butools.mam.GM1StationaryDistr¶

butools.mam.GM1StationaryDistr()¶

Matlab:	pi = GM1StationaryDistr (B, R, K)
Mathematica:	pi = GM1StationaryDistr [B, R, K]
Python/Numpy:	pi = GM1StationaryDistr (B, R, K)

Returns the stationary distribution of the G/M/1 type Markov chain up to a given level K.

Parameters:	A : length(M) list of matrices of shape (N,N) Matrix blocks of the G/M/1 type generator in the regular part, from 0 to M-1. B : length(M) list of matrices of shape (N,N) Matrix blocks of the G/M/1 type generator at the R : matrix, shape (N,N) Matrix R of the G/M/1 type Markov chain K : integer The stationary distribution is returned up to this level.
Returns:	pi : array, shape (1,(K+1)*N) The stationary probability vector up to level K

Examples

For Matlab:

>>> B0 = [0.7, 0.2; 0.3, 0.6];
>>> B1 = [0.3, 0.4; 0.5, 0.2];
>>> B2 = [0.2, 0.4; 0.1, 0.6];
>>> B3 = [0., 0.1; 0.2, 0.];
>>> A0 = [0.1, 0.; 0., 0.1];
>>> A1 = [0., 0.2; 0., 0.2];
>>> A2 = [0., 0.1; 0., 0.];
>>> A3 = [0.3, 0.2; 0.3, 0.2];
>>> A4 = [0., 0.1; 0.2, 0.];
>>> B = {B0, B1, B2, B3};
>>> A = {A0, A1, A2, A3, A4};
>>> R = GM1FundamentalMatrix(A);
>>> disp(R);
      0.10065     0.026961
   0.00065531      0.12569
>>> pi = GM1StationaryDistr(B, R, 300);

For Mathematica:

>>> B0 = {{0.7, 0.2},{0.3, 0.6}};
>>> B1 = {{0.3, 0.4},{0.5, 0.2}};
>>> B2 = {{0.2, 0.4},{0.1, 0.6}};
>>> B3 = {{0., 0.1},{0.2, 0.}};
>>> A0 = {{0.1, 0.},{0., 0.1}};
>>> A1 = {{0., 0.2},{0., 0.2}};
>>> A2 = {{0., 0.1},{0., 0.}};
>>> A3 = {{0.3, 0.2},{0.3, 0.2}};
>>> A4 = {{0., 0.1},{0.2, 0.}};
>>> B = {B0, B1, B2, B3};
>>> A = {A0, A1, A2, A3, A4};
>>> R = GM1FundamentalMatrix[A];
"The evaluation of the iteration required "64" roots\n"
"The evaluation of the iteration required "32" roots\n"
"The evaluation of the iteration required "16" roots\n"
"The evaluation of the iteration required "8" roots\n"
"The evaluation of the iteration required "8" roots\n"
"Final Residual Error for G: "5.551115123125783*^-17
>>> Print[R];
{{0.10065149910973312, 0.026960920607274754},
 {0.0006553100576153258, 0.12568710472819553}}
>>> pi = GM1StationaryDistr[B, R, 300];
"Accumulated mass after "2" iterations: "0.9838720044873233
"Accumulated mass after "3" iterations: "0.9979548824322513
"Accumulated mass after "4" iterations: "0.9997408547470504
"Accumulated mass after "5" iterations: "0.9999671812477241
"Accumulated mass after "6" iterations: "0.9999958456126867
"Accumulated mass after "7" iterations: "0.999999474298702
"Accumulated mass after "8" iterations: "0.9999999334955769
"Accumulated mass after "9" iterations: "0.9999999915886283
"Accumulated mass after "10" iterations: "0.9999999989363275
"Accumulated mass after "11" iterations: "0.9999999998655101
"Accumulated mass after "12" iterations: "0.999999999982997

For Python/Numpy:

>>> B0 = ml.matrix([[0.7, 0.2],[0.3, 0.6]])
>>> B1 = ml.matrix([[0.3, 0.4],[0.5, 0.2]])
>>> B2 = ml.matrix([[0.2, 0.4],[0.1, 0.6]])
>>> B3 = ml.matrix([[0., 0.1],[0.2, 0.]])
>>> A0 = ml.matrix([[0.1, 0.],[0., 0.1]])
>>> A1 = ml.matrix([[0., 0.2],[0., 0.2]])
>>> A2 = ml.matrix([[0., 0.1],[0., 0.]])
>>> A3 = ml.matrix([[0.3, 0.2],[0.3, 0.2]])
>>> A4 = ml.matrix([[0., 0.1],[0.2, 0.]])
>>> B = [B0, B1, B2, B3]
>>> A = [A0, A1, A2, A3, A4]
>>> R = GM1FundamentalMatrix(A)
The Shifted PWCR evaluation of Iteration  1  required  64  roots
The Shifted PWCR evaluation of Iteration  2  required  32  roots
The Shifted PWCR evaluation of Iteration  3  required  16  roots
The Shifted PWCR evaluation of Iteration  4  required  8  roots
The Shifted PWCR evaluation of Iteration  5  required  8  roots
Final Residual Error for G:  5.20417042793e-17
>>> print(R)
[[ 0.10065  0.02696]
 [ 0.00066  0.12569]]
>>> pi = GM1StationaryDistr(B, R, 300)
Accumulated mass after  2  iterations:  0.983872004487
Accumulated mass after  3  iterations:  0.997954882432
Accumulated mass after  4  iterations:  0.999740854747
Accumulated mass after  5  iterations:  0.999967181248
Accumulated mass after  6  iterations:  0.999995845613
Accumulated mass after  7  iterations:  0.999999474299
Accumulated mass after  8  iterations:  0.999999933496
Accumulated mass after  9  iterations:  0.999999991589
Accumulated mass after  10  iterations:  0.999999998936
Accumulated mass after  11  iterations:  0.999999999866
Accumulated mass after  12  iterations:  0.999999999983