butools.mam.FluidSolve

butools.mam.FluidSolve()
Matlab: [mass0, ini, K, clo] = FluidSolve (Fpp, Fpm, Fmp, Fmm, prec)
Mathematica: {mass0, ini, K, clo} = FluidSolve [Fpp, Fpm, Fmp, Fmm, prec]
Python/Numpy: mass0, ini, K, clo = FluidSolve (Fpp, Fpm, Fmp, Fmm, prec)

Returns the parameters of the matrix-exponentially distributed stationary distribution of a canonical Markovian fluid model.

The canonical Markov fluid model is defined by the matrix blocks of the generator of the background Markov chain partitioned according to the sign of the associated fluid rates (i.e., there are “+” and “-” states).

Using the returned 4 parameters the stationary solution can be obtained as follows.

The probability that the fluid level is zero while being in different states of the background process is given by vector mass0.

The density that the fluid level is x while being in different states of the background process is

\[\pi(x)=ini\cdot e^{K x}\cdot clo.\]
Parameters:

Fpp : matrix, shape (Np,Np)

The matrix of transition rates between states having positive fluid rates

Fpm : matrix, shape (Np,Nm)

The matrix of transition rates where the source state has a positive, the destination has a negative fluid rate associated.

Fpm : matrix, shape (Nm,Np)

The matrix of transition rates where the source state has a negative, the destination has a positive fluid rate associated.

Fpp : matrix, shape (Nm,Nm)

The matrix of transition rates between states having negative fluid rates

precision : double, optional

Numerical precision for computing the fundamental matrix. The default value is 1e-14
Returns:

mass0 : matrix, shape (1,Np+Nm)

The stationary probability vector of zero level

ini : matrix, shape (1,Np)

The initial vector of the stationary density

K : matrix, shape (Np,Np)

The matrix parameter of the stationary density

clo : matrix, shape (Np,Np+Nm)

The closing matrix of the stationary density

Examples

For Matlab:

>>> Fpp = [-5., 1.; 2., -3.];
>>> Fpm = [2., 1., 1.; 1., 0., 0.];
>>> Fmm = [-8., 4., 1.; 2., -12., 3.; 2., 0., -2.];
>>> Fmp = [3., 0.; 2., 5.; 0., 0.];
>>> x = 0.7;
>>> [mass0, ini, K, clo] = FluidSolve(Fpp, Fpm, Fmp, Fmm);
Final Residual Error for Psi:    1.1657e-15
>>> disp(mass0);
     0.037514     0.015303     0.097921
>>> disp(ini);
      0.14315     0.076517
>>> disp(K);
       -3.658       1.8258
       3.2553      -2.3502
>>> disp(clo);
            1            0      0.33722      0.16517      0.49761
            0            1       0.3318      0.12995      0.53825
>>> pdfAtX = ini*expm(K*x)*clo;
>>> disp(pdfAtX);
     0.074009     0.070933     0.048493     0.021442     0.075007

For Mathematica:

>>> Fpp = {{-5., 1.},{2., -3.}};
>>> Fpm = {{2., 1., 1.},{1., 0., 0.}};
>>> Fmm = {{-8., 4., 1.},{2., -12., 3.},{2., 0., -2.}};
>>> Fmp = {{3., 0.},{2., 5.},{0., 0.}};
>>> x = 0.7;
>>> {mass0, ini, K, clo} = FluidSolve[Fpp, Fpm, Fmp, Fmm];
"Final Residual Error for Psi: "1.1657341758564144*^-15
>>> Print[mass0];
{0.03751362697958451, 0.0153034356482914, 0.09792058971336806}
>>> Print[ini];
{0.14314775223533632, 0.076517178241457}
>>> Print[K];
{{-3.65799640319986, 1.8258294108775632},
 {3.255293764043593, -2.350237730252557}}
>>> Print[clo];
{{1, 0, 0.33722394414970486, 0.16516588217551262, 0.4976101736747833},
 {0, 1, 0.3317962853815385, 0.12995245394948857, 0.5382512606689742}}
>>> pdfAtX = ini.MatrixExponential[K*x].clo;
>>> Print[pdfAtX];
{0.14314775223533632, 0.076517178241457} . MatrixExponential[{{-2.560597482239902, 1.2780805876142942}, {2.278705634830515, -1.6451664111767899}}] . {{1, 0, 0.33722394414970486, 0.16516588217551262, 0.4976101736747833}, {0, 1, 0.3317962853815385, 0.12995245394948857, 0.5382512606689742}}

For Python/Numpy:

>>> Fpp = ml.matrix([[-5., 1.],[2., -3.]])
>>> Fpm = ml.matrix([[2., 1., 1.],[1., 0., 0.]])
>>> Fmm = ml.matrix([[-8., 4., 1.],[2., -12., 3.],[2., 0., -2.]])
>>> Fmp = ml.matrix([[3., 0.],[2., 5.],[0., 0.]])
>>> x = 0.7
>>> mass0, ini, K, clo = FluidSolve(Fpp, Fpm, Fmp, Fmm)
Final Residual Error for G:  1.7208456881689926e-15
>>> print(mass0)
[[ 0.03751  0.0153   0.09792]]
>>> print(ini)
[[ 0.14315  0.07652]]
>>> print(K)
[[-3.658    1.82583]
 [ 3.25529 -2.35024]]
>>> print(clo)
[[ 1.       0.       0.33722  0.16517  0.49761]
 [ 0.       1.       0.3318   0.12995  0.53825]]
>>> pdfAtX = ini*la.expm(K*x)*clo
>>> print(pdfAtX)
[[ 0.07401  0.07093  0.04849  0.02144  0.07501]]