butools.map.MinimalRepFromMRAP¶

butools.map.MinimalRepFromMRAP()¶

Matlab:	D = MinimalRepFromMRAP(H, how, precision)
Mathematica:	H = MinimalRepFromMRAP[H, how, precision]
Python/Numpy:	H = MinimalRepFromMRAP(H, how, precision)

Returns the minimal representation of a marked rational arrival process.

Parameters:

H : list of matrices of shape (M,M) The list of H0, H1, ..., HK matrices of the marked rational arrival process how : {“obs”, “cont”, “obscont”}, optional Determines how the representation is minimized. “cont” means controllability, “obs” means observability, “obscont” means that the rational arrival process is minimized in both respects. Default value is “obscont”. precision : double, optional Precision used by the Staircase algorithm. The default value is 1e-12.

Returns:

D : list of matrices of shape (M,M) The D0, D1, ..., DK matrices of the minimal representation

References

[R35]

P. Buchholz, M. Telek, “On minimal representation of rational arrival processes.” Madrid Conference on Qeueuing theory (MCQT), June 2010.

Examples

For Matlab:

>>> D0 = [-5., 1., 0; 3., -3., 0; 1., 1., -5.];
>>> D1 = [0, 0, 0.8; 0, 0, 0; 0.2, 0.2, 0.2];
>>> D2 = [0, 0, 3.2; 0, 0, 0; 0.8, 0.8, 0.8];
>>> Dm = {D0, D1, D2};
>>> H = MinimalRepFromMRAP(Dm, 'cont');
>>> disp(H{1});
    -5     1     0
     3    -3     0
     1     1    -5
>>> disp(H{2});
            0            0          0.8
            0            0            0
          0.2          0.2          0.2
>>> disp(H{3});
            0            0          3.2
            0            0            0
          0.8          0.8          0.8
>>> Cm = SimilarityMatrix(H{1}, Dm{1});
>>> err = norm(H{1}*Cm-Cm*Dm{1})+norm(H{2}*Cm-Cm*Dm{2})+norm(H{3}*Cm-Cm*Dm{3});
>>> disp(err);
    4.656e-15
>>> H = MinimalRepFromMRAP(Dm, 'obs');
>>> disp(H{1});
      -4.4074       1.6931
      0.84259      -2.5926
>>> disp(H{2});
      0.40741      0.13545
      0.55741     -0.20741
>>> disp(H{3});
       1.6296       0.5418
       2.2296     -0.82963
>>> Cm = SimilarityMatrix(H{1}, Dm{1});
>>> err = norm(H{1}*Cm-Cm*Dm{1})+norm(H{2}*Cm-Cm*Dm{2})+norm(H{3}*Cm-Cm*Dm{3});
>>> disp(err);
   4.8469e-15
>>> H = MinimalRepFromMRAP(Dm, 'obscont');
>>> disp(H{1});
      -4.4074       1.6931
      0.84259      -2.5926
>>> disp(H{2});
      0.40741      0.13545
      0.55741     -0.20741
>>> disp(H{3});
       1.6296       0.5418
       2.2296     -0.82963
>>> Cm = SimilarityMatrix(H{1}, Dm{1});
>>> err = norm(H{1}*Cm-Cm*Dm{1})+norm(H{2}*Cm-Cm*Dm{2})+norm(H{3}*Cm-Cm*Dm{3});
>>> disp(err);
   4.8469e-15

For Mathematica:

>>> D0 = {{-5., 1., 0},{3., -3., 0},{1., 1., -5.}};
>>> D1 = {{0, 0, 0.8},{0, 0, 0},{0.2, 0.2, 0.2}};
>>> D2 = {{0, 0, 3.2},{0, 0, 0},{0.8, 0.8, 0.8}};
>>> Dm = {D0, D1, D2};
>>> H = MinimalRepFromMRAP[Dm, "cont"];
>>> Print[H[[1]]];
{{-5., 1., 0.},
 {3., -3., 0.},
 {1., 1., -5.}}
>>> Print[H[[2]]];
{{0., 0., 0.8},
 {0., 0., 0.},
 {0.2, 0.2, 0.2}}
>>> Print[H[[3]]];
{{0., 0., 3.2},
 {0., 0., 0.},
 {0.8, 0.8, 0.8}}
>>> Cm = SimilarityMatrix[H[[1]], Dm[[1]]];
>>> err = Norm[H[[1]].Cm-Cm.Dm[[1]]]+Norm[H[[2]].Cm-Cm.Dm[[2]]]+Norm[H[[3]].Cm-Cm.Dm[[3]]];
>>> Print[err];
5.717834399042951*^-15
>>> H = MinimalRepFromMRAP[Dm, "obs"];
>>> Print[H[[1]]];
{{-4.407407407407407, 1.6931216931216932},
 {0.8425925925925922, -2.592592592592593}}
>>> Print[H[[2]]];
{{0.40740740740740733, 0.13544973544973554},
 {0.5574074074074076, -0.20740740740740737}}
>>> Print[H[[3]]];
{{1.6296296296296293, 0.5417989417989422},
 {2.2296296296296303, -0.8296296296296295}}
>>> Cm = SimilarityMatrix[H[[1]], Dm[[1]]];
>>> err = Norm[H[[1]].Cm-Cm.Dm[[1]]]+Norm[H[[2]].Cm-Cm.Dm[[2]]]+Norm[H[[3]].Cm-Cm.Dm[[3]]];
>>> Print[err];
8.2600373258031*^-15
>>> H = MinimalRepFromMRAP[Dm, "obscont"];
>>> Print[H[[1]]];
{{-4.407407407407407, 1.6931216931216932},
 {0.8425925925925922, -2.592592592592593}}
>>> Print[H[[2]]];
{{0.40740740740740733, 0.13544973544973554},
 {0.5574074074074076, -0.20740740740740737}}
>>> Print[H[[3]]];
{{1.6296296296296293, 0.5417989417989422},
 {2.2296296296296303, -0.8296296296296295}}
>>> Cm = SimilarityMatrix[H[[1]], Dm[[1]]];
>>> err = Norm[H[[1]].Cm-Cm.Dm[[1]]]+Norm[H[[2]].Cm-Cm.Dm[[2]]]+Norm[H[[3]].Cm-Cm.Dm[[3]]];
>>> Print[err];
8.2600373258031*^-15

For Python/Numpy:

>>> D0 = ml.matrix([[-5., 1., 0],[3., -3., 0],[1., 1., -5.]])
>>> D1 = ml.matrix([[0, 0, 0.8],[0, 0, 0],[0.2, 0.2, 0.2]])
>>> D2 = ml.matrix([[0, 0, 3.2],[0, 0, 0],[0.8, 0.8, 0.8]])
>>> Dm = [D0, D1, D2]
>>> H = MinimalRepFromMRAP(Dm, "cont")
>>> print(H[0])
[[-5.  1.  0.]
 [ 3. -3.  0.]
 [ 1.  1. -5.]]
>>> print(H[1])
[[ 0.   0.   0.8]
 [ 0.   0.   0. ]
 [ 0.2  0.2  0.2]]
>>> print(H[2])
[[ 0.   0.   3.2]
 [ 0.   0.   0. ]
 [ 0.8  0.8  0.8]]
>>> Cm = SimilarityMatrix(H[0], Dm[0])
>>> err = la.norm(H[0]*Cm-Cm*Dm[0])+la.norm(H[1]*Cm-Cm*Dm[1])+la.norm(H[2]*Cm-Cm*Dm[2])
>>> print(err)
3.65128686018e-15
>>> H = MinimalRepFromMRAP(Dm, "obs")
>>> print(H[0])
[[-4.40741  1.69312]
 [ 0.84259 -2.59259]]
>>> print(H[1])
[[ 0.40741  0.13545]
 [ 0.55741 -0.20741]]
>>> print(H[2])
[[ 1.62963  0.5418 ]
 [ 2.22963 -0.82963]]
>>> Cm = SimilarityMatrix(H[0], Dm[0])
>>> err = la.norm(H[0]*Cm-Cm*Dm[0])+la.norm(H[1]*Cm-Cm*Dm[1])+la.norm(H[2]*Cm-Cm*Dm[2])
>>> print(err)
4.72728076906e-15
>>> H = MinimalRepFromMRAP(Dm, "obscont")
>>> print(H[0])
[[-4.40741  1.69312]
 [ 0.84259 -2.59259]]
>>> print(H[1])
[[ 0.40741  0.13545]
 [ 0.55741 -0.20741]]
>>> print(H[2])
[[ 1.62963  0.5418 ]
 [ 2.22963 -0.82963]]
>>> Cm = SimilarityMatrix(H[0], Dm[0])
>>> err = la.norm(H[0]*Cm-Cm*Dm[0])+la.norm(H[1]*Cm-Cm*Dm[1])+la.norm(H[2]*Cm-Cm*Dm[2])
>>> print(err)
4.72728076906e-15