butools.mam.MG1StationaryDistr¶
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butools.mam.MG1StationaryDistr()¶ Matlab: pi = MG1StationaryDistr (A, B, G, K)Mathematica: pi = MG1StationaryDistr [A, B, G, K]Python/Numpy: pi = MG1StationaryDistr (A, B, G, K)Returns the stationary distribution of the M/G/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 M/G/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 M/G/1 type generator at the boundary, from 0 to M-1.
G : matrix, shape (N,N)
Matrix G of the M/G/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 = [0.1, 0.5; 0.3, 0.4]; >>> B1 = [0., 0.1; 0., 0.]; >>> B2 = [0.2, 0.; 0., 0.2]; >>> B3 = [0., 0.1; 0.1, 0.]; >>> A0 = [0.4, 0.2; 0.3, 0.4]; >>> A1 = [0., 0.1; 0., 0.]; >>> A2 = [0., 0.2; 0., 0.2]; >>> A3 = [0.1, 0.; 0.1, 0.]; >>> B = {B0, B1, B2, B3}; >>> A = {A0, A1, A2, A3}; >>> G = MG1FundamentalMatrix(A); >>> disp(G); 0.60503 0.39497 0.45912 0.54088 >>> pi = MG1StationaryDistr(A, B, G, 300);
For Mathematica:
>>> B0 = {{0.1, 0.5},{0.3, 0.4}}; >>> B1 = {{0., 0.1},{0., 0.}}; >>> B2 = {{0.2, 0.},{0., 0.2}}; >>> B3 = {{0., 0.1},{0.1, 0.}}; >>> A0 = {{0.4, 0.2},{0.3, 0.4}}; >>> A1 = {{0., 0.1},{0., 0.}}; >>> A2 = {{0., 0.2},{0., 0.2}}; >>> A3 = {{0.1, 0.},{0.1, 0.}}; >>> B = {B0, B1, B2, B3}; >>> A = {A0, A1, A2, A3}; >>> G = MG1FundamentalMatrix[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 "16" roots\n" "The evaluation of the iteration required "8" roots\n" "Final Residual Error for G: "1.6653345369377348*^-16 >>> Print[G]; {{0.6050345283244288, 0.39496547167557117}, {0.4591222984767535, 0.5408777015232465}} >>> pi = MG1StationaryDistr[A, B, G, 300]; "Accumulated mass of the first "1" (reblocked) components:"0.3916489588121354 "Accumulated mass of the first "2" (reblocked) components:"0.5693301777261455 "Accumulated mass of the first "3" (reblocked) components:"0.7100355384481125 "Accumulated mass of the first "4" (reblocked) components:"0.80138806057205 "Accumulated mass of the first "5" (reblocked) components:"0.8647665851776781 "Accumulated mass of the first "6" (reblocked) components:"0.9077316038451807 "Accumulated mass of the first "7" (reblocked) components:"0.9370907444252965 "Accumulated mass of the first "8" (reblocked) components:"0.9570975394581935 "Accumulated mass of the first "9" (reblocked) components:"0.9707441149817535 "Accumulated mass of the first "10" (reblocked) components:"0.9800493560702122 "Accumulated mass of the first "11" (reblocked) components:"0.9863950724614493 "Accumulated mass of the first "12" (reblocked) components:"0.9907223698850346 "Accumulated mass of the first "13" (reblocked) components:"0.9936732983453151 "Accumulated mass of the first "14" (reblocked) components:"0.9956856255466052 "Accumulated mass of the first "15" (reblocked) components:"0.9970578944152654 "Accumulated mass of the first "16" (reblocked) components:"0.9979936869664346 "Accumulated mass of the first "17" (reblocked) components:"0.9986318329494369 "Accumulated mass of the first "18" (reblocked) components:"0.9990670044714363 "Accumulated mass of the first "19" (reblocked) components:"0.9993637614250387 "Accumulated mass of the first "20" (reblocked) components:"0.9995661291912358 "Accumulated mass of the first "21" (reblocked) components:"0.9997041300448283 "Accumulated mass of the first "22" (reblocked) components:"0.9997982371051191 "Accumulated mass of the first "23" (reblocked) components:"0.9998624116270072 "Accumulated mass of the first "24" (reblocked) components:"0.9999061742229949 "Accumulated mass of the first "25" (reblocked) components:"0.9999360172939101 "Accumulated mass of the first "26" (reblocked) components:"0.9999563682091505 "Accumulated mass of the first "27" (reblocked) components:"0.999970246129164 "Accumulated mass of the first "28" (reblocked) components:"0.9999797099130591 "Accumulated mass of the first "29" (reblocked) components:"0.9999861635606897 "Accumulated mass of the first "30" (reblocked) components:"0.9999905645030824 "Accumulated mass of the first "31" (reblocked) components:"0.9999935656421364 "Accumulated mass of the first "32" (reblocked) components:"0.9999956122118974 "Accumulated mass of the first "33" (reblocked) components:"0.9999970078312643 "Accumulated mass of the first "34" (reblocked) components:"0.9999979595473771 "Accumulated mass of the first "35" (reblocked) components:"0.9999986085520992 "Accumulated mass of the first "36" (reblocked) components:"0.9999990511285394 "Accumulated mass of the first "37" (reblocked) components:"0.9999993529351345 "Accumulated mass of the first "38" (reblocked) components:"0.9999995587464082 "Accumulated mass of the first "39" (reblocked) components:"0.9999996990954962 "Accumulated mass of the first "40" (reblocked) components:"0.9999997948038905 "Accumulated mass of the first "41" (reblocked) components:"0.9999998600704115 "Accumulated mass of the first "42" (reblocked) components:"0.9999999045776756 "Accumulated mass of the first "43" (reblocked) components:"0.9999999349285588 "Accumulated mass of the first "44" (reblocked) components:"0.9999999556257669 "Accumulated mass of the first "45" (reblocked) components:"0.9999999697398348 "Accumulated mass of the first "46" (reblocked) components:"0.9999999793646553 "Accumulated mass of the first "47" (reblocked) components:"0.9999999859281188 "Accumulated mass of the first "48" (reblocked) components:"0.999999990403948 "Accumulated mass of the first "49" (reblocked) components:"0.9999999934561548 "Accumulated mass of the first "50" (reblocked) components:"0.9999999955375491 "Accumulated mass of the first "51" (reblocked) components:"0.9999999969569165 "Accumulated mass of the first "52" (reblocked) components:"0.9999999979248272 "Accumulated mass of the first "53" (reblocked) components:"0.9999999985848754 "Accumulated mass of the first "54" (reblocked) components:"0.9999999990349828 "Accumulated mass of the first "55" (reblocked) components:"0.999999999341925 "Accumulated mass of the first "56" (reblocked) components:"0.9999999995512383 "Accumulated mass of the first "57" (reblocked) components:"0.9999999996939756 "Accumulated mass of the first "58" (reblocked) components:"0.9999999997913126 "Accumulated mass of the first "59" (reblocked) components:"0.9999999998576897 "Accumulated mass of the first "60" (reblocked) components:"0.9999999999029543
For Python/Numpy:
>>> B0 = ml.matrix([[0.1, 0.5],[0.3, 0.4]]) >>> B1 = ml.matrix([[0., 0.1],[0., 0.]]) >>> B2 = ml.matrix([[0.2, 0.],[0., 0.2]]) >>> B3 = ml.matrix([[0., 0.1],[0.1, 0.]]) >>> A0 = ml.matrix([[0.4, 0.2],[0.3, 0.4]]) >>> A1 = ml.matrix([[0., 0.1],[0., 0.]]) >>> A2 = ml.matrix([[0., 0.2],[0., 0.2]]) >>> A3 = ml.matrix([[0.1, 0.],[0.1, 0.]]) >>> B = [B0, B1, B2, B3] >>> A = [A0, A1, A2, A3] >>> G = MG1FundamentalMatrix(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 16 roots The Shifted PWCR evaluation of Iteration 5 required 8 roots Final Residual Error for G: 1.66533453694e-16 >>> print(G) [[ 0.60503 0.39497] [ 0.45912 0.54088]] >>> pi = MG1StationaryDistr(A, B, G, 300) Accumulated mass of the first 1 (reblocked) components: 0.391648958812 Accumulated mass of the first 2 (reblocked) components: 0.569330177726 Accumulated mass of the first 3 (reblocked) components: 0.710035538448 Accumulated mass of the first 4 (reblocked) components: 0.801388060572 Accumulated mass of the first 5 (reblocked) components: 0.864766585178 Accumulated mass of the first 6 (reblocked) components: 0.907731603845 Accumulated mass of the first 7 (reblocked) components: 0.937090744425 Accumulated mass of the first 8 (reblocked) components: 0.957097539458 Accumulated mass of the first 9 (reblocked) components: 0.970744114982 Accumulated mass of the first 10 (reblocked) components: 0.98004935607 Accumulated mass of the first 11 (reblocked) components: 0.986395072461 Accumulated mass of the first 12 (reblocked) components: 0.990722369885 Accumulated mass of the first 13 (reblocked) components: 0.993673298345 Accumulated mass of the first 14 (reblocked) components: 0.995685625547 Accumulated mass of the first 15 (reblocked) components: 0.997057894415 Accumulated mass of the first 16 (reblocked) components: 0.997993686966 Accumulated mass of the first 17 (reblocked) components: 0.998631832949 Accumulated mass of the first 18 (reblocked) components: 0.999067004471 Accumulated mass of the first 19 (reblocked) components: 0.999363761425 Accumulated mass of the first 20 (reblocked) components: 0.999566129191 Accumulated mass of the first 21 (reblocked) components: 0.999704130045 Accumulated mass of the first 22 (reblocked) components: 0.999798237105 Accumulated mass of the first 23 (reblocked) components: 0.999862411627 Accumulated mass of the first 24 (reblocked) components: 0.999906174223 Accumulated mass of the first 25 (reblocked) components: 0.999936017294 Accumulated mass of the first 26 (reblocked) components: 0.999956368209 Accumulated mass of the first 27 (reblocked) components: 0.999970246129 Accumulated mass of the first 28 (reblocked) components: 0.999979709913 Accumulated mass of the first 29 (reblocked) components: 0.999986163561 Accumulated mass of the first 30 (reblocked) components: 0.999990564503 Accumulated mass of the first 31 (reblocked) components: 0.999993565642 Accumulated mass of the first 32 (reblocked) components: 0.999995612212 Accumulated mass of the first 33 (reblocked) components: 0.999997007831 Accumulated mass of the first 34 (reblocked) components: 0.999997959547 Accumulated mass of the first 35 (reblocked) components: 0.999998608552 Accumulated mass of the first 36 (reblocked) components: 0.999999051129 Accumulated mass of the first 37 (reblocked) components: 0.999999352935 Accumulated mass of the first 38 (reblocked) components: 0.999999558746 Accumulated mass of the first 39 (reblocked) components: 0.999999699095 Accumulated mass of the first 40 (reblocked) components: 0.999999794804 Accumulated mass of the first 41 (reblocked) components: 0.99999986007 Accumulated mass of the first 42 (reblocked) components: 0.999999904578 Accumulated mass of the first 43 (reblocked) components: 0.999999934929 Accumulated mass of the first 44 (reblocked) components: 0.999999955626 Accumulated mass of the first 45 (reblocked) components: 0.99999996974 Accumulated mass of the first 46 (reblocked) components: 0.999999979365 Accumulated mass of the first 47 (reblocked) components: 0.999999985928 Accumulated mass of the first 48 (reblocked) components: 0.999999990404 Accumulated mass of the first 49 (reblocked) components: 0.999999993456 Accumulated mass of the first 50 (reblocked) components: 0.999999995538 Accumulated mass of the first 51 (reblocked) components: 0.999999996957 Accumulated mass of the first 52 (reblocked) components: 0.999999997925 Accumulated mass of the first 53 (reblocked) components: 0.999999998585 Accumulated mass of the first 54 (reblocked) components: 0.999999999035 Accumulated mass of the first 55 (reblocked) components: 0.999999999342 Accumulated mass of the first 56 (reblocked) components: 0.999999999551 Accumulated mass of the first 57 (reblocked) components: 0.999999999694 Accumulated mass of the first 58 (reblocked) components: 0.999999999791 Accumulated mass of the first 59 (reblocked) components: 0.999999999858 Accumulated mass of the first 60 (reblocked) components: 0.999999999903