Beneš method

In queueing theory, a discipline within the mathematical theory of probability, Beneš approach^[1] or Beneš method^[2] is a result for an exact or good approximation to the probability distribution of queue length. It was introduced by Václav E. Beneš in 1963.^[3]

Overview

The method introduces a quantity referred to as the "virtual waiting time" to define the remaining workload in the queue at any time. This process is a step function which jumps upward with new arrivals to the system and otherwise is linear with negative gradient.^[4] By giving a relation for the distribution of unfinished work in terms of the excess work, the difference between arrivals and potential service capacity, it turns a time-dependent virtual waiting time problem into "an integral that, in principle, can be solved."^[5]

References

^ Sivaraman, V.; Chiussi, F. (2000). "Providing end-to-end statistical delay guarantees with earliest deadline first scheduling and per-hop traffic shaping". Proceedings IEEE INFOCOM 2000. Conference on Computer Communications. Nineteenth Annual Joint Conference of the IEEE Computer and Communications Societies (Cat. No.00CH37064). Vol. 2. pp. 631–640. doi:10.1109/INFCOM.2000.832237. ISBN 0-7803-5880-5.
^ Norros, I. (2000). "Queueing Behavior Under Fractional Brownian Traffic". Self-Similar Network Traffic and Performance Evaluation. pp. 101–114. doi:10.1002/047120644X.ch4. ISBN 0471319740.
^ Beneš, V. E. (1963). General Stochastic Processes in the Theory of Queues. Addison Wesley.
^ Reich, E. (1964). "Review: Vaclav E. Benes, General Stochastic Processes in the Theory of Queues". The Annals of Mathematical Statistics. 35 (2): 913–914. doi:10.1214/aoms/1177703602.
^ Van Mieghem, P. (2006). "General queueing theory". Performance Analysis of Communications Networks and Systems. pp. 247–270. doi:10.1017/CBO9780511616488.014. ISBN 9780511616488.

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[1] Sivaraman, V.; Chiussi, F. (2000). "Providing end-to-end statistical delay guarantees with earliest deadline first scheduling and per-hop traffic shaping". Proceedings IEEE INFOCOM 2000. Conference on Computer Communications. Nineteenth Annual Joint Conference of the IEEE Computer and Communications Societies (Cat. No.00CH37064). Vol. 2. pp. 631–640. doi:10.1109/INFCOM.2000.832237. ISBN 0-7803-5880-5.

[2] Norros, I. (2000). "Queueing Behavior Under Fractional Brownian Traffic". Self-Similar Network Traffic and Performance Evaluation. pp. 101–114. doi:10.1002/047120644X.ch4. ISBN 0471319740.

[3] Beneš, V. E. (1963). General Stochastic Processes in the Theory of Queues. Addison Wesley.

[4] Reich, E. (1964). "Review: Vaclav E. Benes, General Stochastic Processes in the Theory of Queues". The Annals of Mathematical Statistics. 35 (2): 913–914. doi:10.1214/aoms/1177703602.

[5] Van Mieghem, P. (2006). "General queueing theory". Performance Analysis of Communications Networks and Systems. pp. 247–270. doi:10.1017/CBO9780511616488.014. ISBN 9780511616488.

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v t e Queueing theory
Single queueing nodes	D/M/1 queue M/D/1 queue M/D/c queue M/M/1 queue Burke's theorem M/M/c queue M/M/∞ queue M/G/1 queue Pollaczek–Khinchine formula Matrix analytic method M/G/k queue G/M/1 queue G/G/1 queue Kingman's formula Lindley equation Fork–join queue Bulk queue
Arrival processes	Poisson point process Markovian arrival process Rational arrival process
Queueing networks	Jackson network Traffic equations Gordon–Newell theorem Mean value analysis Buzen's algorithm Kelly network G-network BCMP network
Service policies	FIFO LIFO Processor sharing Round-robin Shortest job next Shortest remaining time
Key concepts	Continuous-time Markov chain Kendall's notation Little's law Product-form solution Balance equation Quasireversibility Flow-equivalent server method Arrival theorem Decomposition method Beneš method
Limit theorems	Fluid limit Mean-field theory Heavy traffic approximation Reflected Brownian motion
Extensions	Fluid queue Layered queueing network Polling system Adversarial queueing network Loss network Retrial queue
Information systems	Data buffer Erlang (unit) Erlang distribution Flow control (data) Message queue Network congestion Network scheduler Pipeline (software) Quality of service Scheduling (computing) Teletraffic engineering
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Beneš method

Beneš method

Beneš method

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