butools.queues.FluFluQueue¶

butools.queues.FluFluQueue()¶

Matlab:	Ret = FluFluQueue(Qin, Rin, Qout, Rout, srv0stop, ...)
Mathematica:	Ret = FluFluQueue[Qin, Rin, Qout, Rout, srv0stop, ...]
Python/Numpy:	Ret = FluFluQueue(Qin, Rin, Qout, Rout, srv0stop, ...)

Returns various performane measures of a fluid queue with independent fluid arrival and service processes.

Two types of boundary behavior is available. If srv0stop=false, the output process evolves continuously even if the queue is empty. If srv0stop=true, the output process slows down if there is fewer fluid in the queue than it can serve. If the queue is empty and the fluid input rate is zero, the output process freezes till fluid arrives.

Parameters:

Qin : matrix, shape (N,N) The generator of the background Markov chain corresponding to the input process Rin : matrix, shape (N,N) Diagonal matrix containing the fluid input rates associated to the states of the input background process Qout : matrix, shape (N,N) The generator of the background Markov chain corresponding to the output process Rout : matrix, shape (N,N) Diagonal matrix containing the fluid output rates associated to the states of the input background process srv0stop : bool If true, the service output process slows down if there is fewer fluid in the queue than it can serve. If false, the output process evolves continuously. further parameters : The rest of the function parameters specify the options and the performance measures to be computed. The supported performance measures and options in this function are: Parameter name Input parameters Output “flMoms” Number of moments The moments of the fluid level “flDistr” A vector of points The fluid level distribution at the requested points (cdf) “flDistrME” None The vector-matrix parameters of the matrix-exponentially distributed fluid level distribution “flDistrPH” None The vector-matrix parameters of the matrix-exponentially distributed fluid level distribution, converted to a PH representation “stMoms” Number of moments The sojourn time moments of fluid drops “stDistr” A vector of points The sojourn time distribution at the requested points (cummulative, cdf) “stDistrME” None The vector-matrix parameters of the matrix-exponentially distributed sojourn time distribution “stDistrPH” None The vector-matrix parameters of the matrix-exponentially distributed sojourn time distribution, converted to a PH representation “prec” The precision Numerical precision to check if the input is valid and it is also used as a stopping condition when solving the Riccati equation

Returns:

Ret : list of the performance measures Each entry of the list corresponds to a performance measure requested. If there is just a single item, then it is not put into a list.

Notes

“flDistrME” and “stDistrME” behave much better numerically than “flDistrPH” and “stDistrPH”.

References

[R14]

Horvath G, Telek M, “Sojourn times in fluid queues with independent and dependent input and output processes PERFORMANCE EVALUATION 79: pp. 160-181, 2014.

Examples

For Matlab:

>>> Qin = [-2., 1., 1.; 2., -5., 3.; 4., 0., -4.];
>>> vRin = [3.,7.,0.];
>>> Rin = diag(vRin);
>>> Qout = [-4., 1., 3.; 6., -8., 2.; 3., 7., -10.];
>>> vRout = [1.,7.,15.];
>>> Rout = diag(vRout);
>>> [fld, flm] = FluFluQueue(Qin, Rin, Qout, Rout, false, 'flDistr', (0.:0.1:1.), 'flMoms', 5);
Final Residual Error for Psi:    1.0339e-15
>>> disp(fld);
  Columns 1 through 8
       0.3918      0.47163      0.53819      0.59413       0.6415      0.68193      0.71667      0.74673
  Columns 9 through 11
      0.77292      0.79585      0.81605
>>> disp(flm);
       0.5357       1.0765       3.4298        14.87       81.162
>>> [alphap, Ap] = FluFluQueue(Qin, Rin, Qout, Rout, false, 'flDistrPH');
Final Residual Error for Psi:    1.0339e-15
>>> disp(alphap);
      0.45573      0.15247
>>> disp(Ap);
      -2.3405      0.53197
      0.92131      -1.2559
>>> [alpha, A] = FluFluQueue(Qin, Rin, Qout, Rout, false, 'flDistrME');
Final Residual Error for Psi:    1.0339e-15
>>> disp(alpha);
     -0.65561       1.2638
>>> disp(A);
      -2.1425       1.5194
      0.43807      -1.4538
>>> fldFromPH = CdfFromPH(alphap, Ap, (0.:0.1:1.));
>>> disp(fldFromPH);
  Columns 1 through 8
       0.3918      0.47163      0.53819      0.59413       0.6415      0.68193      0.71667      0.74673
  Columns 9 through 11
      0.77292      0.79585      0.81605
>>> flmFromME = MomentsFromME(alpha, A, 5);
>>> disp(flmFromME);
       0.5357       1.0765       3.4298        14.87       81.162
>>> [std, stm] = FluFluQueue(Qin, Rin, Qout, Rout, false, 'stDistr', (0.:0.1:1.), 'stMoms', 5);
Final Residual Error for Psi:    2.0955e-15
>>> disp(std);
  Columns 1 through 8
      0.29838      0.51911      0.66679      0.76705      0.83596      0.88381      0.91733      0.94097
  Columns 9 through 11
      0.95774      0.96968      0.97821
>>> disp(stm);
       0.1948      0.11287      0.10069      0.12158      0.18506
>>> [betap, Bp] = FluFluQueue(Qin, Rin, Qout, Rout, false, 'stDistrPH');
Final Residual Error for Psi:    2.0955e-15
>>> disp(betap);
      0.45285      0.24877
>>> disp(Bp);
      -5.4973      0.83675
       1.4492      -3.7914
>>> [beta, B] = FluFluQueue(Qin, Rin, Qout, Rout, false, 'stDistrME');
Final Residual Error for Psi:    2.0955e-15
>>> disp(beta);
      0.18114      0.52048
>>> disp(B);
      -6.3668       1.6656
     -0.61643      -2.9219
>>> stdFromPH = CdfFromPH(betap, Bp, (0.:0.1:1.));
>>> disp(stdFromPH);
  Columns 1 through 8
      0.29838      0.51911      0.66679      0.76705      0.83596      0.88381      0.91733      0.94097
  Columns 9 through 11
      0.95774      0.96968      0.97821
>>> stmFromME = MomentsFromME(beta, B, 5);
>>> disp(stmFromME);
       0.1948      0.11287      0.10069      0.12158      0.18506
>>> [fld, flm] = FluFluQueue(Qin, Rin, Qout, Rout, true, 'flDistr', (0.:0.1:1.), 'flMoms', 5);
Final Residual Error for Psi:    1.0339e-15
>>> disp(fld);
  Columns 1 through 8
      0.64736      0.68913      0.72467      0.75512       0.7814      0.80423      0.82418      0.84172
  Columns 9 through 11
      0.85721      0.87095      0.88319
>>> disp(flm);
      0.33265      0.68892       2.2198       9.6621        52.81
>>> [alphap, Ap] = FluFluQueue(Qin, Rin, Qout, Rout, true, 'flDistrPH');
Final Residual Error for Psi:    1.0339e-15
>>> disp(alphap);
      0.24142      0.11122
>>> disp(Ap);
      -2.3405      0.73252
      0.66907      -1.2559
>>> [alpha, A] = FluFluQueue(Qin, Rin, Qout, Rout, true, 'flDistrME');
Final Residual Error for Psi:    1.0339e-15
>>> disp(alpha);
     -0.24261      0.59524
>>> disp(A);
      -2.1425       1.5194
      0.43807      -1.4538
>>> fldFromPH = CdfFromPH(alphap, Ap, (0.:0.1:1.));
>>> disp(fldFromPH);
  Columns 1 through 8
      0.64736      0.68913      0.72467      0.75512       0.7814      0.80423      0.82418      0.84172
  Columns 9 through 11
      0.85721      0.87095      0.88319
>>> flmFromME = MomentsFromME(alpha, A, 5);
>>> disp(flmFromME);
      0.33265      0.68892       2.2198       9.6621        52.81
>>> [std, stm] = FluFluQueue(Qin, Rin, Qout, Rout, true, 'stDistr', (0.:0.1:1.), 'stMoms', 5);
Final Residual Error for Psi:    2.0955e-15
>>> disp(std);
  Columns 1 through 8
      0.57864      0.70628      0.79365      0.85412      0.89636      0.92608      0.94712      0.96209
  Columns 9 through 11
      0.97277      0.98041      0.98589
>>> disp(stm);
      0.12096     0.071546     0.064592      0.07852      0.11997
>>> [betap, Bp] = FluFluQueue(Qin, Rin, Qout, Rout, true, 'stDistrPH');
Final Residual Error for Psi:    2.0955e-15
>>> disp(betap);
      0.23831      0.18306
>>> disp(Bp);
      -5.4973      0.83675
       1.4492      -3.7914
>>> [beta, B] = FluFluQueue(Qin, Rin, Qout, Rout, true, 'stDistrME');
Final Residual Error for Psi:    2.0955e-15
>>> disp(beta);
     -0.12204       0.5434
>>> disp(B);
       -5.186       2.4839
      0.66298      -4.1028
>>> stdFromPH = CdfFromPH(betap, Bp, (0.:0.1:1.));
>>> disp(stdFromPH);
  Columns 1 through 8
      0.57864      0.70628      0.79365      0.85412      0.89636      0.92608      0.94712      0.96209
  Columns 9 through 11
      0.97277      0.98041      0.98589
>>> stmFromME = MomentsFromME(beta, B, 5);
>>> disp(stmFromME);
      0.12096     0.071546     0.064592      0.07852      0.11997

For Mathematica:

>>> Qin = {{-2., 1., 1.},{2., -5., 3.},{4., 0., -4.}};
>>> vRin = {3.,7.,0.};
>>> Rin = DiagonalMatrix[vRin];
>>> Qout = {{-4., 1., 3.},{6., -8., 2.},{3., 7., -10.}};
>>> vRout = {1.,7.,15.};
>>> Rout = DiagonalMatrix[vRout];
>>> {fld, flm} = FluFluQueue[Qin, Rin, Qout, Rout, False, "flDistr", Range[0.,1.,0.1], "flMoms", 5];
"Final Residual Error for Psi: "7.008282842946301*^-16
>>> Print[fld];
{0.391800873246579, 0.4716300609142923, 0.5381929941518719, 0.59413043637473, 0.641503602438246, 0.6819269051118623, 0.716669775887913, 0.7467348114111322, 0.7729177924650865, 0.7958538190537734, 0.8160528083141847}
>>> Print[flm];
{0.5357035627676002, 1.0765385576008892, 3.4298090557057153, 14.869885651621978, 81.16233569132378}
>>> {alphap, Ap} = FluFluQueue[Qin, Rin, Qout, Rout, False, "flDistrPH"];
"Final Residual Error for Psi: "7.008282842946301*^-16
>>> Print[alphap];
{0.4557324623147923, 0.15246666443862872}
>>> Print[Ap];
{{-2.3404575203667424, 0.5319666549172487},
 {0.9213094524423505, -1.2559180746181111}}
>>> {alpha, A} = FluFluQueue[Qin, Rin, Qout, Rout, False, "flDistrME"];
"Final Residual Error for Psi: "7.008282842946301*^-16
>>> Print[alpha];
{-0.6556057500800456, 1.2638048768334667}
>>> Print[A];
{{-2.142529166146859, 1.5193788109637358},
 {0.43806809528730795, -1.4538464288379942}}
>>> fldFromPH = CdfFromPH[alphap, Ap, Range[0.,1.,0.1]];
>>> Print[fldFromPH];
{0.391800873246579, 0.4716300609142924, 0.538192994151872, 0.5941304363747301, 0.641503602438246, 0.6819269051118624, 0.716669775887913, 0.7467348114111322, 0.7729177924650866, 0.7958538190537734, 0.8160528083141847}
>>> flmFromME = MomentsFromME[alpha, A, 5];
>>> Print[flmFromME];
{0.5357035627676006, 1.07653855760089, 3.429809055705716, 14.869885651621974, 81.16233569132378}
>>> {std, stm} = FluFluQueue[Qin, Rin, Qout, Rout, False, "stDistr", Range[0.,1.,0.1], "stMoms", 5];
"Final Residual Error for Psi: "1.4988010832439613*^-15
>>> Print[std];
{0.29837541247887833, 0.5191050664654031, 0.6667930660737641, 0.7670494936141816, 0.8359575279199422, 0.8838138562939544, 0.9173341131830883, 0.9409745470882221, 0.9577383239502405, 0.9696768240390095, 0.9782073950108067}
>>> Print[stm];
{0.19480129555185477, 0.11287108072563842, 0.100686427731073, 0.12157602438536942, 0.18505841424939884}
>>> {betap, Bp} = FluFluQueue[Qin, Rin, Qout, Rout, False, "stDistrPH"];
"Final Residual Error for Psi: "1.4988010832439613*^-15
>>> Print[betap];
{0.45285051259891584, 0.24877407492220602}
>>> Print[Bp];
{{-5.4973439722064095, 0.8367525984737355},
 {1.4491661670964533, -3.7914265223267773}}
>>> {beta, B} = FluFluQueue[Qin, Rin, Qout, Rout, False, "stDistrME"];
"Final Residual Error for Psi: "1.4988010832439613*^-15
>>> Print[beta];
{0.18114020503956624, 0.5204843824815555}
>>> Print[B];
{{-6.366843672464282, 1.6656012008589829},
 {-0.6164326032040903, -2.9219268220689054}}
>>> stdFromPH = CdfFromPH[betap, Bp, Range[0.,1.,0.1]];
>>> Print[stdFromPH];
{0.2983754124788781, 0.519105066465403, 0.666793066073764, 0.7670494936141815, 0.8359575279199422, 0.8838138562939544, 0.9173341131830883, 0.9409745470882221, 0.9577383239502405, 0.9696768240390093, 0.9782073950108067}
>>> stmFromME = MomentsFromME[beta, B, 5];
>>> Print[stmFromME];
{0.19480129555185485, 0.11287108072563846, 0.10068642773107304, 0.12157602438536946, 0.18505841424939895}
>>> {fld, flm} = FluFluQueue[Qin, Rin, Qout, Rout, True, "flDistr", Range[0.,1.,0.1], "flMoms", 5];
"Final Residual Error for Psi: "7.008282842946301*^-16
>>> Print[fld];
{0.6473632738053204, 0.6891320433434904, 0.7246660281915851, 0.7551190366324814, 0.7814006612465998, 0.8042313747552501, 0.8241848657852636, 0.8417205998183797, 0.8572088895119173, 0.8709502223792169, 0.8831901837025878}
>>> Print[flm];
{0.33264746858870425, 0.6889173702248804, 2.219774758007364, 9.662108525950131, 52.80956347845118}
>>> {alphap, Ap} = FluFluQueue[Qin, Rin, Qout, Rout, True, "flDistrPH"];
"Final Residual Error for Psi: "7.008282842946301*^-16
>>> Print[alphap];
{0.24141936087976823, 0.11121736531491135}
>>> Print[Ap];
{{-2.3404575203667424, 0.7325208369425898},
 {0.6690675306998952, -1.2559180746181111}}
>>> {alpha, A} = FluFluQueue[Qin, Rin, Qout, Rout, True, "flDistrME"];
"Final Residual Error for Psi: "7.008282842946301*^-16
>>> Print[alpha];
{-0.24260693822391072, 0.5952436644185902}
>>> Print[A];
{{-2.142529166146859, 1.5193788109637358},
 {0.43806809528730795, -1.4538464288379942}}
>>> fldFromPH = CdfFromPH[alphap, Ap, Range[0.,1.,0.1]];
>>> Print[fldFromPH];
{0.6473632738053204, 0.6891320433434904, 0.7246660281915852, 0.7551190366324814, 0.7814006612465998, 0.8042313747552501, 0.8241848657852637, 0.8417205998183797, 0.8572088895119174, 0.8709502223792169, 0.883190183702588}
>>> flmFromME = MomentsFromME[alpha, A, 5];
>>> Print[flmFromME];
{0.3326474685887042, 0.6889173702248805, 2.219774758007364, 9.662108525950126, 52.80956347845115}
>>> {std, stm} = FluFluQueue[Qin, Rin, Qout, Rout, True, "stDistr", Range[0.,1.,0.1], "stMoms", 5];
"Final Residual Error for Psi: "1.4988010832439613*^-15
>>> Print[std];
{0.5786381632358244, 0.7062784286059924, 0.7936484851974519, 0.8541200746555163, 0.8963590382366593, 0.9260821681510065, 0.9471221726585012, 0.9620853669984744, 0.9727657494367457, 0.9804107560683182, 0.9858949950823395}
>>> Print[stm];
{0.12096271585043802, 0.07154626102115877, 0.0645920699157504, 0.07852033701278222, 0.11996554734440769}
>>> {betap, Bp} = FluFluQueue[Qin, Rin, Qout, Rout, True, "stDistrPH"];
"Final Residual Error for Psi: "1.4988010832439613*^-15
>>> Print[betap];
{0.2383068326039564, 0.18305500416021936}
>>> Print[Bp];
{{-5.4973439722064095, 0.8367525984737355},
 {1.4491661670964533, -3.7914265223267773}}
>>> {beta, B} = FluFluQueue[Qin, Rin, Qout, Rout, True, "stDistrME"];
"Final Residual Error for Psi: "1.4988010832439613*^-15
>>> Print[beta];
{-0.12203666944014618, 0.5433985062043218}
>>> Print[B];
{{-5.18601418644967, 2.4838988377244475},
 {0.6629779802695965, -4.102756308083516}}
>>> stdFromPH = CdfFromPH[betap, Bp, Range[0.,1.,0.1]];
>>> Print[stdFromPH];
{0.5786381632358242, 0.7062784286059923, 0.7936484851974519, 0.8541200746555163, 0.8963590382366593, 0.9260821681510065, 0.9471221726585012, 0.9620853669984744, 0.9727657494367457, 0.9804107560683182, 0.9858949950823395}
>>> stmFromME = MomentsFromME[beta, B, 5];
>>> Print[stmFromME];
{0.12096271585043804, 0.07154626102115878, 0.06459206991575042, 0.07852033701278228, 0.11996554734440777}

For Python/Numpy:

>>> Qin = ml.matrix([[-2., 1., 1.],[2., -5., 3.],[4., 0., -4.]])
>>> vRin = ml.matrix([[3.,7.,0.]])
>>> Rin = Diag(vRin)
>>> Qout = ml.matrix([[-4., 1., 3.],[6., -8., 2.],[3., 7., -10.]])
>>> vRout = ml.matrix([[1.,7.,15.]])
>>> Rout = Diag(vRout)
>>> fld, flm = FluFluQueue(Qin, Rin, Qout, Rout, False, "flDistr", np.arange(0.,1.1,0.1), "flMoms", 5)
Final Residual Error for G:  1.2975731600306517e-15
>>> print(fld)
[ 0.3918   0.47163  0.53819  0.59413  0.6415   0.68193  0.71667  0.74673  0.77292  0.79585  0.81605]
>>> print(flm)
[0.53570356276760078, 1.0765385576008903, 3.4298090557057193, 14.869885651621992, 81.162335691323889]
>>> alphap, Ap = FluFluQueue(Qin, Rin, Qout, Rout, False, "flDistrPH")
Final Residual Error for G:  1.2975731600306517e-15
>>> print(alphap)
[[ 0.45573  0.15247]]
>>> print(Ap)
[[-2.34046  0.53197]
 [ 0.92131 -1.25592]]
>>> alpha, A = FluFluQueue(Qin, Rin, Qout, Rout, False, "flDistrME")
Final Residual Error for G:  1.2975731600306517e-15
>>> print(alpha)
[[-0.65561  1.2638 ]]
>>> print(A)
[[-2.14253  1.51938]
 [ 0.43807 -1.45385]]
>>> fldFromPH = CdfFromPH(alphap, Ap, np.arange(0.,1.1,0.1))
>>> print(fldFromPH)
[ 0.3918   0.47163  0.53819  0.59413  0.6415   0.68193  0.71667  0.74673  0.77292  0.79585  0.81605]
>>> flmFromME = MomentsFromME(alpha, A, 5)
>>> print(flmFromME)
[0.53570356276760067, 1.0765385576008901, 3.4298090557057166, 14.869885651621992, 81.162335691323918]
>>> std, stm = FluFluQueue(Qin, Rin, Qout, Rout, False, "stDistr", np.arange(0.,1.1,0.1), "stMoms", 5)
Final Residual Error for G:  1.8457457784393227e-15
>>> print(std)
[[ 0.29838  0.51911  0.66679  0.76705  0.83596  0.88381  0.91733  0.94097  0.95774  0.96968  0.97821]]
>>> print(stm)
[0.19480129555185471, 0.1128710807256384, 0.10068642773107296, 0.12157602438536934, 0.18505841424939878]
>>> betap, Bp = FluFluQueue(Qin, Rin, Qout, Rout, False, "stDistrPH")
Final Residual Error for G:  1.8457457784393227e-15
>>> print(betap)
[[ 0.45285  0.24877]]
>>> print(Bp)
[[-5.49734  0.83675]
 [ 1.44917 -3.79143]]
>>> beta, B = FluFluQueue(Qin, Rin, Qout, Rout, False, "stDistrME")
Final Residual Error for G:  1.8457457784393227e-15
>>> print(beta)
[[ 0.18114  0.52048]]
>>> print(B)
[[-6.36684  1.6656 ]
 [-0.61643 -2.92193]]
>>> stdFromPH = CdfFromPH(betap, Bp, np.arange(0.,1.1,0.1))
>>> print(stdFromPH)
[ 0.29838  0.51911  0.66679  0.76705  0.83596  0.88381  0.91733  0.94097  0.95774  0.96968  0.97821]
>>> stmFromME = MomentsFromME(beta, B, 5)
>>> print(stmFromME)
[0.19480129555185474, 0.11287108072563844, 0.10068642773107304, 0.12157602438536946, 0.18505841424939901]
>>> fld, flm = FluFluQueue(Qin, Rin, Qout, Rout, True, "flDistr", np.arange(0.,1.1,0.1), "flMoms", 5)
Final Residual Error for G:  1.2975731600306517e-15
>>> print(fld)
[ 0.64736  0.68913  0.72467  0.75512  0.7814   0.80423  0.82418  0.84172  0.85721  0.87095  0.88319]
>>> print(flm)
[0.33264746858870425, 0.68891737022488087, 2.2197747580073659, 9.6621085259501385, 52.809563478451246]
>>> alphap, Ap = FluFluQueue(Qin, Rin, Qout, Rout, True, "flDistrPH")
Final Residual Error for G:  1.2975731600306517e-15
>>> print(alphap)
[[ 0.24142  0.11122]]
>>> print(Ap)
[[-2.34046  0.73252]
 [ 0.66907 -1.25592]]
>>> alpha, A = FluFluQueue(Qin, Rin, Qout, Rout, True, "flDistrME")
Final Residual Error for G:  1.2975731600306517e-15
>>> print(alpha)
[[-0.24261  0.59524]]
>>> print(A)
[[-2.14253  1.51938]
 [ 0.43807 -1.45385]]
>>> fldFromPH = CdfFromPH(alphap, Ap, np.arange(0.,1.1,0.1))
>>> print(fldFromPH)
[ 0.64736  0.68913  0.72467  0.75512  0.7814   0.80423  0.82418  0.84172  0.85721  0.87095  0.88319]
>>> flmFromME = MomentsFromME(alpha, A, 5)
>>> print(flmFromME)
[0.3326474685887042, 0.68891737022488064, 2.219774758007365, 9.6621085259501367, 52.809563478451238]
>>> std, stm = FluFluQueue(Qin, Rin, Qout, Rout, True, "stDistr", np.arange(0.,1.1,0.1), "stMoms", 5)
Final Residual Error for G:  1.8457457784393227e-15
>>> print(std)
[[ 0.57864  0.70628  0.79365  0.85412  0.89636  0.92608  0.94712  0.96209  0.97277  0.98041  0.98589]]
>>> print(stm)
[0.12096271585043798, 0.071546261021158727, 0.064592069915750366, 0.078520337012782174, 0.11996554734440762]
>>> betap, Bp = FluFluQueue(Qin, Rin, Qout, Rout, True, "stDistrPH")
Final Residual Error for G:  1.8457457784393227e-15
>>> print(betap)
[[ 0.23831  0.18306]]
>>> print(Bp)
[[-5.49734  0.83675]
 [ 1.44917 -3.79143]]
>>> beta, B = FluFluQueue(Qin, Rin, Qout, Rout, True, "stDistrME")
Final Residual Error for G:  1.8457457784393227e-15
>>> print(beta)
[[-0.12204  0.5434 ]]
>>> print(B)
[[-5.18601  2.4839 ]
 [ 0.66298 -4.10276]]
>>> stdFromPH = CdfFromPH(betap, Bp, np.arange(0.,1.1,0.1))
>>> print(stdFromPH)
[ 0.57864  0.70628  0.79365  0.85412  0.89636  0.92608  0.94712  0.96209  0.97277  0.98041  0.98589]
>>> stmFromME = MomentsFromME(beta, B, 5)
>>> print(stmFromME)
[0.12096271585043798, 0.071546261021158755, 0.064592069915750394, 0.078520337012782215, 0.11996554734440766]