Fluid limit

In queueing theory, a discipline within the mathematical theory of probability, a fluid limit, fluid approximation or fluid analysis of a stochastic model is a deterministic real-valued process which approximates the evolution of a given stochastic process, usually subject to some scaling or limiting criteria.

Fluid limits were first introduced by Thomas G. Kurtz publishing a law of large numbers and central limit theorem for Markov chains.^[1]^[2] It is known that a queueing network can be stable, but have an unstable fluid limit.^[3]

References

↑ Pakdaman, K.; Thieullen, M.; Wainrib, G. (2010). "Fluid limit theorems for stochastic hybrid systems with application to neuron models". Advances in Applied Probability. 42 (3): 761. arXiv:1001.2474. doi:10.1239/aap/1282924062.
↑ Kurtz, T. G. (1971). "Limit Theorems for Sequences of Jump Markov Processes Approximating Ordinary Differential Processes". Journal of Applied Probability. Applied Probability Trust. 8 (2): 344–356. JSTOR 3211904.
↑ Bramson, M. (1999). "A stable queueing network with unstable fluid model". The Annals of Applied Probability. 9 (3): 818. doi:10.1214/aoap/1029962815. JSTOR 2667284.

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 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 first 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

Category

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