We present the explicit construction of a stable queue with several servers and impatient customers, under stationary ergodic assumptions. Using a stochastic comparison of the (multivariate) workload sequence with two monotonic stochastic recursions, we propose a sufficient condition of existence of a unique stationary state of the system using Renovation theory. Whenever this condition is relaxed we use extension techniques to prove the existence of a stationary state in some cases.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  • Anantharam VV, Konstantopoulos T (1997) Stationary solutions of stochastic recursions describing discrete time events systems. Stochastic Process Appl 68:181–194

    Article  MathSciNet  MATH  Google Scholar 

  • Anantharam VV, Konstantopoulos T (1999) A correction and some additional remarks on Stationary solutions of stochastic recursions describing discrete time events systems. Stoch Process Appl 80(2):271– 278

    Article  MATH  Google Scholar 

  • Baccelli F, Boyer P, Hébuterne G (1984) Single-server queues with impatient customers. Adv Appl Probab 16(4):887–905

    Article  MathSciNet  MATH  Google Scholar 

  • Baccelli F, Brémaud P (2002) Elements of Queueing Theory, vol 2. Springer, Berlin

    MATH  Google Scholar 

  • Baccelli F, Hébuterne G (1981) On queues with impatient customers. In: Performance’81, pp 159–179

  • Blanchet J, Dong J (2015) Perfect sampling for infinite server and loss systems. Adv Appl Probab 47(3):761–786

    Article  MathSciNet  MATH  Google Scholar 

  • Borovkov AA (1984) Asymptotic Methods in Queueing Theory. J. Wiley, New York

    MATH  Google Scholar 

  • Borovkov AA, Foss S (1992) Stochastic recursive sequences and their generalizations. Sib Adv Math 2(1):16–81

    Google Scholar 

  • Brandt A, Franken P (1990) B lisek Stationary Stochastic Models. Akademie-Verlag/Wiley, Hoboken

    Google Scholar 

  • Dai JG (1995) On positive Harris recurrence of multiclass queueing networks: a unified approach via fluid limit models. Ann Appl Probab 5:49–77

    Article  MathSciNet  MATH  Google Scholar 

  • Decreusefond L, Moyal P (2012) Stochastic modeling and analysis of telecom networks. ISTE Wiley

  • Flipo D (1983) Steady state of loss systems. In: French. Comptes rendus de l’Académie des Sciences de Paris, Ser. I, vol 297, no 6

  • Flipo D (1988) Charge Stationnaire d’une File d’attente à Rejet. Application au Cas indépendant. In: French. Annales scientifiques de l’université de Clermont-Ferrand 2, tome. 92, Probabilité,s et Applications vol 7. pp 47–74

  • Kiefer J, Wolfowitz J (1955) On the theory of queues with many servers. Trans Amer Math Soc 78:1–18

    Article  MathSciNet  MATH  Google Scholar 

  • Lisek B (1982) A method for solving a class of recursive stochastic equation. Zeitschrift fur Wahrsch. 60:151–162

    Article  MathSciNet  MATH  Google Scholar 

  • Loynes RM (1962) The stability of queues with non-independent interarrivals and service times. Proc Camb Philos Soc 58:497–520

    Article  MathSciNet  MATH  Google Scholar 

  • Moyal P (2008) Comparison of service disciplines in real-time queues. Oper Res Lett 36(4):496–499

    Article  MathSciNet  MATH  Google Scholar 

  • Moyal P (2010) The queue with impatience: construction of the stationary workload under FIFO. J Appl Probab 47(4):498–512

    Article  MathSciNet  MATH  Google Scholar 

  • Moyal P (2013) On queues with impatience: stability, and the optimality of Earliest Deadline First. Queueing Syst: Theory Appl 75(2-4):211–242

    Article  MathSciNet  MATH  Google Scholar 

  • Moyal P (2015) A generalized backwards scheme for solving non monotonic stochastic recursions. Ann Appl Probab 25(2):582–599

    Article  MathSciNet  MATH  Google Scholar 

  • Moyal P (2017) On the stability of a class of non-monotonic systems of parallel queues. Discrete Events Dynamic Systems 27(1):85–107

    Article  MathSciNet  MATH  Google Scholar 

  • Robert Ph (2003) Stochastic Networks and Queues. Springer, Berlin

    Book  Google Scholar 

  • Rybko AN, Stolyar AL (1992) Ergodicity of stochastic processes describing the operations of open queueing networks. Probl Inf Transm 28:3–26. (in Russian)

    MATH  Google Scholar 

  • Neveu J (1983) Construction de files d’attente stationnaires. In: French. Lecture Notes in Control and Information Sciences, vol 60. Springer, pp 31–41

  • Propp JG, Wilson DB (1996) Exact sampling with coupled Markov chains and applications to stastical mechanics. Random Struct Algoritm 9(1-2):223–252

    Article  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Laboratoire de Mathématiques Appliquées de Compiègne, Université de Technologie de Compiègne, Compiègne, France

    Pascal Moyal

  2. Institut Elie Cartan de Lorraine, Université de Lorraine, Nancy, France

    Pascal Moyal

Authors
  1. Pascal Moyal
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Pascal Moyal.

Additional information

Publisher’s note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Moyal, P. Coupling in the queue with impatience: case of several servers. Discrete Event Dyn Syst 29, 145–162 (2019). https://doi.org/10.1007/s10626-019-00280-2

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10626-019-00280-2

Keywords

Search

Navigation