butools.mam.QBDSolve¶

butools.mam.QBDSolve()¶

Matlab:	[pi0, R] = QBDSolve (B, L, F, L0, prec)
Mathematica:	{pi0, R} = QBDSolve [B, L, F, L0, prec]
Python/Numpy:	pi0, R = QBDSolve (B, L, F, L0, prec)

Returns the parameters of the matrix-geometrically distributed stationary distribution of a QBD.

Using vector pi0 and matrix R provided by this function the stationary solution can be obtained by

\[\pi_k=\pi_0 R^k.\]

Parameters:

B : matrix, shape (N,N) The matrix corresponding to backward transitions L : matrix, shape (N,N) The matrix corresponding to local transitions F : matrix, shape (N,N) The matrix corresponding to forward transitions L0 : matrix, shape (N,N) The matrix corresponding to local transitions at level zero precision : double, optional The fundamental matrix R is computed up to this precision. The default value is 1e-14

Returns:

pi0 : matrix, shape (1,N) The stationary probability vector of level zero R : matrix, shape (N,N) The matrix parameter of the matrix geometrical distribution of the QBD

Examples

For Matlab:

>>> B = [0., 0.; 3., 4.];
>>> L = [-6., 5.; 3., -12.];
>>> F = [1., 0.; 2., 0.];
>>> L0 = [-6., 5.; 6., -8.];
>>> [pi0, R] = QBDSolve(B, L, F, L0);
>>> disp(pi0);
      0.22992      0.18681
>>> disp(R);
      0.27839      0.14286
      0.55678      0.28571

For Mathematica:

>>> B = {{0., 0.},{3., 4.}};
>>> L = {{-6., 5.},{3., -12.}};
>>> F = {{1., 0.},{2., 0.}};
>>> L0 = {{-6., 5.},{6., -8.}};
>>> {pi0, R} = QBDSolve[B, L, F, L0];
"Final Residual Error for G: "5.551115123125783*^-17
"Final Residual Error for R: "0.027036455607884147
>>> Print[pi0];
{0.22992392223161465, 0.18681318681318687}
>>> Print[R];
{{0.27838827838827834, 0.14285714285714282},
 {0.5567765567765567, 0.28571428571428564}}

For Python/Numpy:

>>> B = ml.matrix([[0., 0.],[3., 4.]])
>>> L = ml.matrix([[-6., 5.],[3., -12.]])
>>> F = ml.matrix([[1., 0.],[2., 0.]])
>>> L0 = ml.matrix([[-6., 5.],[6., -8.]])
>>> pi0, R = QBDSolve(B, L, F, L0)
Final Residual Error for G:  1.38777878078e-16
Final Residual Error for R:  5.55111512313e-17
>>> print(pi0)
[[ 0.22992  0.18681]]
>>> print(R)
[[ 0.27839  0.14286]
 [ 0.55678  0.28571]]