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The stability of open queueing networks

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  • Sigman, Karl

Abstract

The stability of open Jackson networks is established where service times are i.i.d. general distribution, exogeneous interarrival times are i.i.d. general distribution, and the routing is Markovian. The service time distributions are only required to have finite first moment. The system is modeled (at arrival epochs) as a general state space Markov chain. Explicit regeneration points are found (even in the case when the system never empties) and the chain is shown to be Harris ergodic if standard rate conditions are enforced, that is, if at each node, the long run average amount of work per unit time that arrives exogenously destined for that node is strictly less than one. In addition, we prove that if the system is modeled in continuous time then convergence to a steady-state occurs in total variation if the interarrival time distribution is spread-out. Extensions of the results to multi-server nodes, non-Markovian routing and Markov modulated arrivals are given.

Suggested Citation

  • Sigman, Karl, 1990. "The stability of open queueing networks," Stochastic Processes and their Applications, Elsevier, vol. 35(1), pages 11-25, June.
    • Handle: RePEc:eee:spapps:v:35:y:1990:i:1:p:11-25
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    Cited by:

    1. Down, Douglas G. & Karakostas, George, 2008. "Maximizing throughput in queueing networks with limited flexibility," European Journal of Operational Research, Elsevier, vol. 187(1), pages 98-112, May.
    2. Amarjit Budhiraja & Chihoon Lee, 2009. "Stationary Distribution Convergence for Generalized Jackson Networks in Heavy Traffic," Mathematics of Operations Research, INFORMS, vol. 34(1), pages 45-56, February.
    3. Sigrún Andradóttir & Hayriye Ayhan & Douglas G. Down, 2007. "Compensating for Failures with Flexible Servers," Operations Research, INFORMS, vol. 55(4), pages 753-768, August.
    4. Dimitris Bertsimas & David Gamarnik & Alexander Anatoliy Rikun, 2011. "Performance Analysis of Queueing Networks via Robust Optimization," Operations Research, INFORMS, vol. 59(2), pages 455-466, April.
    5. B. Curtis Eaves & Peter W. Glynn & Uriel G. Rothblum, 2004. "The Mean Number-in-System Vector Range for Multiclass Queueing Networks," Mathematics of Operations Research, INFORMS, vol. 29(1), pages 182-190, February.
    6. Bertsimas, Dimitris & Gamarnik, David. & Tsitsiklis, John N., 1995. "Stability conditions for multiclass fluid queueing networks," Working papers 3790-95., Massachusetts Institute of Technology (MIT), Sloan School of Management.
    7. Jose Blanchet & Xinyun Chen, 2019. "Perfect Sampling of Generalized Jackson Networks," Mathematics of Operations Research, INFORMS, vol. 44(2), pages 693-714, May.
    8. Sigrún Andradóttir & Hayriye Ayhan & Douglas G. Down, 2003. "Dynamic Server Allocation for Queueing Networks with Flexible Servers," Operations Research, INFORMS, vol. 51(6), pages 952-968, December.
    9. Ward Whitt & Wei You, 2020. "Heavy-traffic limits for stationary network flows," Queueing Systems: Theory and Applications, Springer, vol. 95(1), pages 53-68, June.

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