LARGE DEVIATIONS BOUNDS FOR A POLLING SYSTEM WITH MARKOVIAN ON/OFF SOURCES AND BERNOULLI SERVICE SCHEDULE

In this paper we consider a large deviations problem for a discrete-time polling system consisting of two-parallel queues and a single server. The arrival process of each queue is a superposition of traffic streams generated by a number of mutually independent and identical Markovian on/off sources, and the single server serves the two queues according to the so-called Bernoulli service schedule. Using the large de- viations techniques, we derive the upper and lower bounds of the probability that the queue length of each queue exceeds a certain level (i.e., the buffer overflow probabil- ity). These results have important implications for traffic management of high-speed communication networks such as call admission control and bandwidth allocation.

[1]  Nick G. Duffield,et al.  Exponential bounds for queues with Markovian arrivals , 1994, Queueing Syst. Theory Appl..

[2]  高木 英明,et al.  Analysis of polling systems , 1986 .

[3]  Ward Whitt,et al.  Tail probabilities with statistical multiplexing and effective bandwidths in multi-class queues , 1993, Telecommun. Syst..

[4]  Nick G. Duffield,et al.  Large deviations, the shape of the loss curve, and economies of scale in large multiplexers , 1995, Queueing Syst. Theory Appl..

[5]  James A. Bucklew,et al.  Large Deviation Techniques in Decision, Simulation, and Estimation , 1990 .

[6]  Cheng-Shang Chang,et al.  Stability, queue length, and delay of deterministic and stochastic queueing networks , 1994, IEEE Trans. Autom. Control..

[7]  Herwig Bruneel,et al.  Storage Requirements in ATM Switching Elements with Correlated Arrivals and Independent Uniform Routing , 1996, Perform. Evaluation.

[8]  Cheng-Shang Chang,et al.  Effective bandwidths of departure processes from queues with time varying capacities , 1995, Proceedings of INFOCOM'95.

[9]  John N. Tsitsiklis,et al.  Large deviations analysis of the generalized processor sharing policy , 1999, Queueing Syst. Theory Appl..

[10]  Amir Dembo,et al.  Large Deviations Techniques and Applications , 1998 .

[11]  G. Parmigiani.  Large Deviation Techniques in Decision, Simulation and Estimation , 1992 .

[12]  Hideaki Takagi,et al.  Queuing analysis of polling models , 1988, CSUR.

[13]  Herwig Bruneel.  Queueing behavior of statistical multiplexers with correlated inputs , 1988, IEEE Trans. Commun..

[14]  Leandros Tassiulas,et al.  Dynamic server allocation to parallel queues with randomly varying connectivity , 1993, IEEE Trans. Inf. Theory.

[15]  Onno Boxma,et al.  Two queues with alternating service and switching times , 1987 .

[16]  Yoshitaka Takahashi,et al.  ANALYSIS OF A TWO-CLASS PRIORITY QUEUE WITH BERNOULLI SCHEDULES , 1992 .

[17]  P. Glynn,et al.  Logarithmic asymptotics for steady-state tail probabilities in a single-server queue , 1994, Journal of Applied Probability.

[18]  M. Eisenberg.  Two Queues with Alternating Service , 1979 .

[19]  Zhi-Li Zhang.  Large deviations and the generalized processor sharing scheduling for a two-queue system , 1997, Queueing Syst. Theory Appl..

[20]  R. M. Loynes,et al.  The stability of a queue with non-independent inter-arrival and service times , 1962, Mathematical Proceedings of the Cambridge Philosophical Society.

[21]  Martin Eisenberg.  Two Queues with Changeover Times , 1971, Oper. Res..

[22]  John N. Tsitsiklis,et al.  Asymptotic buffer overflow probabilities in multiclass multiplexers: an optimal control approach , 1998, IEEE Trans. Autom. Control..

[23]  C H Chang,et al.  EFFECTIVE BANDWIDTH IN HIGHSPEED DIGITAL NETWORKS , 1995 .

[24]  Stability , 1973 .

[25]  Wei Feng,et al.  A two-queue model with Bernoulli service schedule and switching times , 1998, Queueing Syst. Theory Appl..

[26]  Arnaud de La Fortelle,et al.  Large Deviations Rate Function for Polling Systems , 2002, Queueing Syst. Theory Appl..

[27]  Theodore E. Tedijanto,et al.  Exact Results for the Cyclic-Service Queue with a Bernoulli Schedule , 1990, Perform. Evaluation.

[28]  Alan Weiss,et al.  An Introduction to Large Deviations for Communication Networks , 1995, IEEE J. Sel. Areas Commun..

[29]  Cheng-Shang Chang,et al.  Sample path large deviations and intree networks , 1995, Queueing Syst. Theory Appl..

[30]  Debasis Mitra,et al.  Effective bandwidth of general Markovian traffic sources and admission control of high speed networks , 1993, IEEE INFOCOM '93 The Conference on Computer Communications, Proceedings.

[31]  Zhi-Li Zhang,et al.  Large deviations and the generalized processor sharing scheduling for a multiple-queue system , 1998, Queueing Syst. Theory Appl..