Abstract
Fluid models have recently become an important tool for the study of open multiclass queueing networks. We are interested in a family of such models, which we refer to as head-of-the-line proportional processor sharing (HLPPS) fluid models. Here, the fraction of time spent serving a class present at a station is proportional to the quantity of the class there, with all of the service going into the “first customer” of each class. To study such models, we employ an entropy function associated with the state of the system. The corresponding estimates show that if the traffic intensity function is at most 1, then such fluid models converge exponentially fast to equilibria. When the traffic intensity function is strictly less than 1, the limit is always the empty state and occurs after a finite time. A consequence is that generalized HLPPS networks with traffic intensity strictly less than 1 are positive Harris recurrent. Related results for FIFO fluid models of Kelly type were obtained in Bramson [4].
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References
F. Baskett, K.M. Chandy, R.R. Muntz and F.G. Palacios, Open, closed and mixed networks of queues with different classes of customers,J. ACM 22 (1975) 248–260.
M. Bramson, Instability of FIFO queueing networks,The Annals of Applied Probability 4 (1994) 414–431.
M. Bramson, Instability of FIFO queueing networks with quick service times,The Annals of Applied Probability 4 (1994) 693–718.
M. Bramson, Convergence to equilibria for fluid models of FIFO queueing networks,Queueing Systems 22 (1996) 5–45.
M. Bramson, Convergence to equilibria for fluid models of certain FIFO and processor sharing queueing networks, in:Stochastic Networks: Theory and Applications, RSS Lecture Notes in Statistics (Oxford Univ. Press, Oxford, England, 1996) pp. 1–18.
H. Chen, Fluid approximations and stability of multiclass queueing networks: work-conserving disciplines,The Annals of Applied Probability 5 (1995) 637–655.
H. Chen and H. Zhang, Diffusion approximations for re-entrant lines with a first-buffer-first-served priority discipline,Queueing Systems 23 (1996) 177–195 (this issue).
J. Dai, On the positive Harris recurrence for multiclass queueing networks,The Annals of Applied Probability 5 (1995) 49–77.
J. Dai and T. Kurtz, A multiclass station with Markovian feedback in heavy traffic,Mathematics of Operations Research 20 (1995) 721–742.
J. Dai and S. Meyn, Stability and convergence of moments for multiclass queueing networks via fluid models,IEEE Trans. Automat. Control 40 (1995) 1889–1904.
J. Dai and Y. Wang, Nonexistence of Brownian models for certain multiclass queueing networks,Queueing Systems 13 (1993) 41–46.
V. Dumas, Unstable cycles in fluid Bramson networks, Rapport de Recherche 2318, INRIA (August, 1994).
J.M. Harrison, Brownian models of queueing networks with heterogeneous customer populations, in:Stochastic Differential Systems, Stochastic Control Theory and their Applications, IMA Volumes in Mathematics and its Applications, Vol. 10 (Springer-Verlag, New York, 1988) pp. 147–186.
J.M. Harrison, Balanced fluid models of multiclass queueing networks: a heavy traffic conjecture, in:Stochastic Networks, IMA Volumes in Mathematics and its Applications, Vol. 71 (Springer-Verlag, New York, 1995) pp. 1–20.
J.M. Harrison and V. Nguyen, The QNET method for two-moment analysis of open queueing networks,Queueing Systems 6 (1990) 1–32.
J.M. Harrison and V. Nguyen, Brownian models of multiclass queueing networks: current status and open problems,Queueing Systems 13 (1993) 5–40.
D.P. Johnson, Diffusion approximations for optimal filtering of jump processes and for queueing networks, PhD Thesis, University of Wisconsin (1983).
F.P. Kelly, Networks of queues with customers of different types,Journal of Applied Probability 12 (1975) 542–554.
F.P. Kelly,Reversibility and Stochastic Networks (Wiley, New York, 1979).
S.H. Lu and P.R. Kumar, Distributed scheduling based on due dates and buffer priorities,IEEE Trans. Automat. Control 36 (1991) 1406–1416.
I.P. Natanson,Theory of Functions of a Real Variable, Vol. 1 (Ungar, New York, 1964).
M.I. Reiman, Open queueing networks in heavy traffic,Mathematics of Operations Research 9 (1984) 441–458.
M.I. Reiman, A multiclass feedback queue in heavy traffic,Advances in Applied Probability 9 (1988) 179–207.
S. Rybko and A. Stolyar, Ergodicity of stochastic processes that describe the functioning of open queueing networks,Problems of Information Transmission 28 (1992) 3–26 (in Russian).
T.I. Seidman, “First come, first served” can be unstable!,IEEE Trans. Automat. Control 39 (1994) 2166–2171.
A. Stolyar, On the stability of multiclass queueing networks, in:Proceedings of the 2nd Conference on Telecommunication Systems — Modeling and Analysis, Nashville (March 24–27, 1994) pp. 1020–1028.
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Partially supported by NSF Grants DMS-93-00612 and DMS-93-04580. The paper was written while the author was in residence at the Institute for Advanced Study.
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Bramson, M. Convergence to equilibria for fluid models of head-of-the-line proportional processor sharing queueing networks. Queueing Syst 23, 1–26 (1996). https://doi.org/10.1007/BF01206549
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DOI: https://doi.org/10.1007/BF01206549