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003ooooooccooooo003ooooooccooooo003ooooooccooooo003ooooooccooooo003ooooooccooooo 003ooooooccooooo003ooooooccooooo003ooooooccooooo003ooooooccooooo003ooooooccooooo 003ooooooccooooo003ooooooccooooo003ooooooccooooo003ooooooccooooo003ooooooccooooo 003ooooooccooooo003ooooooccooooo003ooooooccooooo003ooooooccooooo003ooooooccooooo 0000\ \>"], ImageRangeCache->{{{0, 314.312}, {209.188, 0}} -> {-2.00384, -1.00002, \ 0.0143414, 0.0143414}}], Cell[CellGroupData[{ Cell["\<\ EXPLOOT project Predicting the capacity in a logistic model\ \>", "Subsubtitle"], Cell[TextData[StyleBox["http://we.vub.ac.be/exploot/website", FontColor->RGBColor[0, 0, 1], FontVariations->{"Underline"->True}]], "Subsubsection", CellDingbat->None] }, Open ]], Cell["\<\ I. Cnop icnop@vub.ac.be\ \>", "Subsubtitle"], Cell[CellGroupData[{ Cell["Dotcom crashes: can we predict the capacity in a system?", "Subtitle"], Cell[CellGroupData[{ Cell["Introduction", "Subsection"], Cell["\<\ At the end of the nineties , nobody would have believed that the \ portable phone (GSM) exponential expansion could level off. Already in the Fall 2000, shares of Nokia and Ericsson saw their equities \ decline. After its 1Q 2001, Siemens reported a loss exceeding 300 M DEM (about 150 M \ Euro) in its GSM production & sales division. Price competition between manufacturers is fierce and investments (including \ underwritten debt) in third generation content delivery technology is still \ not paying off. The reason is that unlimited expansion has changed to a replacement economy \ in the original markets. \ \>", "Text"] }, Closed]], Cell[CellGroupData[{ Cell["Is this maximum capacity predictable? The answer is: YES !", \ "Subsubtitle"], Cell["\<\ The market for portable (mobile cell phone or GSM) use is a typical \ example of the logistic equation. Its success can be explained by the fact that users tend to incite non-users \ to join in, thus creating a nonlinear multiplier effect. The logistic equation can be solved exactly (up to rescaling of the \ time-axis) if the capacity of the system is known. The method below was originally developed to answer another problem: to \ recover the asymptotic capacity in population estimates for several countries \ and continents published in graphical form in the seventies by UNESCO, \ relying on census figures from 1950 to 1970. This problem was complicated by the fact that only the graphical output was \ available and original census figures where not.\ \>", "Text"], Cell[CellGroupData[{ Cell["Solution of the logistic model", "Subsection"], Cell[CellGroupData[{ Cell["General solution", "Subsubsection"], Cell["\<\ We assume increase per time unit (say one month) is proportinal to \ the number of portable users multiplied by the number of non users:\ \>", \ "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(sol\ = DSolve[\ \(n'\)[t]\ \[Equal] r\ n[t]\ \((\ 1\ - \ n[t]/K)\), n[t], t]\)], "Input"], Cell[BoxData[ \({{n[ t] \[Rule] \(\[ExponentialE]\^\(r\ t + K\ C[1]\)\ K\)\/\(\(-1\) + \ \[ExponentialE]\^\(r\ t + K\ C[1]\)\)}}\)], "Output"] }, Closed]], Cell[CellGroupData[{ Cell[BoxData[ \(n[t] /. First[%] // TraditionalForm\)], "Input"], Cell[BoxData[ FormBox[ FractionBox[ RowBox[{ SuperscriptBox["\[ExponentialE]", RowBox[{\(r\ t\), "+", RowBox[{"K", " ", SubscriptBox[ TagBox["c", C], "1"]}]}]], " ", "K"}], RowBox[{\(-1\), "+", SuperscriptBox["\[ExponentialE]", RowBox[{\(r\ t\), "+", RowBox[{"K", " ", SubscriptBox[ TagBox["c", C], "1"]}]}]]}]], TraditionalForm]], "Output"] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell["Initial condition and parametrisation", "Subsubsection"], Cell["\<\ Let us start with 1 k potential customers and 10 users to begin \ with. 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