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I am trying to use Mathematica to verify simulation results with an analytic result.

The analytic results are performance measures of the M/D/1 queue, which unfortunately uses the factorial function as shown below.

The problem I am having is that as utilization gets high ($\alpha \rightarrow 1$), then everything blows up and the results become unusable and are obviously wrong.

Can someone help me use the functions below so that they can compute the things I need in Mathematica?

Here is my code:

Pn[0, \[Alpha]_] := 1 - \[Alpha];
Pn[1, \[Alpha]_] := (1 - \[Alpha]) (E^\[Alpha] - 1);
Pn[n_Integer?NonNegative, \[Alpha]_] := (1 - \[Alpha])*
  Sum[E^(j*\[Alpha])*(-1)^(n - 
       j)*(((j*\[Alpha] + n - j)*(j*\[Alpha])^(n - j - 1))/(n - 
         j)!), {j, 0, n}]

maxRho = 0.998;
c = 1/(mean*100)*maxRho;
Block[{$MaxExtraPrecision = 5000},
 For[i = 0, i <= 100, i++,
  lam = i*c + 0.001;
  Xn = 1000;
  (*lambda=0.3;*)
  \[Alpha] = lam;
  n = (1 - \[Alpha]) Xn;
  \[Lambda] = \[Alpha];
  F = (1 - Pn[k, \[Lambda]]/\!\(
\*UnderoverscriptBox[\(\[Sum]\), \(x = 1\), \(k\)]\(Pn[
       x, \[Lambda]]\)\))^First[Integer[n]];
  R = 1 - F;
  EX = \!\(
\*UnderoverscriptBox[\(\[Sum]\), \(k = 1\), \(30\)]R\);
  Print["(", lam, ",", First[First[EX]], ")"]
  ]
 ]

First::normal: Nonatomic expression expected at position 1 in First[1.39351].

(0.001,1.39351)

First::normal: Nonatomic expression expected at position 1 in First[2.01588].

(0.01098,2.01588)

General::munfl: 1.11022*10^-16^979.04 is too small to represent as a normalized machine number; precision may be lost.

First::normal: Nonatomic expression expected at position 1 in First[2.07202].

General::stop: Further output of First::normal will be suppressed during this calculation.

(0.02096,2.07202)

(0.03094,2.15184)

General::munfl: 1.11022*10^-16^959.08 is too small to represent as a normalized machine number; precision may be lost.

(0.04092,2.25282)

(0.0509,2.3669)

(0.06088,2.48577)

(0.07086,2.60208)

(0.08084,2.71029)

(0.09082,2.80709)

(0.1008,2.8914)

(0.11078,2.96398)

General::munfl: 1.11022*10^-16^879.24 is too small to represent as a normalized machine number; precision may be lost.

General::stop: Further output of General::munfl will be suppressed during this calculation.

(0.12076,3.02689)

(0.13074,3.08281)

(0.14072,3.13459)

(0.1507,3.18476)

(0.16068,3.23538)

(0.17066,3.28792)

(0.18064,3.34327)

(0.19062,3.40183)

(0.2006,3.46363)

(0.21058,3.52839)

(0.22056,3.59566)

(0.23054,3.66485)

(0.24052,3.73537)

(0.2505,3.80664)

(0.26048,3.87815)

(0.27046,3.94954)

(0.28044,4.02056)

(0.29042,4.09113)

(0.3004,4.16132)

(0.31038,4.23133)

(0.32036,4.30149)

(0.33034,4.37215)

(0.34032,4.44374)

(0.3503,4.51662)

(0.36028,4.59118)

(0.37026,4.66765)

(0.38024,4.74624)

(0.39022,4.82712)

(0.4002,4.9102)

(0.41018,4.99555)

(0.42016,5.08302)

(0.43014,5.17285)

(0.44012,5.2647)

(0.4501,5.35885)

(0.46008,5.45487)

(0.47006,5.55297)

(0.48004,5.65391)

(0.49002,5.7572)

(0.5,5.86342)

(0.50998,5.97264)

(0.51996,6.08366)

(0.52994,6.20112)

(0.53992,6.32168)

(0.5499,6.44275)

(0.55988,6.56648)

(0.56986,6.70172)

(0.57984,6.84117)

(0.58982,6.97977)

(0.5998,7.12799)

(0.60978,7.27335)

(0.61976,7.43172)

(0.62974,7.6146)

(0.63972,7.75012)

(0.6497,7.98637)

(0.65968,8.13822)

(0.66966,8.36521)

(0.67964,8.46801)

(0.68962,8.63453)

(0.6996,9.03215)

(0.70958,9.37622)

(0.71956,9.6912)

(0.72954,9.61503)

(0.73952,9.70622)

(0.7495,8.88941)

(0.75948,10.7016)

(0.76946,10.425)

(0.77944,10.7229)

(0.78942,11.7255)

(0.7994,9.18956)

(0.80938,11.3586)

(0.81936,4.34082)

(0.82934,-16224.1)

(0.83932,12.8142)

(0.8493,-234.757)

(0.85928,6.62275)

(0.86926,15.2752)

(0.87924,13.2532)

(0.88922,16.7048)

(0.8992,17.376)

(0.90918,12.8485)

(0.91916,-5.48706*10^6)

(0.92914,-579624.)

(0.93912,-117.103)

(0.9491,20.4013)

(0.95908,20.973)

(0.96906,18.0114)

(0.97904,-252.336)

(0.98902,16.2527)

(0.999,3.95909)

Here are the simulation results. These should be just a tad lower than the values obtained above...

(0.001, 1.38589716783818)
(0.01098, 1.93374314103297)
(0.02096, 2.04522385202462)
(0.03094, 2.12186645396796)
(0.04092, 2.22403849126566)
(0.0509, 2.30691299397109)
(0.06088, 2.4378943443334)
(0.07086, 2.517177035023)
(0.08084, 2.57433346612195)
(0.09082, 2.67158769982665)
(0.1008, 2.73730997183621)
(0.11078, 2.81456659676033)
(0.12076, 2.88046155754555)
(0.13074, 2.92834154807784)
(0.14072, 2.99152596657689)
(0.1507, 3.07758958278677)
(0.16068, 3.11452150883068)
(0.17066, 3.22186780211217)
(0.18064, 3.28585314018323)
(0.19062, 3.34311179028806)
(0.2006, 3.37947576048867)
(0.21058, 3.43904742811983)
(0.22056, 3.55115038582248)
(0.23054, 3.60564960241238)
(0.24052, 3.67602522287391)
(0.2505, 3.73716344456839)
(0.26048, 3.8023064739795)
(0.27046, 3.83022560528692)
(0.28044, 3.94579480981448)
(0.29042, 3.97752100875677)
(0.3004, 4.11174789714349)
(0.31038, 4.10941122204524)
(0.32036, 4.25769629491418)
(0.33034, 4.28543667053131)
(0.34032, 4.33550740258635)
(0.3503, 4.42321705517266)
(0.36028, 4.54665402485875)
(0.37026, 4.63709534155703)
(0.38024, 4.66087141744613)
(0.39022, 4.81662374476191)
(0.4002, 4.86464050588849)
(0.41018, 4.95909206371304)
(0.42016, 5.0479785077863)
(0.43014, 5.10824300657774)
(0.44012, 5.20201717430431)
(0.4501, 5.29614975368933)
(0.46008, 5.36298433627588)
(0.47006, 5.49571576491656)
(0.48004, 5.6751101667503)
(0.49002, 5.67510989318424)
(0.5, 5.82865734860674)
(0.50998, 5.87748869917215)
(0.51996, 6.06715682675748)
(0.52994, 6.20824576522561)
(0.53992, 6.32709817320504)
(0.5499, 6.43811298970349)
(0.55988, 6.54654046655082)
(0.56986, 6.68678926260718)
(0.57984, 6.80169359902264)
(0.58982, 6.9518458553012)
(0.5998, 7.18302700779921)
(0.60978, 7.26791401965148)
(0.61976, 7.46058198390404)
(0.62974, 7.6028334326128)
(0.63972, 7.73442299789392)
(0.6497, 7.95705167019446)
(0.65968, 8.10187610383737)
(0.66966, 8.2822881207868)
(0.67964, 8.51473176758009)
(0.68962, 8.80623111370376)
(0.6996, 8.99592207159753)
(0.70958, 9.05867682860675)
(0.71956, 9.39082704752084)
(0.72954, 9.71967711010753)
(0.73952, 10.0616975101211)
(0.7495, 10.3163942606361)
(0.75948, 10.4803904667259)
(0.76946, 10.9672798131673)
(0.77944, 11.1338450777013)
(0.78942, 11.6669130099429)
(0.7994, 12.1099037780777)
(0.80938, 12.3838005979187)
(0.81936, 12.9501380959976)
(0.82934, 13.5238685220341)
(0.83932, 13.7657134896756)
(0.8493, 14.3022168202217)
(0.85928, 15.146915943203)
(0.86926, 15.9529380044323)
(0.87924, 16.5836713009528)
(0.88922, 17.6962236798752)
(0.8992, 18.3914440210623)
(0.90918, 19.7524583648123)
(0.91916, 21.1159814023742)
(0.92914, 22.4887559250303)
(0.93912, 23.7662819041863)
(0.9491, 27.3198186537483)
(0.95908, 30.0755919036298)
(0.96906, 34.546908244586)
(0.97904, 40.2033189750874)
Improve this question
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5
  • $\begingroup$ Your code doesn't work for me, but in general if you want to increase precision, you need to use high precision or exact numbers, not machine numbers as you are using here. Once you use a machine number like 0.998 or 0.001, you are stuck with machine precision for the entire calculation. $\endgroup$
    – Bill Watts
    Jun 25 at 0:37
  • 3
    $\begingroup$ Integer is the head for integers, it is not a function. What are you trying to do with First[Integer[n]]? Are you trying to Round the value of n? Find the Floor for n? Something else? $\endgroup$
    – Bob Hanlon
    Jun 25 at 4:36
  • $\begingroup$ "mean" is not defined. $\endgroup$
    – Daniel Huber
    Jun 25 at 7:09
  • $\begingroup$ Sorry, the mean is 1.0 $\endgroup$
    – PiE
    Jun 25 at 7:42
  • $\begingroup$ I'm just trying to take the integral value of the number - But Ceil[] could work too $\endgroup$
    – PiE
    Jun 25 at 7:42

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