butools.ph.MEOrder¶
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butools.ph.MEOrder()¶ Matlab: order = MEOrder(alpha, A, kind, prec)Mathematica: order = MEOrder[alpha, A, kind, prec]Python/Numpy: order = MEOrder(alpha, A, kind, prec)Returns the order of the ME distribution (which is not necessarily equal to the size of the representation).
Parameters: alpha : vector, shape (1,M)
The initial vector of the matrix-exponential distribution.
A : matrix, shape (M,M)
The matrix parameter of the matrix-exponential distribution.
kind : {‘obs’, ‘cont’, ‘obscont’, ‘moment’}, optional
Determines which order is computed. Possibilities: ‘obs’: observability, ‘cont’: controllability, ‘obscont’: the minimum of observability and controllability order, ‘moment’: moment order (which is the default).
prec : double, optional
Precision used to detect if the determinant of the Hankel matrix is zero (in case of kind=”moment” only), or the tolerance for the rank calculation. The default value is 1e-10.
Returns: order : int
The order of ME distribution
References
[R24] P. Buchholz, M. Telek, “On minimal representation of rational arrival processes.” Madrid Conference on Qeueuing theory (MCQT), June 2010. Examples
For Matlab:
>>> a = [1.0/6,1.0/6,1.0/6,1.0/6,1.0/6,1.0/6]; >>> A = [-1., 0., 0., 0., 0., 0.; 0.5, -2., 1., 0., 0., 0.; 1., 0., -3., 1., 0., 0.; 1., 0., 1., -4., 1., 0.; 4., 0., 0., 0., -5., 0.; 5., 0., 0., 0., 0., -6.]; >>> co = MEOrder(a, A, 'cont'); >>> disp(co); 2 >>> oo = MEOrder(a, A, 'obs'); >>> disp(oo); 6 >>> coo = MEOrder(a, A, 'obscont'); >>> disp(coo); 2 >>> mo = MEOrder(a, A, 'moment'); >>> disp(mo); 2 >>> a = [2.0/3,1.0/3]; >>> A = [-1., 1.; 0., -3.]; >>> co = MEOrder(a, A, 'cont'); >>> disp(co); 2 >>> oo = MEOrder(a, A, 'obs'); >>> disp(oo); 1 >>> coo = MEOrder(a, A, 'obscont'); >>> disp(coo); 1 >>> mo = MEOrder(a, A, 'moment'); >>> disp(mo); 1 >>> b = [0.2,0.3,0.5]; >>> B = [-1., 0., 0.; 0., -3., 1.; 0., -1., -3.]; >>> [a, A] = MonocyclicPHFromME(b, B); >>> disp(a); Columns 1 through 6 0.0055089 0.0090301 0.016938 0.015216 0.0053543 0.0087356 Columns 7 through 9 0.052486 0.22657 0.66016 >>> disp(A); Columns 1 through 6 -1 1 0 0 0 0 0 -2.4226 2.4226 0 0 0 0 0 -2.4226 2.4226 0 0 0 0.26232 0 -2.4226 2.1603 0 0 0 0 0 -4.2414 4.2414 0 0 0 0 0 -4.2414 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 Columns 7 through 9 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 4.2414 0 0 -4.2414 4.2414 0 0 -4.2414 4.2414 0 0 -4.2414 >>> co = MEOrder(a, A, 'cont'); >>> disp(co); 9 >>> oo = MEOrder(a, A, 'obs'); >>> disp(oo); 3 >>> coo = MEOrder(a, A, 'obscont'); >>> disp(coo); 3 >>> mo = MEOrder(a, A, 'moment'); >>> disp(mo); 3
For Mathematica:
>>> a = {1.0/6,1.0/6,1.0/6,1.0/6,1.0/6,1.0/6}; >>> A = {{-1., 0., 0., 0., 0., 0.},{0.5, -2., 1., 0., 0., 0.},{1., 0., -3., 1., 0., 0.},{1., 0., 1., -4., 1., 0.},{4., 0., 0., 0., -5., 0.},{5., 0., 0., 0., 0., -6.}}; >>> co = MEOrder[a, A, "cont"]; >>> Print[co]; 2 >>> oo = MEOrder[a, A, "obs"]; >>> Print[oo]; 6 >>> coo = MEOrder[a, A, "obscont"]; >>> Print[coo]; 2 >>> mo = MEOrder[a, A, "moment"]; >>> Print[mo]; 2 >>> a = {2.0/3,1.0/3}; >>> A = {{-1., 1.},{0., -3.}}; >>> co = MEOrder[a, A, "cont"]; >>> Print[co]; 2 >>> oo = MEOrder[a, A, "obs"]; >>> Print[oo]; 1 >>> coo = MEOrder[a, A, "obscont"]; >>> Print[coo]; 1 >>> mo = MEOrder[a, A, "moment"]; >>> Print[mo]; 1 >>> b = {0.2,0.3,0.5}; >>> B = {{-1., 0., 0.},{0., -3., 1.},{0., -1., -3.}}; >>> {a, A} = MonocyclicPHFromME[b, B]; >>> Print[a]; {0.00550893408977846, 0.00903007832853331, 0.016937512518639578, 0.015215980106503445, 0.005354337535618665, 0.008735592607040744, 0.05248568615571608, 0.22657249403204927, 0.6601593846261203} >>> Print[A]; {{-1., 1., 0., 0., 0., 0., 0., 0., 0.}, {0., -2.4226497308103743, 2.4226497308103743, 0., 0., 0., 0., 0., 0.}, {0., 0., -2.4226497308103743, 2.4226497308103743, 0., 0., 0., 0., 0.}, {0., 0.2623172489622428, 0., -2.4226497308103743, 2.1603324818481315, 0., 0., 0., 0.}, {0., 0., 0., 0., -4.241399978863847, 4.241399978863847, 0., 0., 0.}, {0., 0., 0., 0., 0., -4.241399978863847, 4.241399978863847, 0., 0.}, {0., 0., 0., 0., 0., 0., -4.241399978863847, 4.241399978863847, 0.}, {0., 0., 0., 0., 0., 0., 0., -4.241399978863847, 4.241399978863847}, {0., 0., 0., 0., 0., 0., 0., 0., -4.241399978863847}} >>> co = MEOrder[a, A, "cont"]; >>> Print[co]; 9 >>> oo = MEOrder[a, A, "obs"]; >>> Print[oo]; 3 >>> coo = MEOrder[a, A, "obscont"]; >>> Print[coo]; 3 >>> mo = MEOrder[a, A, "moment"]; >>> Print[mo]; 3
For Python/Numpy:
>>> a = ml.matrix([[1.0/6,1.0/6,1.0/6,1.0/6,1.0/6,1.0/6]]) >>> A = ml.matrix([[-1., 0., 0., 0., 0., 0.],[0.5, -2., 1., 0., 0., 0.],[1., 0., -3., 1., 0., 0.],[1., 0., 1., -4., 1., 0.],[4., 0., 0., 0., -5., 0.],[5., 0., 0., 0., 0., -6.]]) >>> co = MEOrder(a, A, "cont") >>> print(co) 2 >>> oo = MEOrder(a, A, "obs") >>> print(oo) 6 >>> coo = MEOrder(a, A, "obscont") >>> print(coo) 2 >>> mo = MEOrder(a, A, "moment") >>> print(mo) 2 >>> a = ml.matrix([[2.0/3,1.0/3]]) >>> A = ml.matrix([[-1., 1.],[0., -3.]]) >>> co = MEOrder(a, A, "cont") >>> print(co) 2 >>> oo = MEOrder(a, A, "obs") >>> print(oo) 1 >>> coo = MEOrder(a, A, "obscont") >>> print(coo) 1 >>> mo = MEOrder(a, A, "moment") >>> print(mo) 1 >>> b = ml.matrix([[0.2,0.3,0.5]]) >>> B = ml.matrix([[-1., 0., 0.],[0., -3., 1.],[0., -1., -3.]]) >>> a, A = MonocyclicPHFromME(b, B) >>> print(a) [[ 0.00551 0.00903 0.01694 0.01522 0.00535 0.00874 0.05249 0.22657 0.66016]] >>> print(A) [[-1. 1. 0. 0. 0. 0. 0. 0. 0. ] [ 0. -2.42265 2.42265 0. 0. 0. 0. 0. 0. ] [ 0. 0. -2.42265 2.42265 0. 0. 0. 0. 0. ] [ 0. 0.26232 0. -2.42265 2.16033 0. 0. 0. 0. ] [ 0. 0. 0. 0. -4.2414 4.2414 0. 0. 0. ] [ 0. 0. 0. 0. 0. -4.2414 4.2414 0. 0. ] [ 0. 0. 0. 0. 0. 0. -4.2414 4.2414 0. ] [ 0. 0. 0. 0. 0. 0. 0. -4.2414 4.2414 ] [ 0. 0. 0. 0. 0. 0. 0. 0. -4.2414 ]] >>> co = MEOrder(a, A, "cont") >>> print(co) 9 >>> oo = MEOrder(a, A, "obs") >>> print(oo) 3 >>> coo = MEOrder(a, A, "obscont") >>> print(coo) 3 >>> mo = MEOrder(a, A, "moment") >>> print(mo) 3