// Workers AI · dad joke modeWhat did fluid limit say to its date? You're my boundary.

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. 8 (2). Applied Probability Trust: 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.

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[1] 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.

[2] Kurtz, T. G. (1971). "Limit Theorems for Sequences of Jump Markov Processes Approximating Ordinary Differential Processes". Journal of Applied Probability. 8 (2). Applied Probability Trust: 344–356. JSTOR 3211904.

[3] 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.

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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
Category

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References