Fluid queue
In
The model was first introduced by
Fluid queues have been used to model the performance of a
Model description
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 ri 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]
The operator is a
The model is a particular type of
Stationary distribution
The stationary distribution is a
The additive decomposition method is numerically stable and separates the eigenvalues necessary for computation using Schur decomposition.[23][24]
On/off model
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 a
the stationary distribution can be computed explicitly and is given by[6]
and average fluid level[25]
Busy period
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
write W*(s) for the Laplace–Stieltjes transform of the busy period distribution, then[29]
which gives the mean busy period[30]
In this case, of a single on/off source, the busy period distribution is known to be a
There are two main approaches to solving for the busy period in general, using either spectral decomposition or an iterative recurrent method.[32] A
Example
For example, if a fluid queue with service rate μ = 2 is fed by an on/off source with parameters α = 2, β = 1 and λ = 3 then the fluid queue has busy period with mean 1 and variance 5/3.
Loss rate
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]
Mountain process
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 a G/M/1 queue.[35][36]
Networks of fluid queues
The stationary distribution of two tandem fluid queues has been computed and shown not to exhibit a
Feedback fluid queues
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.
Second order fluid queues
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]
External links
- BuTools, a Mathematicaimplementation of some of the above results.
- PevaTools, MATLAB code for multi-regime models
- Fluid flow models tutorial by V. Ramaswami at MAM8
References
- JSTOR 1427040.
- ^ S2CID 6733796.
- S2CID 19379411.
- S2CID 123591340.
- S2CID 121190780.
- ^ ISBN 978-0-8493-8076-1.
- Moran, P. A. P.(1954). "A probability theory of dams and storage systems". Aust. J. Appl. Sci. 5: 116–124.
- .
- S2CID 42193342.
- ^ S2CID 16836549.
- S2CID 14416842.
- .
- doi:10.1109/49.76636.
- doi:10.1109/49.76633.
- .
- S2CID 863180.
- JSTOR 1426410.
- ^ Ramaswami, V. Smith, D.; Hey, P (eds.). "Matrix analytic methods for stochastic fluid flows". Teletraffic Engineering in a Competitive World (Proceedings of the 16th International Teletraffic Congress). Elsevier Science B.V.
- S2CID 120102947.
- JSTOR 3215314.
- .
- ^ .
- JSTOR 3216036.
- ISBN 978-3-642-35979-8.
- ^ .
- ^ .
- .
- .
- ^ .
- ^ S2CID 3482641.
- .
- S2CID 53498442.
- .
- .
- .
- S2CID 9973624.
- .
- .
- S2CID 16150704.
- .
- S2CID 53363967.
- S2CID 19329962.
- .
- S2CID 1735120.