Fluid queue

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 state i we write r_i for the net fluid arrival rate in this state (input less output). When the buffer contains fluid, if we write X(t) for the fluid level at time t,[20]

${\displaystyle {\frac {\mathrm {d} X(t)}{\mathrm {d} t}}={\begin{cases}r_{i}&{\text{ if }}X(t)>0\\\max(r_{i},0)&{\text{ if }}X(t)=0.\end{cases}}}$

The operator is a continuous time Markov chain and is usually called the environment process, background process[21] or driving process.[6] As the process X represents the level of fluid in the buffer it can only take non-negative values.

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

The stationary distribution is a phase-type distribution[2] as first shown by Asmussen[22] and can be computed using matrix-analytic methods.[10]

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

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] The Laplace–Stieltjes transform of the busy period distribution is known for the fluid queue with infinite buffer[27][28][29] and the expected 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

${\displaystyle Q={\begin{pmatrix}-\alpha &\alpha \\\beta &-\beta \end{pmatrix}}}$

write W*(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 the mean 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 a decreasing 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] A quadratically convergent algorithm for computing points of the transform was published by Ahn and Ramaswami.[33]

The stationary distribution of two tandem fluid queues has been computed and shown not to exhibit a product 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 ordered Schur 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 a reflected 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, a MATLAB, Python and Mathematica implementation of some of the above results.
• PevaTools, MATLAB code for multi-regime models
• Fluid flow models tutorial by V. Ramaswami at MAM8