butools.queues.MMAPPH1FCFS

butools.queues.MMAPPH1FCFS()
Matlab: Ret = MMAPPH1FCFS(D, sigma, S, ...)
Mathematica: Ret = MMAPPH1FCFS[D, sigma, S, ...]
Python/Numpy: Ret = MMAPPH1FCFS(D, sigma, S, ...)

Returns various performane measures of a MMAP[K]/PH[K]/1 first-come-first-serve queue, see [R29].

Parameters:

D : list of matrices of shape (N,N), length (K+1)

The D0...DK matrices of the arrival process.

sigma : list of row vectors, length (K)

The list containing the initial probability vectors of the service time distributions of the various customer types. The length of the

vectors does not have to be the same.

S : list of square matrices, length (K)

The transient generators of the phase type distributions representing the service time of the jobs belonging to various types.

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
“ncMoms” Number of moments The moments of the number of customers
“ncDistr” Upper limit K The distribution of the number of customers from level 0 to level K-1
“stMoms” Number of moments The sojourn time moments
“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 continuous PH representation
“prec” The precision Numerical precision used as a stopping condition when solving the Riccati equation
“classes” Vector of integers Only the performance measures belonging to these classes are returned. If not given, all classes are analyzed.

(The quantities related to the number of customers in the system include the customer in the server, and the sojourn time related quantities include the service times as well)
Returns:

Ret : list of the performance measures

Each entry of the list corresponds to a performance measure requested. Each entry is a matrix, where the columns belong to the various job types. If there is just a single item, then it is not put into a list.

References

[R29](1, 2) Qiming He, “Analysis of a continuous time SM[K]/PH[K]/1/FCFS queue: Age process, sojourn times, and queue lengths”, Journal of Systems Science and Complexity, 25(1), pp 133-155, 2012.

Examples

For Matlab:

>>> D0 = [-5.49, 0., 1.15, 0.; 0., -2.29, 0., 0.; 0., 0.08, -1.32, 0.; 0.72, 1.17, 0.7, -7.07];
>>> D1 = [0.25, 0.38, 0.64, 0.; 0., 0., 0., 1.09; 0., 1.24, 0., 0.; 0.37, 0., 0., 0.];
>>> D2 = [0.3, 1.0, 0., 0.48; 0., 0.2, 0., 0.; 0., 0., 0., 0.; 0.61, 0., 0., 0.2];
>>> D3 = [0., 0.98, 0., 0.31; 0., 0., 1.0, 0.; 0., 0., 0., 0.; 1.1, 0.84, 0.33, 1.03];
>>> sigma3 = [0.83333,0.11404,0.05263];
>>> S3 = [-3., 0., 0.; 0.73077, -0.73077, 0.; 0., 0.5, -0.5];
>>> sigma2 = [1.];
>>> S2 = [-2.];
>>> sigma1 = [0.25,0.75];
>>> S1 = [-2.5, 2.5; 0., -10.];
>>> [ncm1, ncd1, ncm2, ncd2, ncm3, ncd3] = MMAPPH1FCFS({D0, D1, D2, D3}, {sigma1, sigma2, sigma3}, {S1, S2, S3}, 'ncMoms', 3, 'ncDistr', 500);
Final Residual Error for Psi:    4.0523e-15
>>> distrPoints = [1., 5., 10.];
>>> [stm1, std1, stm2, std2, stm3, std3] = MMAPPH1FCFS({D0, D1, D2, D3}, {sigma1, sigma2, sigma3}, {S1, S2, S3}, 'stMoms', 3, 'stDistr', distrPoints);
Final Residual Error for Psi:    4.0523e-15
>>> disp(stm1);
        5.847       84.706       1866.7
>>> disp(std1);
      0.28789      0.60379      0.79933
>>> disp(stm2);
       6.3613       91.529       2014.4
>>> disp(std2);
      0.20213      0.57229       0.7835
>>> disp(stm3);
       6.4108       92.984       2049.2
>>> disp(std3);
      0.21755       0.5651      0.77972
>>> [betap1, Bp1, betap2, Bp2, betap3, Bp3] = MMAPPH1FCFS({D0, D1, D2, D3}, {sigma1, sigma2, sigma3}, {S1, S2, S3}, 'stDistrPH');
Final Residual Error for Psi:    4.0523e-15
>>> [beta1, B1, beta2, B2, beta3, B3] = MMAPPH1FCFS({D0, D1, D2, D3}, {sigma1, sigma2, sigma3}, {S1, S2, S3}, 'stDistrME');
Final Residual Error for Psi:    4.0523e-15
>>> stdFromPH1 = CdfFromPH(betap1, Bp1, distrPoints);
>>> disp(stdFromPH1);
      0.28789      0.60379      0.79933
>>> stmFromME1 = MomentsFromME(beta1, B1, 3);
>>> disp(stmFromME1);
        5.847       84.706       1866.7
>>> stdFromPH2 = CdfFromPH(betap2, Bp2, distrPoints);
>>> disp(stdFromPH2);
      0.20213      0.57229       0.7835
>>> stmFromME2 = MomentsFromME(beta2, B2, 3);
>>> disp(stmFromME2);
       6.3613       91.529       2014.4
>>> stdFromPH3 = CdfFromPH(betap3, Bp3, distrPoints);
>>> disp(stdFromPH3);
      0.21755       0.5651      0.77972
>>> stmFromME3 = MomentsFromME(beta3, B3, 3);
>>> disp(stmFromME3);
       6.4108       92.984       2049.2

For Mathematica:

>>> D0 = {{-5.49, 0., 1.15, 0.},{0., -2.29, 0., 0.},{0., 0.08, -1.32, 0.},{0.72, 1.17, 0.7, -7.07}};
>>> D1 = {{0.25, 0.38, 0.64, 0.},{0., 0., 0., 1.09},{0., 1.24, 0., 0.},{0.37, 0., 0., 0.}};
>>> D2 = {{0.3, 1.0, 0., 0.48},{0., 0.2, 0., 0.},{0., 0., 0., 0.},{0.61, 0., 0., 0.2}};
>>> D3 = {{0., 0.98, 0., 0.31},{0., 0., 1.0, 0.},{0., 0., 0., 0.},{1.1, 0.84, 0.33, 1.03}};
>>> sigma3 = {0.83333,0.11404,0.05263};
>>> S3 = {{-3., 0., 0.},{0.73077, -0.73077, 0.},{0., 0.5, -0.5}};
>>> sigma2 = {1.};
>>> S2 = {{-2.}};
>>> sigma1 = {0.25,0.75};
>>> S1 = {{-2.5, 2.5},{0., -10.}};
>>> {ncm1, ncd1, ncm2, ncd2, ncm3, ncd3} = MMAPPH1FCFS[{D0, D1, D2, D3}, {sigma1, sigma2, sigma3}, {S1, S2, S3}, "ncMoms", 3, "ncDistr", 500];
"Final Residual Error for Psi: "7.105427357601002*^-15
>>> distrPoints = {1., 5., 10.};
>>> {stm1, std1, stm2, std2, stm3, std3} = MMAPPH1FCFS[{D0, D1, D2, D3}, {sigma1, sigma2, sigma3}, {S1, S2, S3}, "stMoms", 3, "stDistr", distrPoints];
"Final Residual Error for Psi: "7.105427357601002*^-15
>>> Print[stm1];
{5.847006216024009, 84.70619836863656, 1866.7093911950303}
>>> Print[std1];
{0.28789222215503396, 0.6037889208096201, 0.7993288816416165}
>>> Print[stm2];
{6.3613219350713415, 91.52937282421111, 2014.4076848738787}
>>> Print[std2];
{0.20213123601937755, 0.5722851898998502, 0.7834982143099977}
>>> Print[stm3];
{6.410840761114221, 92.98444287168671, 2049.1524939597166}
>>> Print[std3];
{0.21755284454422186, 0.5650994743810849, 0.7797186180777291}
>>> {betap1, Bp1, betap2, Bp2, betap3, Bp3} = MMAPPH1FCFS[{D0, D1, D2, D3}, {sigma1, sigma2, sigma3}, {S1, S2, S3}, "stDistrPH"];
"Final Residual Error for Psi: "7.105427357601002*^-15
>>> {beta1, B1, beta2, B2, beta3, B3} = MMAPPH1FCFS[{D0, D1, D2, D3}, {sigma1, sigma2, sigma3}, {S1, S2, S3}, "stDistrME"];
"Final Residual Error for Psi: "7.105427357601002*^-15
>>> stdFromPH1 = CdfFromPH[betap1, Bp1, distrPoints];
>>> Print[stdFromPH1];
{0.2878922221550334, 0.6037889208096203, 0.7993288816416169}
>>> stmFromME1 = MomentsFromME[beta1, B1, 3];
>>> Print[stmFromME1];
{5.847006216024715, 84.70619836865984, 1866.7093911957213}
>>> stdFromPH2 = CdfFromPH[betap2, Bp2, distrPoints];
>>> Print[stdFromPH2];
{0.2021312360193771, 0.5722851898998502, 0.7834982143099979}
>>> stmFromME2 = MomentsFromME[beta2, B2, 3];
>>> Print[stmFromME2];
{6.361321935071883, 91.52937282422667, 2014.4076848742416}
>>> stdFromPH3 = CdfFromPH[betap3, Bp3, distrPoints];
>>> Print[stdFromPH3];
{0.21755284454422208, 0.5650994743810853, 0.7797186180777295}
>>> stmFromME3 = MomentsFromME[beta3, B3, 3];
>>> Print[stmFromME3];
{6.410840761113892, 92.98444287168762, 2049.152493959693}

For Python/Numpy:

>>> D0 = ml.matrix([[-5.49, 0., 1.15, 0.],[0., -2.29, 0., 0.],[0., 0.08, -1.32, 0.],[0.72, 1.17, 0.7, -7.07]])
>>> D1 = ml.matrix([[0.25, 0.38, 0.64, 0.],[0., 0., 0., 1.09],[0., 1.24, 0., 0.],[0.37, 0., 0., 0.]])
>>> D2 = ml.matrix([[0.3, 1.0, 0., 0.48],[0., 0.2, 0., 0.],[0., 0., 0., 0.],[0.61, 0., 0., 0.2]])
>>> D3 = ml.matrix([[0., 0.98, 0., 0.31],[0., 0., 1.0, 0.],[0., 0., 0., 0.],[1.1, 0.84, 0.33, 1.03]])
>>> sigma3 = ml.matrix([[0.83333,0.11404,0.05263]])
>>> S3 = ml.matrix([[-3., 0., 0.],[0.73077, -0.73077, 0.],[0., 0.5, -0.5]])
>>> sigma2 = ml.matrix([[1.]])
>>> S2 = ml.matrix([[-2.]])
>>> sigma1 = ml.matrix([[0.25,0.75]])
>>> S1 = ml.matrix([[-2.5, 2.5],[0., -10.]])
>>> ncm1, ncd1, ncm2, ncd2, ncm3, ncd3 = MMAPPH1FCFS([D0, D1, D2, D3], [sigma1, sigma2, sigma3], [S1, S2, S3], "ncMoms", 3, "ncDistr", 500)
Final Residual Error for G:  4.121702978920894e-15
>>> distrPoints = [1., 5., 10.]
>>> stm1, std1, stm2, std2, stm3, std3 = MMAPPH1FCFS([D0, D1, D2, D3], [sigma1, sigma2, sigma3], [S1, S2, S3], "stMoms", 3, "stDistr", distrPoints)
Final Residual Error for G:  4.121702978920894e-15
>>> print(stm1)
[5.8470062160235257, 84.706198368623802, 1866.7093911946247]
>>> print(std1)
[ 0.28789  0.60379  0.79933]
>>> print(stm2)
[6.3613219350708565, 91.529372824197793, 2014.4076848734521]
>>> print(std2)
[ 0.20213  0.57229  0.7835 ]
>>> print(stm3)
[6.4108407611137332, 92.984442871673338, 2049.1524939592864]
>>> print(std3)
[ 0.21755  0.5651   0.77972]
>>> betap1, Bp1, betap2, Bp2, betap3, Bp3 = MMAPPH1FCFS([D0, D1, D2, D3], [sigma1, sigma2, sigma3], [S1, S2, S3], "stDistrPH")
Final Residual Error for G:  4.121702978920894e-15
>>> beta1, B1, beta2, B2, beta3, B3 = MMAPPH1FCFS([D0, D1, D2, D3], [sigma1, sigma2, sigma3], [S1, S2, S3], "stDistrME")
Final Residual Error for G:  4.121702978920894e-15
>>> stdFromPH1 = CdfFromPH(betap1, Bp1, distrPoints)
>>> print(stdFromPH1)
[ 0.28789  0.60379  0.79933]
>>> stmFromME1 = MomentsFromME(beta1, B1, 3)
>>> print(stmFromME1)
[5.847006216026557, 84.7061983687164, 1866.7093911974439]
>>> stdFromPH2 = CdfFromPH(betap2, Bp2, distrPoints)
>>> print(stdFromPH2)
[ 0.20213  0.57229  0.7835 ]
>>> stmFromME2 = MomentsFromME(beta2, B2, 3)
>>> print(stmFromME2)
[6.3613219350690784, 91.52937282416201, 2014.4076848721443]
>>> stdFromPH3 = CdfFromPH(betap3, Bp3, distrPoints)
>>> print(stdFromPH3)
[ 0.21755  0.5651   0.77972]
>>> stmFromME3 = MomentsFromME(beta3, B3, 3)
>>> print(stmFromME3)
[6.4108407611132225, 92.984442871662935, 2049.1524939590845]