butools.map.MRAPFromMoments¶

butools.map.MRAPFromMoments()¶

Matlab:	H = MRAPFromMoments(moms, Nm)
Mathematica:	H = MRAPFromMoments[moms, Nm]
Python/Numpy:	H = MRAPFromMoments(moms, Nm)

Creates a marked rational arrival process that has the same marginal and lag-1 joint moments as given (see [R32]).

Parameters:	moms : vector of doubles The list of marginal moments. To obtain a marked rational process of order M, 2*M-1 marginal moments are required. Nm : list of matrices, shape (M,M) The list of lag-1 joint moment matrices. The length of the list determines K, the number of arrival types of the rational process.
Returns:	H : list of matrices, shape (M,M) The H0, H1, ..., HK matrices of the marked rational process

Notes

There is no guarantee that the returned matrices define a valid stochastic process. The joint densities may be negative.

References

[R32]

(1, 2) Andras Horvath, Gabor Horvath, Miklos Telek, “A traffic based decomposition of two-class queueing networks with priority service,” Computer Networks 53:(8) pp. 1235-1248. (2009)

Examples

For Matlab:

>>> G0 = [-1.05, 0.03, 0.07; 0.19, -1.63, 0.06; 0, 0.2, -1.03];
>>> G1 = [0.16, 0.11, 0; 0.1, 0.16, 0; 0.27, 0, 0.19];
>>> G2 = [0.01, 0.09, 0.13; 0.26, 0.21, 0.05; 0, 0.16, 0.07];
>>> G3 = [0.19, 0.06, 0.2; 0.17, 0.16, 0.27; 0, 0, 0.14];
>>> G = {G0, G1, G2, G3};
>>> moms = MarginalMomentsFromMRAP(G, 5);
>>> disp(moms);
      0.99805       2.0262       6.2348       25.738        133.3
>>> Nm = LagkJointMomentsFromMRAP(G, 2, 1);
>>> [Nm1, Nm2, Nm3] = Nm{:};
>>> disp(Nm1);
         0.34      0.34055      0.69322
      0.34966      0.35186      0.71829
      0.72396      0.73079       1.4947
>>> disp(Nm2);
      0.29553      0.27734      0.54078
      0.29028      0.27211      0.53013
      0.58391      0.54682       1.0646
>>> disp(Nm3);
      0.36447      0.38016      0.79224
      0.35811      0.37446      0.78148
      0.71836      0.75239       1.5716
>>> H = MRAPFromMoments(moms, Nm);
>>> disp(H{1});
      -1.9473       3.0344      -2.1704
     -0.33434     -0.88118      0.21355
     -0.33363      0.21321     -0.88152
>>> disp(H{2});
      0.22304       7.4962      -7.3973
     0.092805       7.9131      -7.6658
     0.086985       7.8804      -7.6261
>>> disp(H{3});
      0.49395      0.92951      -1.0798
       0.4418      -2.4454       2.2995
      0.44568      -2.3908       2.2415
>>> disp(H{4});
     -0.14271      -5.4356       5.9959
     -0.19985      -5.1102       5.6761
     -0.19945      -5.1791       5.7429
>>> BuToolsCheckPrecision = 10.^-10;
>>> rmoms = MarginalMomentsFromMRAP(H, 5);
>>> disp(rmoms);
      0.99805       2.0262       6.2348       25.738        133.3
>>> rNm = LagkJointMomentsFromMRAP(H, 2, 1);
>>> [rNm1, rNm2, rNm3] = rNm{:};
>>> disp(rNm1);
         0.34      0.34055      0.69322
      0.34966      0.35186      0.71829
      0.72396      0.73079       1.4947
>>> disp(rNm2);
      0.29553      0.27734      0.54078
      0.29028      0.27211      0.53013
      0.58391      0.54682       1.0646
>>> disp(rNm3);
      0.36447      0.38016      0.79224
      0.35811      0.37446      0.78148
      0.71836      0.75239       1.5716

For Mathematica:

>>> G0 = {{-1.05, 0.03, 0.07},{0.19, -1.63, 0.06},{0, 0.2, -1.03}};
>>> G1 = {{0.16, 0.11, 0},{0.1, 0.16, 0},{0.27, 0, 0.19}};
>>> G2 = {{0.01, 0.09, 0.13},{0.26, 0.21, 0.05},{0, 0.16, 0.07}};
>>> G3 = {{0.19, 0.06, 0.2},{0.17, 0.16, 0.27},{0, 0, 0.14}};
>>> G = {G0, G1, G2, G3};
>>> moms = MarginalMomentsFromMRAP[G, 5];
>>> Print[moms];
{0.998048613345479, 2.0262345014785748, 6.2347773609141575, 25.738432370871866, 133.30214790750443}
>>> Nm = LagkJointMomentsFromMRAP[G, 2, 1];
>>> {Nm1, Nm2, Nm3} = Nm;
>>> Print[Nm1];
{{0.3400009192827923, 0.3405452798817129, 0.6932173701035182},
 {0.3496600253904083, 0.3518559816976867, 0.7182912226288702},
 {0.7239624620818725, 0.7307864272366612, 1.494690822742042}}
>>> Print[Nm2];
{{0.29553270348533534, 0.27734235602329654, 0.5407773515893262},
 {0.29028125712379704, 0.27210886385024, 0.5301325803470373},
 {0.5839079003431904, 0.5468180031775005, 1.064574978879992}}
>>> Print[Nm3];
{{0.36446637723187236, 0.3801609774404695, 0.7922397797857303},
 {0.35810733083127366, 0.37445996131218023, 0.7814764837751109},
 {0.7183641390535117, 0.752387606929777, 1.5716457698900985}}
>>> H = MRAPFromMoments[moms, Nm];
>>> Print[H[[1]]];
{{-1.9473051937398436, 3.0344139261695524, -2.1703554254825432},
 {-0.3343442044701893, -0.8811768825595621, 0.2135477889258592},
 {-0.33362593020289716, 0.2132067478864758, -0.8815179235989457}}
>>> Print[H[[2]]];
{{0.22303732415660393, 7.496153934451286, -7.397276128273006},
 {0.09280493616992504, 7.913084312262072, -7.665823186685884},
 {0.0869846290873446, 7.880402325125033, -7.626121636429161}}
>>> Print[H[[3]]];
{{0.4939487332101481, 0.9295081378932082, -1.079810419889327},
 {0.4418009261624444, -2.4454040931686905, 2.2994668648207153},
 {0.44567916840142563, -2.3907771669346403, 2.241455359959218}}
>>> Print[H[[4]]];
{{-0.1427050168593569, -5.435557714889001, 5.995947843257454},
 {-0.19984735284082544, -5.110198412374302, 5.6760893037571805},
 {-0.19945217229000395, -5.17913683023653, 5.742903429225407}}
>>> BuTools`CheckPrecision = 10.^-10;
>>> rmoms = MarginalMomentsFromMRAP[H, 5];
>>> Print[rmoms];
{0.9980486133447148, 2.026234501476063, 6.23477736090465, 25.738432370828296, 133.30214790726575}
>>> rNm = LagkJointMomentsFromMRAP[H, 2, 1];
>>> {rNm1, rNm2, rNm3} = rNm;
>>> Print[rNm1];
{{0.3400009192835025, 0.3405452798822086, 0.6932173701042554},
 {0.3496600253876201, 0.3518559816946487, 0.7182912226223763},
 {0.7239624620661935, 0.7307864272203464, 1.4946908227080655}}
>>> Print[rNm2];
{{0.29553270348466576, 0.27734235602249724, 0.5407773515875407},
 {0.2902812571227895, 0.2721088638491218, 0.5301325803446262},
 {0.5839079003408321, 0.5468180031749403, 1.0645749788745391}}
>>> Print[rNm3];
{{0.3644663772275738, 0.3801609774359047, 0.7922397797761231},
 {0.35810733082500956, 0.374459961305627, 0.7814764837614341},
 {0.7183641390383446, 0.752387606913997, 1.5716457698572626}}

For Python/Numpy:

>>> G0 = ml.matrix([[-1.05, 0.03, 0.07],[0.19, -1.63, 0.06],[0, 0.2, -1.03]])
>>> G1 = ml.matrix([[0.16, 0.11, 0],[0.1, 0.16, 0],[0.27, 0, 0.19]])
>>> G2 = ml.matrix([[0.01, 0.09, 0.13],[0.26, 0.21, 0.05],[0, 0.16, 0.07]])
>>> G3 = ml.matrix([[0.19, 0.06, 0.2],[0.17, 0.16, 0.27],[0, 0, 0.14]])
>>> G = [G0, G1, G2, G3]
>>> moms = MarginalMomentsFromMRAP(G, 5)
>>> print(moms)
[0.99804861334547901, 2.0262345014785748, 6.2347773609141584, 25.738432370871866, 133.30214790750446]
>>> Nm = LagkJointMomentsFromMRAP(G, 2, 1)
>>> Nm1, Nm2, Nm3 = Nm
>>> print(Nm1)
[[ 0.34     0.34055  0.69322]
 [ 0.34966  0.35186  0.71829]
 [ 0.72396  0.73079  1.49469]]
>>> print(Nm2)
[[ 0.29553  0.27734  0.54078]
 [ 0.29028  0.27211  0.53013]
 [ 0.58391  0.54682  1.06457]]
>>> print(Nm3)
[[ 0.36447  0.38016  0.79224]
 [ 0.35811  0.37446  0.78148]
 [ 0.71836  0.75239  1.57165]]
>>> H = MRAPFromMoments(moms, Nm)
>>> print(H[0])
[[-1.94731  3.03441 -2.17036]
 [-0.33434 -0.88118  0.21355]
 [-0.33363  0.21321 -0.88152]]
>>> print(H[1])
[[ 0.22304  7.49615 -7.39728]
 [ 0.0928   7.91308 -7.66582]
 [ 0.08698  7.8804  -7.62612]]
>>> print(H[2])
[[ 0.49395  0.92951 -1.07981]
 [ 0.4418  -2.4454   2.29947]
 [ 0.44568 -2.39078  2.24146]]
>>> print(H[3])
[[-0.14271 -5.43556  5.99595]
 [-0.19985 -5.1102   5.67609]
 [-0.19945 -5.17914  5.7429 ]]
>>> butools.checkPrecision = 10.**-10
>>> rmoms = MarginalMomentsFromMRAP(H, 5)
>>> print(rmoms)
[0.9980486133458466, 2.026234501479784, 6.2347773609187414, 25.738432370892887, 133.30214790761983]
>>> rNm = LagkJointMomentsFromMRAP(H, 2, 1)
>>> rNm1, rNm2, rNm3 = rNm
>>> print(rNm1)
[[ 0.34     0.34055  0.69322]
 [ 0.34966  0.35186  0.71829]
 [ 0.72396  0.73079  1.49469]]
>>> print(rNm2)
[[ 0.29553  0.27734  0.54078]
 [ 0.29028  0.27211  0.53013]
 [ 0.58391  0.54682  1.06457]]
>>> print(rNm3)
[[ 0.36447  0.38016  0.79224]
 [ 0.35811  0.37446  0.78148]
 [ 0.71836  0.75239  1.57165]]