Inqueueing theory, a discipline within the mathematicaltheory of probability, afluid queue (fluid model,^[1]fluid flow model^[2] orstochastic fluid model^[3]) is a mathematical model used to describe the fluid level in a reservoir subject to randomly determined periods of filling and emptying. The termdam theory was used in earlier literature for these models. The model has been used to approximate discrete models, model the spread ofwildfires,^[4] inruin theory^[5] and to model high speed data networks.^[6] The model applies theleaky bucket algorithm to a stochastic source.

The model was first introduced byPat Moran in 1954 where a discrete-time model was considered.^[7][8][9] Fluid queues allow arrivals to be continuous rather than discrete, as in models like theM/M/1 andM/G/1 queues.

Fluid queues have been used to model the performance of anetwork switch,^[10] arouter,^[11] theIEEE 802.11 protocol,^[12]Asynchronous Transfer Mode (the intended technology forB-ISDN),^[13][14]peer-to-peer file sharing,^[15]optical burst switching,^[16] and has applications in civil engineering when designingdams.^[17] The process is closely connected toquasi-birth–death processes, for which efficient solution methods are known.^[18][19]

A fluid queue can be viewed as a large tank, typically assumed to be of infinite capacity, connected to a series of pipes that pour fluid in to the tank and a series of pumps which remove fluid from the tank. An operator controls the pipes and pumps controlling the rate at which fluid pours in to the buffer and the rate at which fluid leaves. When the operator puts the system in to statei we writer_i for the net fluid arrival rate in this state (input less output). When the buffer contains fluid, if we writeX(t) for the fluid level at timet,^[20]

The operator is acontinuous time Markov chain and is usually called theenvironment process,background process^[21] ordriving process.^[6] As the processX represents the level of fluid in the buffer it can only take non-negative values.

The model is a particular type ofpiecewise deterministic Markov process and can also be viewed as aMarkov reward model with boundary conditions.

The stationary distribution is aphase-type distribution^[2] as first shown by Asmussen^[22] and can be computed usingmatrix-analytic methods.^[10]

The additive decomposition method is numerically stable and separates the eigenvalues necessary for computation usingSchur decomposition.^[23][24]

For a simple system where service has a constant rate μ and arrival fluctuate between rates λ and 0 (in states 1 and 2 respectively) according to acontinuous time Markov chain with generator matrix

the stationary distribution can be computed explicitly and is given by^[6]

${\displaystyle F(x,1)=\frac{\beta }{\alpha +\beta }\left(1-e^{\left(\frac{\beta }{\mu }-\frac{\alpha }{\lambda -\mu }\right)x}\right)}$

${\displaystyle F(x,2)=\frac{\alpha }{\alpha +\beta }-\frac{\beta \left(\lambda -\mu \right)}{\alpha +\beta }{e}^{\left(\frac{\beta }{\mu }-\frac{\alpha }{\lambda -\mu }\right)x}}$

and average fluid level^[25]

${\displaystyle \frac{(\lambda -\mu )\beta }{(\mu (\alpha +\beta )-\beta \lambda )(\alpha +\beta )}(\mu ,\lambda -\mu ).}$

The busy period is the period of time measured from the instant that fluid first arrives in the buffer (X(t) becomes non-zero) until the buffer is again empty (X(t) returns to zero). In earlier literature it is sometimes referred to as the wet period (of the dam).^[26] TheLaplace–Stieltjes transform of the busy period distribution is known for the fluid queue with infinite buffer^[27][28][29] and theexpected busy period in the case of a finite buffer and arrivals as instantaneous jumps.^[26]

For an infinite buffer with constant service rate μ and arrivals at rates λ and 0, modulated by a continuous time Markov chain with parameters

writeW*(s) for the Laplace–Stieltjes transform of the busy period distribution, then^[29]

${\displaystyle W^{\ast }(s)=\frac{\beta \lambda +s\lambda -\beta \mu +\alpha \mu -\sqrt{4\beta \alpha \mu (\mu -\lambda )+(s\lambda +\beta (\lambda -\mu )+\alpha \mu {)}^2}}{2\beta (\lambda -\mu )}}$

which gives themean busy period^[30]

${\displaystyle \mathbb{E}(W)=\frac{\lambda }{\alpha \mu +\beta (\lambda -\mu )}.}$

In this case, of a single on/off source, the busy period distribution is known to be adecreasing failure rate function which means that the longer a busy period has lasted the longer it is likely to last.^[31]

There are two main approaches to solving for the busy period in general, using either spectral decomposition or an iterative recurrent method.^[32]Aquadratically convergent algorithm for computing points of the transform was published by Ahn and Ramaswami.^[33]

For example, if a fluid queue with service rateμ = 2 is fed by an on/off source with parametersα = 2,β = 1 andλ =  3 then the fluid queue has busy period with mean 1 and variance 5/3.

In a finite buffer the rate at which fluid is lost (rejected from the system due to a full buffer) can be computed using Laplace-Stieltjes transforms.^[34]

The term mountain process has been coined to describe the maximum buffer content process value achieved during a busy period and can be computed using results from aG/M/1 queue.^[35][36]

The stationary distribution of two tandem fluid queues has been computed and shown not to exhibit aproduct form stationary distribution in nontrivial cases.^{[25][30][37][38][39]}

A feedback fluid queue is a model where the model parameters (transition rate matrix and drift vector) are allowed to some extent to depend on the buffer content. Typically the buffer content is partitioned and the parameters depend on which partition the buffer content process is in.^[40] The orderedSchur factorization can be used to efficiently compute the stationary distribution of such a model.^[41]

Second order fluid queues (sometimes called Markov modulated diffusion processes or fluid queues with Brownian noise^[42]) consider areflected Brownian motion with parameters controlled by a Markov process.^[22][43] Two different types of boundary conditions are commonly considered: absorbing and reflecting.^[44]

• BuTools, aMATLAB,Python andMathematica implementation of some of the above results.
• PevaTools,MATLAB code for multi-regime models
• Fluid flow models tutorial by V. Ramaswami at MAM8

• Queueing theory