Abstract
A stationary regime for polling systems with general ergodic (G/G) arrival processes at each station is constructed. Mutual independence of the arrival processes is not required. It is shown that the stationary workload so constructed is minimal in the stochastic ordering sense. In the model considered the server switches from station to station in a Markovian fashion, and a specific service policy is applied to each queue. Our hypotheses cover the purely gated, thea-limited, the binomial-gated and other policies. As a by-product we obtain sufficient conditions for the stationary regime of aG/G/1/∞ queue with multiple server vacations (see Doshi [11]) to be ergodic.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
E. Altman and S. Foss, Polling on a graph with general arrival and service time, INRIA Report No. 1992.
E. Altman, S. Foss, E.R. Riehl and S. Stidham, Pathwise stability and performance bounds for generalized vacation and polling systems, Research Report UNC/OR TR93-8, University of North Carolina at Chapel Hill, USA, submitted to Oper. Res.
E. Altman, P. Konstantopoulos and Z. Liu, Stability, monotonicity and invariant quantities in general polling systems, Queueing Systems 11 (1992) 35–57.
S. Asmussen,Applied Probability and Queues (Wiley, Chichester, 1987).
F. Baccelli and P. Brémaud,Elements of Queueing Theory (Springer, New York, 1994).
F. Baccelli and S. Foss, On the saturation rule for the stability of queues, submitted to J. Appl. Prob.
A.A. Borovkov and R. Schassberger, Ergodicity of queueing systems, Stoch. Proc. Appl. 50 (1994) 253–262.
A. Brandt, P. Franken and B. Lisek,Stationary Stochastic Models (Wiley, Chichester, 1992).
I.P. Cornfeld, S.V. Fomin and Ya.G. Sinai,Ergodic Theory (Springer, New York, 1982).
D.J. Daley and D. Vere-Jones,An Introduction to the Theory of Point Processes (Springer, New York, 1988).
B. Doshi, Generalizations of the stochastic decomposition results for single server queues with vacations, Commun. Stat. Stoch. Models 6 (1990) 307–333.
S. Foss and N. Chernova, Ergodic properties of polling systems, Preprint (1994).
C. Fricker and M.R. Jaïbi, Monotonicity and stability of periodic polling models, Queueing Systems 15 (1993).
L. Georgiadis and W. Szpankowski, Stability of token passing rings, Queueing Systems 11 (1992) 7–33.
D.P. Kroese and V. Schmidt, A continuous polling system with general service times, Ann. Appl. Prob. 2 (1992) 906–927.
H. Levy, Analysis of cyclic polling systems with binomial-gated service, in:Performance of Distributed and Parallel Systems, eds. T. Hasewaga, H. Takagi and Y. Takahashi (Elsevier, 1989).
L. Massoulié, Stability of non-Markovian polling systems, INRIA Report No. 2148 (1993).
J. Neveu, Sur les mesures de Palm de deux processus ponctuels stationnaires, Z. Wahrs. 34 (1976) 199–203.
J.A.C. Resing, Polling systems and multi-type branching processes, Queueing Systems 13 (1993) 409–426.
H. Takagi, Queueing analysis of polling models: an update, in:Stochastic Analysis of Computer and Communications Systems, ed. H. Takagi (Elsevier Science Publ., Amsterdam 1990) pp. 267–318.
W. Whitt, Bivariate distributions with given marginals, Ann. Stat. 4 (1976) 1280–1289.
V.S. Zhdanov and E.A. Saksonov, Conditions of existence of steady-state modes in cyclic queueing systems, Autom. and Remote Contr. 40 (1979) 176–179.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Massoulié, L. Stability of non-Markovian polling systems. Queueing Syst 21, 67–95 (1995). https://doi.org/10.1007/BF01158575
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF01158575