Performance of Multiclass Markovian Queueing Networks Via Piecewise Linear Lyapunov Functions

queueing networks under a stable policy. We propose a general methodology based on Lyapunov functions for the performance analysis of infinite state Markov chains and apply it specifically to Markovian multiclass queueing networks. We establish a deeper connection between stability and performance of such networks by showing that if there exist linear and piecewise linear Lyapunov functions that show stability, then these Lyapunov functions can be used to establish geometric-type lower and upper bounds on the tail probabilities, and thus bounds on the expectation of the queue lengths. As an example of our results, for a reentrant line queueing network with two processing stations operating under a work-conserving policy, we

[1]  Steven A. Lippman,et al.  Applying a New Device in the Optimization of Exponential Queuing Systems , 1975, Oper. Res..

[2]  Erol Gelenbe,et al.  Analysis and Synthesis of Computer Systems , 1980 .

[3]  J. P. Gray.  Addresses , 1869, Appetite.

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

[5]  John N. Tsitsiklis,et al.  Optimization of multiclass queuing networks: polyhedral and nonlinear characterizations of achievable performance , 1994 .

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

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

[8]  Sean P. Meyn,et al.  Stability of queueing networks and scheduling policies , 1993, Proceedings of 32nd IEEE Conference on Decision and Control.

[9]  Ruth J. Williams,et al.  Lyapunov Functions for Semimartingale Reflecting Brownian Motions , 1994 .

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

[11]  P. R. Kumar,et al.  Performance bounds for queueing networks and scheduling policies , 1994, IEEE Trans. Autom. Control..

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

[13]  Sean P. Meyn,et al.  Stability of acyclic multiclass queueing networks , 1995, IEEE Trans. Autom. Control..

[14]  Hakeem,et al.  CORRECTION , 1995, The Lancet.

[15]  Sean P. Meyn,et al.  Stability of queueing networks and scheduling policies , 1995, IEEE Trans. Autom. Control..

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

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

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

[19]  P. Kumar,et al.  The throughput of irreducible closed Markovian queueing networks: functional bounds, asymptotic loss, efficiency, and the Harrison-Wein conjectures , 1997 .

[20]  Panganamala Ramana Kumar,et al.  The Delay of Open Markovian Queueing Networks: Uniform Functional Bounds, Heavy Traffic Pole Multiplicities, and Stability , 1997, Math. Oper. Res..

[21]  Sean P. Meyn,et al.  Piecewise linear test functions for stability and instability of queueing networks , 1997, Queueing Syst. Theory Appl..

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

[23]  Ann Appl,et al.  On the Positive Harris Recurrence for Multiclass Queueing Networks: a Uniied Approach via Uid Limit Models , 1999 .

[24]  Dimitris Bertsimas,et al.  Optimization of Multiclass Queueing Networks with Changeover Times Via the Achievable Region Approach: Part I, the Single-Station Case , 1999 .

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