butools.dmap.DMRAPFromMoments¶

butools.dmap.DMRAPFromMoments()¶

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

Creates a discrete marked rational arrival process that has the same marginal and lag-1 joint moments as given (see [R9]).

Parameters:	moms : vector of doubles The list of marginal moments. To obtain a discrete 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 discrete rational process.
Returns:	H : list of matrices, shape (M,M) The H0, H1, ..., HK matrices of the discrete marked rational process

References

[R9]	(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 = [0.34, 0, 0; 0.06, 0.05, 0.03; 0.11, 0.13, 0];
>>> G1 = [0.3, 0, 0; 0.16, 0.18, 0.05; 0.15, 0.04, 0.09];
>>> G2 = [0, 0.01, 0; 0.1, 0.07, 0.08; 0.13, 0.12, 0.13];
>>> G3 = [0.35, 0, 0; 0, 0.18, 0.04; 0.06, 0.03, 0.01];
>>> G = {G0, G1, G2, G3};
>>> moms = MarginalMomentsFromDMRAP(G, 5);
>>> disp(moms);
       1.5037       3.0278       8.4243       31.097       143.88
>>> Nm = LagkJointMomentsFromDMRAP(G, 2, 1);
>>> [Nm1, Nm2, Nm3] = Nm{:};
>>> disp(Nm1);
      0.45395      0.68525       1.3856
      0.68283       1.0318       2.0887
       1.3755       2.0807       4.2171
>>> disp(Nm2);
     0.026281      0.03323     0.053055
     0.035051     0.043866     0.068917
     0.060653      0.07477      0.11464
>>> disp(Nm3);
      0.51977      0.78522       1.5891
      0.78582       1.1881       2.4067
       1.5917       2.4087       4.8838
>>> H = DMRAPFromMoments(moms, Nm);
>>> disp(H{1});
     0.067795      0.67922     -0.42509
    -0.018716    0.0035507      0.35386
    -0.019902     0.039925      0.31865
>>> disp(H{2});
      0.19512       1.6571      -1.5453
   -0.0081004      0.18194      0.12658
   -0.0076905      0.11526      0.19294
>>> disp(H{3});
      0.25787       3.8341      -4.0555
      0.12827      -1.3508       1.2347
      0.12666      -1.4076       1.2929
>>> disp(H{4});
      0.17441     -0.11574      0.27606
    -0.014006    -0.013671      0.37636
    -0.012887      -0.0175      0.37926
>>> BuToolsCheckPrecision = 10.^-10;
>>> rmoms = MarginalMomentsFromDMRAP(H, 5);
>>> disp(rmoms);
       1.5037       3.0278       8.4243       31.097       143.88
>>> rNm = LagkJointMomentsFromDMRAP(H, 2, 1);
>>> [rNm1, rNm2, rNm3] = rNm{:};
>>> disp(rNm1);
      0.45395      0.68525       1.3856
      0.68283       1.0318       2.0887
       1.3755       2.0807       4.2171
>>> disp(rNm2);
     0.026281      0.03323     0.053055
     0.035051     0.043866     0.068917
     0.060653      0.07477      0.11464
>>> disp(rNm3);
      0.51977      0.78522       1.5891
      0.78582       1.1881       2.4067
       1.5917       2.4087       4.8838
>>> disp(norm(moms-rmoms));
   3.0056e-11

For Mathematica:

>>> G0 = {{0.34, 0, 0},{0.06, 0.05, 0.03},{0.11, 0.13, 0}};
>>> G1 = {{0.3, 0, 0},{0.16, 0.18, 0.05},{0.15, 0.04, 0.09}};
>>> G2 = {{0, 0.01, 0},{0.1, 0.07, 0.08},{0.13, 0.12, 0.13}};
>>> G3 = {{0.35, 0, 0},{0, 0.18, 0.04},{0.06, 0.03, 0.01}};
>>> G = {G0, G1, G2, G3};
>>> moms = MarginalMomentsFromDMRAP[G, 5];
>>> Print[moms];
{1.503697331491185, 3.027841857350894, 8.424305390832199, 31.097386717744087, 143.8804184010114}
>>> Nm = LagkJointMomentsFromDMRAP[G, 2, 1];
>>> {Nm1, Nm2, Nm3} = Nm;
>>> Print[Nm1];
{{0.4539462822484643, 0.6852510653292481, 1.3856372122857255},
 {0.6828250179169082, 1.0317922106199087, 2.088678354109492},
 {1.3754547360883858, 2.08069063568443, 4.2170625241275665}}
>>> Print[Nm2];
{{0.02628103195085287, 0.03323044515488185, 0.05305519981654366},
 {0.0350514750959702, 0.04386614712772344, 0.06891661923475792},
 {0.06065278939554716, 0.07477024474361887, 0.11464021882832742}}
>>> Print[Nm3];
{{0.5197726858006831, 0.7852158210070554, 1.5891494452486254},
 {0.7858208384783067, 1.1881008368374677, 2.4066502002309207},
 {1.5917343318669612, 2.4086996142807697, 4.883779522055594}}
>>> H = DMRAPFromMoments[moms, Nm];
>>> Print[H[[1]]];
{{0.06779513528098983, 0.6792233137807228, -0.42509134379512803},
 {-0.018716173794511234, 0.0035506821092199867, 0.3538644199083848},
 {-0.019901518585921547, 0.039924500868717246, 0.31865418278856805}}
>>> Print[H[[2]]];
{{0.1951207776434683, 1.6570695160216928, -1.5452971870081456},
 {-0.008100374639338296, 0.18194378529369715, 0.12657574230343016},
 {-0.007690549936086862, 0.115257691431907, 0.19293543687217607}}
>>> Print[H[[3]]];
{{0.2578685382565766, 3.834117264599513, -4.055530267739243},
 {0.12827076956352768, -1.350769988779902, 1.2347002335875743},
 {0.12665640477215312, -1.4076139251314146, 1.2929014505253633}}
>>> Print[H[[4]]];
{{0.1744074874552913, -0.11574488139467576, 0.27606164689950674},
 {-0.014005700448463549, -0.013671167607753887, 0.3763577725021605},
 {-0.012887066531690031, -0.01750028735204978, 0.3792636802754714}}
>>> BuTools`CheckPrecision = 10.^-10;
>>> rmoms = MarginalMomentsFromDMRAP[H, 5];
>>> Print[rmoms];
{1.5036973314912396, 3.0278418573511203, 8.424305390833059, 31.09738671774769, 143.88041840102892}
>>> rNm = LagkJointMomentsFromDMRAP[H, 2, 1];
>>> {rNm1, rNm2, rNm3} = rNm;
>>> Print[rNm1];
{{0.4539462822461965, 0.6852510653258723, 1.3856372122790048},
 {0.6828250179124524, 1.0317922106132444, 2.0886783540961558},
 {1.3754547360759344, 2.080690635665735, 4.217062524089992}}
>>> Print[rNm2];
{{0.026281031950898093, 0.033230445154958294, 0.053055199816716886},
 {0.03505147509598383, 0.043866147127753385, 0.06891661923484005},
 {0.06065278939548402, 0.07477024474353555, 0.11464021882818698}}
>>> Print[rNm3];
{{0.519772685800297, 0.7852158210065349, 1.5891494452477044},
 {0.7858208384782196, 1.1881008368374257, 2.406650200231027},
 {1.5917343318688073, 2.408699614283732, 4.88377952206196}}
>>> Print[Norm[moms-rmoms]];
1.7896738102973677*^-11

For Python/Numpy:

>>> G0 = ml.matrix([[0.34, 0, 0],[0.06, 0.05, 0.03],[0.11, 0.13, 0]])
>>> G1 = ml.matrix([[0.3, 0, 0],[0.16, 0.18, 0.05],[0.15, 0.04, 0.09]])
>>> G2 = ml.matrix([[0, 0.01, 0],[0.1, 0.07, 0.08],[0.13, 0.12, 0.13]])
>>> G3 = ml.matrix([[0.35, 0, 0],[0, 0.18, 0.04],[0.06, 0.03, 0.01]])
>>> G = [G0, G1, G2, G3]
>>> moms = MarginalMomentsFromDMRAP(G, 5)
>>> print(moms)
[1.503697331491185, 3.0278418573508938, 8.424305390832199, 31.097386717744087, 143.88041840101141]
>>> Nm = LagkJointMomentsFromDMRAP(G, 2, 1)
>>> Nm1, Nm2, Nm3 = Nm
>>> print(Nm1)
[[ 0.45395  0.68525  1.38564]
 [ 0.68283  1.03179  2.08868]
 [ 1.37545  2.08069  4.21706]]
>>> print(Nm2)
[[ 0.02628  0.03323  0.05306]
 [ 0.03505  0.04387  0.06892]
 [ 0.06065  0.07477  0.11464]]
>>> print(Nm3)
[[ 0.51977  0.78522  1.58915]
 [ 0.78582  1.1881   2.40665]
 [ 1.59173  2.4087   4.88378]]
>>> H = DMRAPFromMoments(moms, Nm)
>>> print(H[0])
[[ 0.0678   0.67922 -0.42509]
 [-0.01872  0.00355  0.35386]
 [-0.0199   0.03992  0.31865]]
>>> print(H[1])
[[ 0.19512  1.65707 -1.5453 ]
 [-0.0081   0.18194  0.12658]
 [-0.00769  0.11526  0.19294]]
>>> print(H[2])
[[ 0.25787  3.83412 -4.05553]
 [ 0.12827 -1.35077  1.2347 ]
 [ 0.12666 -1.40761  1.2929 ]]
>>> print(H[3])
[[ 0.17441 -0.11574  0.27606]
 [-0.01401 -0.01367  0.37636]
 [-0.01289 -0.0175   0.37926]]
>>> butools.checkPrecision = 10.**-10
>>> rmoms = MarginalMomentsFromDMRAP(H, 5)
>>> print(rmoms)
[1.5036973314922903, 3.0278418573555848, 8.4243053908503818, 31.09738671782155, 143.88041840139212]
>>> rNm = LagkJointMomentsFromDMRAP(H, 2, 1)
>>> rNm1, rNm2, rNm3 = rNm
>>> print(rNm1)
[[ 0.45395  0.68525  1.38564]
 [ 0.68283  1.03179  2.08868]
 [ 1.37545  2.08069  4.21706]]
>>> print(rNm2)
[[ 0.02628  0.03323  0.05306]
 [ 0.03505  0.04387  0.06892]
 [ 0.06065  0.07477  0.11464]]
>>> print(rNm3)
[[ 0.51977  0.78522  1.58915]
 [ 0.78582  1.1881   2.40665]
 [ 1.59173  2.4087   4.88378]]
>>> print(la.norm(np.array(moms)-np.array(rmoms)))
3.8896478315352875e-10