Stability and Heavy Traffic Limits for Queueing Networks

We discuss here two topics of recent interest in queueing theory. The first is the question of when strictly subcritical queueing networks are stable. Namely, given a network whose stations all serve customers more quickly than the long-term rate at which customers visit the system, when is the underlying Markov process positive recurrent? The other topic is the existence of heavy traffic limits for queueing networks. That is, when does a sequence of networks, under diffusive scaling, converge to a reflecting Brownian motion?

[1]  J. Tsitsiklis,et al.  Stability conditions for multiclass fluid queueing networks , 1996, IEEE Trans. Autom. Control..

[2]  Sean P. Meyn,et al.  Stability and convergence of moments for multiclass queueing networks via fluid limit models , 1995, IEEE Trans. Autom. Control..

[3]  J. Michael Harrison,et al.  The QNET method for two-moment analysis of open queueing networks , 1990, Queueing Syst. Theory Appl..

[4]  Hong Chen,et al.  Existence Condition for the Diffusion Approximations of Multiclass Priority Queueing Networks , 2001, Queueing Syst. Theory Appl..

[5]  Ralph L. Disney,et al.  Applied Probability— Computer Science: The Interface , 1982, Progress in Computer Science.

[6]  R. J. Williams,et al.  Existence and uniqueness of semimartingale reflecting Brownian motions in an orthant , 1993 .

[7]  Thomas Bonald,et al.  Window flow control in FIFO networks with cross traffic , 1999, Queueing Syst. Theory Appl..

[8]  M. Reiman.  A multiclass feedback queue in heavy traffic , 1988 .

[9]  John H. Vande Vate,et al.  Stability of a three‐station fluid network , 1999, Queueing Syst. Theory Appl..

[10]  Yang Wang,et al.  Nonexistence of Brownian models for certain multiclass queueing networks , 1993, Queueing Syst. Theory Appl..

[11]  J. Harrison,et al.  Reflected Brownian Motion on an Orthant , 1981 .

[12]  Vincent Dumas,et al.  A multiclass network with non-linear, non-convex, non-monotonic stability conditions , 1997, Queueing Syst. Theory Appl..

[13]  Ruth J. Williams,et al.  Diffusion approximations for open multiclass queueing networks: sufficient conditions involving state space collapse , 1998, Queueing Syst. Theory Appl..

[14]  J. M. Harrison,et al.  Brownian Models of Feedforward Queueing Networks: Quasireversibility and Product Form Solutions , 1992 .

[15]  Richard L. Tweedie,et al.  Markov Chains and Stochastic Stability , 1993, Communications and Control Engineering Series.

[16]  Maury Bramson,et al.  Convergence to equilibria for fluid models of head-of-the-line proportional processor sharing queueing networks , 1996, Queueing Syst. Theory Appl..

[17]  Baruch Awerbuch,et al.  Universal stability results for greedy contention-resolution protocols , 1996, Proceedings of 37th Conference on Foundations of Computer Science.

[18]  Serguei Foss,et al.  A stability criterion via fluid limits and its application to a polling system , 1999, Queueing Syst. Theory Appl..

[19]  Jr. C. Humes,et al.  A regulator stabilization technique: Kumar-Seidman revisited , 1994, IEEE Trans. Autom. Control..

[20]  Hong Chen,et al.  Discrete Flow Networks: Bottleneck Analysis and Fluid Approximations , 1991, Math. Oper. Res..

[21]  S. Orey.  Lecture Notes on Limit Theorems for Markov Chain Transition Probabilities , 1971 .

[22]  F. G. Foster.  On the Stochastic Matrices Associated with Certain Queuing Processes , 1953 .

[23]  Maury Bramson,et al.  Stability of Earliest-Due-Date, First-Served Queueing Networks , 2001, Queueing Syst. Theory Appl..

[24]  W. Whitt.  Large Fluctuations in a Deterministic Multiclass Network of Queues , 1993 .

[25]  John Michael Harrison,et al.  Balanced fluid models of multiclass queueing networks: a heavy traffic conjecture , 1995 .

[26]  P. R. Kumar,et al.  Re-entrant lines , 1993, Queueing Syst. Theory Appl..

[27]  Alexander L. Stolyar,et al.  The stability of a flow merge point with non-interleaving cut-through scheduling disciplines , 1999, IEEE INFOCOM '99. Conference on Computer Communications. Proceedings. Eighteenth Annual Joint Conference of the IEEE Computer and Communications Societies. The Future is Now (Cat. No.99CH36320).

[28]  Jean C. Walrand,et al.  A probabilistic look at networks of quasireversible queues , 1981, 1981 20th IEEE Conference on Decision and Control including the Symposium on Adaptive Processes.

[29]  Ruth J. Williams,et al.  An invariance principle for semimartingale reflecting Brownian motions in an orthant , 1998, Queueing Syst. Theory Appl..

[30]  John J. Hasenbein,et al.  Necessary conditions for global stability of multiclass queueing networks , 1997, Oper. Res. Lett..

[31]  Guy Pujolle,et al.  Introduction to queueing networks , 1987 .

[32]  K. Meyer.  The Output of a Queueing System , 1981 .

[33]  Martin I. Reiman,et al.  Open Queueing Networks in Heavy Traffic , 1984, Math. Oper. Res..

[34]  Zhang Hanqin,et al.  MULTIPLE CHANNEL QUEUES IN HEAVY TRAFFIC , 1990 .

[35]  Elena Yudovina,et al.  Stochastic networks , 1995, Physics Subject Headings (PhySH).

[36]  Sheldon M. Ross,et al.  Stochastic Processes , 2018, Gauge Integral Structures for Stochastic Calculus and Quantum Electrodynamics.

[37]  Mark H. Davis.  Markov Models and Optimization , 1995 .

[38]  Maury Bramson,et al.  State space collapse with application to heavy traffic limits for multiclass queueing networks , 1998, Queueing Syst. Theory Appl..

[39]  Sean P. Meyn,et al.  State-Dependent Criteria for Convergence of Markov Chains , 1994 .

[40]  K. Sigman.  The stability of open queueing networks , 1990 .

[41]  Allan Borodin,et al.  Adversarial queueing theory , 1996, STOC '96.

[42]  Ronald W. Wolff,et al.  Stochastic Modeling and the Theory of Queues , 1989 .

[43]  R. R. P. Jackson,et al.  Queueing Systems with Phase Type Service , 1954 .

[44]  John H. Vande Vate,et al.  The Stability of Two-Station Multitype Fluid Networks , 2000, Oper. Res..

[45]  Ruth J. Williams,et al.  A multiclass closed queueing network with unconventional heavy traffic behavior , 1996 .

[46]  E. Nummelin.  General irreducible Markov chains and non-negative operators: List of symbols and notation , 1984 .

[47]  R. Getoor.  Transience and recurrence of Markov processes , 1980 .

[48]  Frank Kelly,et al.  Networks of queues with customers of different types , 1975, Journal of Applied Probability.

[49]  Mark H. A. Davis.  Piecewise‐Deterministic Markov Processes: A General Class of Non‐Diffusion Stochastic Models , 1984 .

[50]  Allen J. Scott,et al.  Applications of Queueing Theory , 1972 .

[51]  M. Bramson.  Instability of FIFO Queueing Networks , 1994 .

[52]  Thomas I. Seidman,et al.  "First come, first served" can be unstable! , 1994, IEEE Trans. Autom. Control..

[53]  Martin I. Reiman,et al.  Some diffusion approximations with state space collapse , 1984 .

[54]  A. Borovkov.  Limit Theorems for Queueing Networks. I , 1987 .

[55]  Ruth J. Williams.  Semimartingale reflecting Brownian motions in the orthant , 1995 .

[56]  William P. Peterson,et al.  A Heavy Traffic Limit Theorem for Networks of Queues with Multiple Customer Types , 1991, Math. Oper. Res..

[57]  A. N. Rybko,et al.  On the ergodicity of stochastic processes describing functioning of open queueing networks , 1992 .

[58]  Joanna Mitro.  General theory of markov processes , 1991 .

[59]  P. R. Kumar,et al.  Stable distributed real-time scheduling of flexible manufacturing/assembly/disassembly systems , 1988, Proceedings of the 27th IEEE Conference on Decision and Control.

[60]  Sean P. Meyn,et al.  Stability of Generalized Jackson Networks , 1994 .

[61]  R. Serfozo.  Introduction to Stochastic Networks , 1999 .

[62]  J. Azema,et al.  Mesure invariante sur les classes récurrentes des processus de Markov , 1967 .

[63]  E. Reich.  Waiting Times When Queues are in Tandem , 1957 .

[64]  P. R. Kumar,et al.  Distributed scheduling based on due dates and buffer priorities , 1991 .

[65]  Hong Chen.  Fluid Approximations and Stability of Multiclass Queueing Networks: Work-Conserving Disciplines , 1995 .

[66]  Ruth J. Williams,et al.  A boundary property of semimartingale reflecting Brownian motions , 1988 .

[67]  Anthony Unwin,et al.  Reversibility and Stochastic Networks , 1980 .

[68]  J. Michael Harrison,et al.  Brownian models of multiclass queueing networks: Current status and open problems , 1993, Queueing Syst. Theory Appl..

[69]  Thomas G. Kurtz,et al.  A multiclass Station with Markovian Feedback in Heavy Traffic , 1995, Math. Oper. Res..

[70]  J. Wendelberger.  Adventures in Stochastic Processes , 1993 .

[71]  K. Mani Chandy,et al.  Open, Closed, and Mixed Networks of Queues with Different Classes of Customers , 1975, JACM.

[72]  A. Stolyar.  On the Stability of Multiclass Queueing Networks: A Relaxed SuÆcient Condition via Limiting Fluid Processes , .

[73]  S. Meyn,et al.  Stability of Markovian processes II: continuous-time processes and sampled chains , 1993, Advances in Applied Probability.

[74]  T. E. Harris.  The Existence of Stationary Measures for Certain Markov Processes , 1956 .

[75]  S. Meyn,et al.  Stability of Markovian processes III: Foster–Lyapunov criteria for continuous-time processes , 1993, Advances in Applied Probability.

[76]  A. Shiryaev,et al.  Limit Theorems for Stochastic Processes , 1987 .

[77]  Maury Bramson,et al.  Convergence to equilibria for fluid models of FIFO queueing networks , 1996, Queueing Syst. Theory Appl..

[78]  J. Harrison,et al.  The QNET Method for Two-Moment Analysis of Closed Manufacturing Systems , 1993 .

[79]  J. Dai,et al.  Heavy Traffic Limits for Some Queueing Networks , 2001 .

[80]  Sean P. Meyn,et al.  Piecewise linear test functions for stability of queueing networks , 1994, Proceedings of 1994 33rd IEEE Conference on Decision and Control.

[81]  J. Dai.  On Positive Harris Recurrence of Multiclass Queueing Networks: A Unified Approach Via Fluid Limit Models , 1995 .

[82]  A. Barbour.  Networks of queues and the method of stages , 1976, Advances in Applied Probability.

[83]  M. Bramson.  Instability of FIFO Queueing Networks with Quick Service Times , 1994 .

[84]  Sean P. Meyn,et al.  Generalized Resolvents and Harris Recurrence of Markov Processes , 1992 .

[85]  J. Michael Harrison,et al.  Brownian Models of Queueing Networks with Heterogeneous Customer Populations , 1988 .

[86]  F. Knight.  On the ray topology , 1984 .

[87]  E. Nummelin,et al.  A splitting technique for Harris recurrent Markov chains , 1978 .

[88]  Hong Chen,et al.  Diffusion approximations for re-entrant lines with a first-buffer-first-served priority discipline , 1996, Queueing Syst. Theory Appl..

[89]  H. Kushner.  Heavy Traffic Analysis of Controlled Queueing and Communication Networks , 2001 .

[90]  M. Bramson.  A stable queueing network with unstable fluid model , 1999 .

[91]  P. R. Kumar,et al.  Dynamic instabilities and stabilization methods in distributed real-time scheduling of manufacturing systems , 1989, Proceedings of the 28th IEEE Conference on Decision and Control,.

[92]  François Baccelli,et al.  Ergodicity of Jackson-type queueing networks , 1994, Queueing Syst. Theory Appl..

[93]  Sean P. Meyn.  Transience of Multiclass Queueing Networks Via Fluid Limit Models , 1995 .

[94]  D. Botvich,et al.  Ergodicity of conservative communication networks , 1994 .

[95]  G. Dai.  A Fluid-limit Model Criterion for Instability of Multiclass Queueing Networks , 1996 .

[96]  W. Whitt.  Weak convergence theorems for priority queues: preemptive-resume discipline , 1971, Journal of Applied Probability.

[97]  Maury Bramson,et al.  Stability of two families of queueing networks and a discussion of fluid limits , 1998, Queueing Syst. Theory Appl..

[98]  D. Iglehart,et al.  Multiple channel queues in heavy traffic. II: sequences, networks, and batches , 1970, Advances in Applied Probability.

[99]  Avishai Mandelbaum,et al.  On Harris Recurrence in Continuous Time , 1994, Math. Oper. Res..

[100]  J. Harrison.  The diffusion approximation for tandem queues in heavy traffic , 1978, Advances in Applied Probability.