Abstract
In this paper we show how the stability criteria for the model proposed in MacPhee and Müller (Queueing Syst 52(3):215–229, 2006) can be applied to queueing networks with re-entrant lines. The model considered has Poisson arrival streams, servers that can be configured in various ways, exponential service times and Markov feedback of completed jobs. The stability criteria are expressed in terms of the mean drifts of the process under the various server configurations. For models with re-entrant lines we impose here a boundary sojourn condition to ensure adequate control of the process when one or more queues are empty. We show with some examples, including the generalised Lu–Kumar network discussed in Niño-Mora and Glazebrook (J Appl Probab 37(3):890–899, 2000), how our results can be applied.
Similar content being viewed by others
References
M. Bramson, “Instability of FIFO queueing networks,” The Annals of Applied Probability vol. 4(2) pp. 414–431, 1994.
J. G. Dai, “On positive Harris recurrence of multiclass queueing networks: a unified approach via fluid limit models,” The Annals of Applied Probability vol. 5(1) pp. 49–77, 1995.
D. Down and S. P. Meyn, “Piecewise linear test functions for stability and instability of queueing networks,” Queueing Systems Theory and Applications vol. 27(3–4) pp. 205–226, 1997.
G. Fayolle, V. A. Malishev, and M. V. Menshikov, Topics in the Constructive Theory of Countable Markov Chains, Cambridge University Press: Cambridge, 1995.
S. H. Lu and P. R. Kumar, “Distributed scheduling based on due dates and buffer priorities,” IEEE Transactions on Automatic Control vol. 36(12) pp. 1406–1416, 1991.
I. M. MacPhee and L. J. Müller, “Stability classification of a controlled multi-queue system with many service and routing regimes,” Queueing Systems vol. 52(3) pp. 215–229, 2006.
J. Niño-Mora and K. D. Glazebrook, “Assessing an intuitive condition for stability under a range of traffic conditions via a generalised Lu-Kumar network,” Journal of Applied Probability vol. 37 (3) pp. 890–899, 2000.
R. F. Serfozo, “An equivalence between continuous and discrete time Markov decision processes,” Operations Research vol. 27 pp. 616–620, 1979.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
MacPhee, I.M., Müller, L.J. Stability Criteria for Multi-class Queueing Networks with Re-entrant Lines. Methodol Comput Appl Probab 9, 377–388 (2007). https://doi.org/10.1007/s11009-007-9032-8
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11009-007-9032-8