butools.queues.QBDQueue¶
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butools.queues.QBDQueue()¶ Matlab: Ret = QBDQueue(B, L, F, L0, ...)Mathematica: Ret = QBDQueue[B, L, F, L0, ...]Python/Numpy: Ret = QBDQueue(B, L, F, L0, ...)Returns various performane measures of a continuous time QBD queue.
QBD queues have a background continuous time Markov chain with generator Q whose the transitions can be partitioned into three sets: transitions accompanied by an arrival of a new job (F, forward), transitions accompanied by the service of the current job in the server (B, backward) and internal transitions (L, local). Thus we have Q=B+L+F. L0 is the matrix of local transition rates if the queue is empty.
Parameters: B : matrix, shape(N,N)
Transitions of the background process accompanied by the service of the current job in the server
L : matrix, shape(N,N)
Internal transitions of the background process that do not generate neither arrival nor service
F : matrix, shape(N,N)
Transitions of the background process accompanied by an arrival of a new job
L0 : matrix, shape(N,N)
Internal transitions of the background process when there are no jobs in the queue
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 “ncDistrMG” None The vector-matrix parameters of the matrix-geometric distribution of the number of customers in the system “ncDistrDPH” None The vector-matrix parameters of the matrix-geometric distribution of the number of customers in the system, converted to a discrete PH representation “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 matrix-quadratic equation (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. If there is just a single item, then it is not put into a list.
Notes
“ncDistrMG” and “stDistrMG” behave much better numerically than “ncDistrDPH” and “stDistrPH”.
Examples
For Matlab:
>>> B = [6., 1., 0.; 0., 4., 1.; 2., 0., 0.]; >>> F = [0., 1., 1.; 5., 0., 0.; 1., 3., 0.]; >>> L = [-14., 3., 2.; 0., -14., 4.; 3., 1., -10.]; >>> L0 = L+B; >>> disp(L0); -8 4 2 0 -10 5 5 1 -10 >>> [ncd, ncm] = QBDQueue(B, L, F, L0, 'ncDistr', 11, 'ncMoms', 5); >>> disp(ncd); Columns 1 through 8 0.29094 0.20433 0.14547 0.10358 0.073709 0.052461 0.037338 0.026574 Columns 9 through 11 0.018913 0.013461 0.0095805 >>> disp(ncm); 2.46 14.608 128.88 1515.9 22288 >>> [alphap, Ap] = QBDQueue(B, L, F, L0, 'ncDistrDPH'); >>> disp(alphap); 0.28256 0.22386 0.20264 >>> disp(Ap); 0.11832 0.4103 0.18373 0.196 0.19477 0.32261 0.27566 0.22146 0.21229 >>> [alpha, A] = QBDQueue(B, L, F, L0, 'ncDistrMG'); >>> disp(alpha); 0.042581 -0.27854 0.94503 >>> disp(A); -0.0056928 -0.63242 1.3313 0.069475 -0.36133 0.99875 0.037353 -0.21854 0.8924 >>> ncdFromDPH = PmfFromDPH(alphap, Ap, (0:1:10)); >>> disp(ncdFromDPH); Columns 1 through 8 0.29094 0.20433 0.14547 0.10358 0.073709 0.052461 0.037338 0.026574 Columns 9 through 11 0.018913 0.013461 0.0095805 >>> ncmFromMG = MomentsFromMG(alpha, A, 5); >>> disp(ncmFromMG); 2.46 14.608 128.88 1515.9 22288 >>> [std, stm] = QBDQueue(B, L, F, L0, 'stDistr', (0.:0.1:1.), 'stMoms', 5); >>> disp(std); Columns 1 through 8 1.1102e-16 0.14225 0.25715 0.35489 0.43933 0.51262 0.5763 0.63165 Columns 9 through 11 0.67977 0.72161 0.75797 >>> disp(stm); 0.70236 1.0017 2.1463 6.1325 21.903 >>> [betap, Bp] = QBDQueue(B, L, F, L0, 'stDistrPH'); >>> disp(betap); Columns 1 through 8 0.52899 0 0 0 0.35507 0 0 0 Column 9 0.11594 >>> disp(Bp); Columns 1 through 8 -12.591 2.362 0.50976 4.2349 0.39366 0.084961 3 0 2.001 -12.662 0.95667 0.3335 4.2231 0.15944 0 3 2.8354 0.93483 -13.595 0.47257 0.15581 4.0674 0 0 5 0 0 -13.061 1.5746 0.33984 4.2349 0.39366 0 5 0 1.334 -13.108 0.63778 0.3335 4.2231 0 0 5 1.8903 0.62322 -13.73 0.47257 0.15581 4.4697 0.78732 0.16992 4 0 0 -10 0 0.667 4.4461 0.31889 0 4 0 0 -10 0.94514 0.31161 4.1349 0 0 4 0 0 Column 9 0 0 3 0.084961 0.15944 4.0674 0 0 -10 >>> [beta, B] = QBDQueue(B, L, F, L0, 'stDistrME'); >>> disp(beta); Columns 1 through 8 0.17391 0.47826 -0.71739 0.71739 -1.2391 1.2391 -1.2391 0.52174 Column 9 1.0652 >>> disp(B); Columns 1 through 8 -12.591 -0.41431 -3.408 2.165 -13.136 15 -13.409 12.066 0.5517 -13.315 -8.0386 7.3408 -10.01 10.714 -6.3994 0.033031 0.20504 -1.8844 -15.359 1.8939 -4.0612 3.4626 -0.19879 0.022021 -0.28868 1.3811 -2.5957 -11.073 -3.7559 3.4626 -1.3717 -3.2512 0.48659 -1.3755 -1.3906 5.4522 -17.351 -1.5374 2.853 -3.4525 1.5997 -0.77852 -1.6145 5.5725 -4.1872 -15.012 -0.7327 0.5475 1.0779 -2.1431 1.8972 1.9933 -1.6703 0.26853 -14.463 0.5475 2.0033 -1.9938 1.8972 1.9933 -5.9052 4.3998 -4.3594 -9.4525 0.59557 0.82156 -1.1028 4.9933 -6.4916 4.9354 -4.3085 0.5475 Column 9 1.3431 6.6716 4.4477 8.8939 9.8705 9.8705 9.8705 9.8705 -0.12955 >>> stdFromPH = CdfFromPH(betap, Bp, (0.:0.1:1.)); >>> disp(stdFromPH); Columns 1 through 8 1.1102e-16 0.14225 0.25715 0.35489 0.43933 0.51262 0.5763 0.63165 Columns 9 through 11 0.67977 0.72161 0.75797 >>> stmFromME = MomentsFromME(beta, B, 5); >>> disp(stmFromME); 0.70236 1.0017 2.1463 6.1325 21.903
For Mathematica:
>>> B = {{6., 1., 0.},{0., 4., 1.},{2., 0., 0.}}; >>> F = {{0., 1., 1.},{5., 0., 0.},{1., 3., 0.}}; >>> L = {{-14., 3., 2.},{0., -14., 4.},{3., 1., -10.}}; >>> L0 = L+B; >>> Print[L0]; {{-8., 4., 2.}, {0., -10., 5.}, {5., 1., -10.}} >>> {ncd, ncm} = QBDQueue[B, L, F, L0, "ncDistr", 11, "ncMoms", 5]; "Final Residual Error for G: "1.249000902703301*^-16 "Final Residual Error for R: "8.326672684688674*^-17 >>> Print[ncd]; {0.29093602691449844, 0.2043276077989581, 0.14546751846779368, 0.10357801953919166, 0.07370933599101509, 0.0524613602236208, 0.037337898930337544, 0.02657416066003302, 0.018913402582327217, 0.013461073324799929, 0.009580534061633792} >>> Print[ncm]; {2.460029520212033, 14.60791905179449, 128.87860443055757, 1515.888454046844, 22287.969412532402} >>> {alphap, Ap} = QBDQueue[B, L, F, L0, "ncDistrDPH"]; "Final Residual Error for G: "1.249000902703301*^-16 "Final Residual Error for R: "8.326672684688674*^-17 >>> Print[alphap]; {0.28256467199124746, 0.22386225497958323, 0.2026370461146711} >>> Print[Ap]; {{0.11831531605275415, 0.41029505226571794, 0.18373196776426398}, {0.19600449960093794, 0.19477428983994485, 0.32260956940537433}, {0.2756583766062947, 0.2214649539635013, 0.2122872279368174}} >>> {alpha, A} = QBDQueue[B, L, F, L0, "ncDistrMG"]; "Final Residual Error for G: "1.249000902703301*^-16 "Final Residual Error for R: "8.326672684688674*^-17 >>> Print[alpha]; {0.042581006723761944, -0.27854425286936896, 0.9450272192311088} >>> Print[A]; {{-0.005692752239919885, -0.6324216674588319, 1.3312918216215348}, {0.06947485017082566, -0.3613333778272245, 0.9987453751370272}, {0.037353395792620064, -0.21853933991282004, 0.8924029638966606}} >>> ncdFromDPH = PmfFromDPH[alphap, Ap, Range[0,10,1]]; >>> Print[ncdFromDPH]; {0.29093602691449827, 0.2043276077989581, 0.14546751846779366, 0.10357801953919163, 0.07370933599101509, 0.0524613602236208, 0.03733789893033755, 0.026574160660033017, 0.018913402582327217, 0.013461073324799929, 0.009580534061633792} >>> ncmFromMG = MomentsFromMG[alpha, A, 5]; >>> Print[ncmFromMG]; {2.4600295202120344, 14.607919051794505, 128.87860443055771, 1515.8884540468462, 22287.969412532442} >>> {std, stm} = QBDQueue[B, L, F, L0, "stDistr", Range[0.,1.,0.1], "stMoms", 5]; "Final Residual Error for G: "1.249000902703301*^-16 "Final Residual Error for R: "8.326672684688674*^-17 >>> Print[std]; {-2.220446049250313*^-16, 0.1422502687156617, 0.257145752951746, 0.35488604834530724, 0.4393330753410507, 0.5126194621799739, 0.5763000198215283, 0.6316534259881905, 0.6797734673117404, 0.7216066395715914, 0.7579746780193366} >>> Print[stm]; {0.702356254321406, 1.0017175985977291, 2.1462963079180923, 6.1325315761778585, 21.90316181223906} >>> {betap, Bp} = QBDQueue[B, L, F, L0, "stDistrPH"]; "Final Residual Error for G: "1.249000902703301*^-16 "Final Residual Error for R: "8.326672684688674*^-17 >>> Print[betap]; {0.5289855072463769, 0., 0., 0., 0.35507246376811596, 0., 0., 0., 0.11594202898550725} >>> Print[Bp]; {{-12.590793810464493, 2.361969158122816, 0.5097641466892873, 4.2348676982559175, 0.39366152635380264, 0.08496069111488122, 3., 0., 0.}, {2.0010093466117524, -12.661585604321022, 0.956667718183873, 0.3335015577686254, 4.223069065946496, 0.15944461969731216, 0., 3., 0.}, {2.8354199578367134, 0.9348326530713538, -13.595359582237387, 0.47256999297278557, 0.15580544217855896, 4.067440069627102, 0., 0., 3.}, {5., 0., 0., -13.060529206976328, 1.5746461054152106, 0.3398427644595249, 4.2348676982559175, 0.39366152635380264, 0.08496069111488122}, {0., 5., 0., 1.3340062310745016, -13.107723736214014, 0.6377784787892486, 0.3335015577686254, 4.223069065946496, 0.15944461969731216}, {0., 0., 5., 1.8902799718911423, 0.6232217687142358, -13.730239721491591, 0.47256999297278557, 0.15580544217855896, 4.067440069627102}, {4.469735396511836, 0.7873230527076053, 0.16992138222976244, 4., 0., 0., -10., 0., 0.}, {0.6670031155372508, 4.446138131892993, 0.3188892393946243, 0., 4., 0., 0., -10., 0.}, {0.9451399859455711, 0.3116108843571179, 4.1348801392542045, 0., 0., 4., 0., 0., -10.}} >>> {beta, B} = QBDQueue[B, L, F, L0, "stDistrME"]; "Final Residual Error for G: "1.249000902703301*^-16 "Final Residual Error for R: "8.326672684688674*^-17 >>> Print[beta]; {0.17391304347826098, 0.4782608695652174, -0.7173913043478262, 0.7173913043478262, -1.239130434782609, 1.239130434782609, -1.239130434782609, 0.5217391304347829, 1.065217391304348} >>> Print[B]; {{-12.590793810464493, -0.4143079390446338, -3.4079678715964405, 2.165044054462539, -13.135614274299147, 15., -13.409206189535507, 12.066062929481042, 1.3431432600544637}, {0.5517026304648383, -13.31487770573633, -8.038587534717966, 7.340812756512537, -10.010346278592026, 10.71360053252702, -6.399351616213842, 0.03303146474052099, 6.671571630027232}, {0.20503537103555214, -1.884385705275542, -15.359058356478641, 1.8938751710083572, -4.061197338939663, 3.462593438602556, -0.19879434118212647, 0.022020976493681843, 4.447714420018154}, {-0.28868062527080385, 1.3811139661569207, -2.5957222593063154, -11.072961911866576, -3.7559458603857516, 3.462593438602556, -1.371745011542437, -3.2511664850390023, 8.893852551911149}, {0.48658568115248757, -1.3754727933458817, -1.3906370069902225, 5.452192557308922, -17.35130544262314, -1.537406561397444, 2.8529886820342716, -3.4524984742617457, 9.870450847557183}, {1.5997319672256904, -0.7785202333216761, -1.6144942169993, 5.572472782556839, -4.187199250885652, -15.012243707418406, -0.7327033789210926, 0.5475015257382534, 9.870450847557183}, {1.0778774878655475, -2.143055839126959, 1.8972066351776808, 1.9933318607527575, -1.6702917941051352, 0.2685286269197684, -14.462943100412684, 0.5475015257382534, 9.870450847557183}, {2.0032584178625217, -1.9938176991209073, 1.8972066351776808, 1.9933318607527575, -5.905159492361053, 4.399819340414528, -4.359366115651527, -9.452498474261745, 9.870450847557183}, {0.5955687292136896, 0.8215616781767565, -1.1027933648223192, 4.9933318607527575, -6.4916348275412075, 4.935419429169031, -4.308490869225876, 0.5475015257382534, -0.12954915244281562}} >>> stdFromPH = CdfFromPH[betap, Bp, Range[0.,1.,0.1]]; >>> Print[stdFromPH]; {-2.220446049250313*^-16, 0.14225026871566182, 0.25714575295174624, 0.354886048345307, 0.4393330753410508, 0.5126194621799739, 0.576300019821528, 0.6316534259881904, 0.6797734673117404, 0.7216066395715919, 0.7579746780193367} >>> stmFromME = MomentsFromME[beta, B, 5]; >>> Print[stmFromME]; {0.7023562543214075, 1.0017175985977365, 2.1462963079181128, 6.132531576177928, 21.903161812239368}
For Python/Numpy:
>>> B = ml.matrix([[6., 1., 0.],[0., 4., 1.],[2., 0., 0.]]) >>> F = ml.matrix([[0., 1., 1.],[5., 0., 0.],[1., 3., 0.]]) >>> L = ml.matrix([[-14., 3., 2.],[0., -14., 4.],[3., 1., -10.]]) >>> L0 = L+B >>> print(L0) [[ -8. 4. 2.] [ 0. -10. 5.] [ 5. 1. -10.]] >>> ncd, ncm = QBDQueue(B, L, F, L0, "ncDistr", 11, "ncMoms", 5) Final Residual Error for G: 9.71445146547e-17 Final Residual Error for R: 1.11022302463e-16 >>> print(ncd) [ 0.29094 0.20433 0.14547 0.10358 0.07371 0.05246 0.03734 0.02657 0.01891 0.01346 0.00958] >>> print(ncm) [2.4600295202120317, 14.607919051794482, 128.87860443055746, 1515.8884540468425, 22287.969412532377] >>> alphap, Ap = QBDQueue(B, L, F, L0, "ncDistrDPH") Final Residual Error for G: 9.71445146547e-17 Final Residual Error for R: 1.11022302463e-16 >>> print(alphap) [[ 0.28256 0.22386 0.20264]] >>> print(Ap) [[ 0.11832 0.4103 0.18373] [ 0.196 0.19477 0.32261] [ 0.27566 0.22146 0.21229]] >>> alpha, A = QBDQueue(B, L, F, L0, "ncDistrMG") Final Residual Error for G: 9.71445146547e-17 Final Residual Error for R: 1.11022302463e-16 >>> print(alpha) [[ 0.04258 -0.27854 0.94503]] >>> print(A) [[-0.00569 -0.63242 1.33129] [ 0.06947 -0.36133 0.99875] [ 0.03735 -0.21854 0.8924 ]] >>> ncdFromDPH = PmfFromDPH(alphap, Ap, np.arange(0,11.0,1)) >>> print(ncdFromDPH) [ 0.29094 0.20433 0.14547 0.10358 0.07371 0.05246 0.03734 0.02657 0.01891 0.01346 0.00958] >>> ncmFromMG = MomentsFromMG(alpha, A, 5) >>> print(ncmFromMG) [2.4600295202120313, 14.607919051794475, 128.87860443055737, 1515.8884540468407, 22287.969412532337] >>> std, stm = QBDQueue(B, L, F, L0, "stDistr", np.arange(0.,1.1,0.1), "stMoms", 5) Final Residual Error for G: 9.71445146547e-17 Final Residual Error for R: 1.11022302463e-16 >>> print(std) [ 1.11022e-16 1.42250e-01 2.57146e-01 3.54886e-01 4.39333e-01 5.12619e-01 5.76300e-01 6.31653e-01 6.79773e-01 7.21607e-01 7.57975e-01] >>> print(stm) [0.70235625432140525, 1.0017175985977307, 2.1462963079180937, 6.1325315761778585, 21.903161812239034] >>> betap, Bp = QBDQueue(B, L, F, L0, "stDistrPH") Final Residual Error for G: 9.71445146547e-17 Final Residual Error for R: 1.11022302463e-16 >>> print(betap) [[ 0.52899 0. 0. 0. 0.35507 0. 0. 0. 0.11594]] >>> print(Bp) [[-12.59079 2.36197 0.50976 4.23487 0.39366 0.08496 3. 0. 0. ] [ 2.00101 -12.66159 0.95667 0.3335 4.22307 0.15944 0. 3. 0. ] [ 2.83542 0.93483 -13.59536 0.47257 0.15581 4.06744 0. 0. 3. ] [ 5. 0. 0. -13.06053 1.57465 0.33984 4.23487 0.39366 0.08496] [ 0. 5. 0. 1.33401 -13.10772 0.63778 0.3335 4.22307 0.15944] [ 0. 0. 5. 1.89028 0.62322 -13.73024 0.47257 0.15581 4.06744] [ 4.46974 0.78732 0.16992 4. 0. 0. -10. 0. 0. ] [ 0.667 4.44614 0.31889 0. 4. 0. 0. -10. 0. ] [ 0.94514 0.31161 4.13488 0. 0. 4. 0. 0. -10. ]] >>> beta, B = QBDQueue(B, L, F, L0, "stDistrME") Final Residual Error for G: 9.71445146547e-17 Final Residual Error for R: 1.11022302463e-16 >>> print(beta) [[ 0.17391 0.47826 -0.71739 0.71739 -1.23913 1.23913 -1.23913 0.52174 1.06522]] >>> print(B) [[-12.59079 -0.41431 -3.40797 2.16504 -13.13561 15. -13.40921 12.06606 1.34314] [ 0.5517 -13.31488 -8.03859 7.34081 -10.01035 10.7136 -6.39935 0.03303 6.67157] [ 0.20504 -1.88439 -15.35906 1.89388 -4.0612 3.46259 -0.19879 0.02202 4.44771] [ -0.28868 1.38111 -2.59572 -11.07296 -3.75595 3.46259 -1.37175 -3.25117 8.89385] [ 0.48659 -1.37547 -1.39064 5.45219 -17.35131 -1.53741 2.85299 -3.4525 9.87045] [ 1.59973 -0.77852 -1.61449 5.57247 -4.1872 -15.01224 -0.7327 0.5475 9.87045] [ 1.07788 -2.14306 1.89721 1.99333 -1.67029 0.26853 -14.46294 0.5475 9.87045] [ 2.00326 -1.99382 1.89721 1.99333 -5.90516 4.39982 -4.35937 -9.4525 9.87045] [ 0.59557 0.82156 -1.10279 4.99333 -6.49163 4.93542 -4.30849 0.5475 -0.12955]] >>> stdFromPH = CdfFromPH(betap, Bp, np.arange(0.,1.1,0.1)) >>> print(stdFromPH) [ 1.11022e-16 1.42250e-01 2.57146e-01 3.54886e-01 4.39333e-01 5.12619e-01 5.76300e-01 6.31653e-01 6.79773e-01 7.21607e-01 7.57975e-01] >>> stmFromME = MomentsFromME(beta, B, 5) >>> print(stmFromME) [0.70235625432140614, 1.001717598597732, 2.1462963079180968, 6.1325315761778718, 21.903161812239095]