(* Content-type: application/mathematica *) (*** Wolfram Notebook File ***) (* http://www.wolfram.com/nb *) (* CreatedBy='Mathematica 6.0' *) (*CacheID: 234*) (* Internal cache information: NotebookFileLineBreakTest NotebookFileLineBreakTest NotebookDataPosition[ 145, 7] NotebookDataLength[ 20364, 577] NotebookOptionsPosition[ 18176, 513] NotebookOutlinePosition[ 18595, 531] CellTagsIndexPosition[ 18552, 528] WindowFrame->Normal*) (* Beginning of Notebook Content *) Notebook[{ Cell[TextData[StyleBox["2301520 Fundamental of AMCS\nSimulation", FontWeight->"Bold", FontColor->RGBColor[0, 0, 1]]], "Title", CellChangeTimes->{{3.51415193884375*^9, 3.514151941796875*^9}, { 3.546811806796875*^9, 3.5468118124375*^9}, {3.576900100234375*^9, 3.576900105296875*^9}, {3.6131251928536477`*^9, 3.6131251998852663`*^9}, 3.6131262729460135`*^9, {3.644665273697053*^9, 3.6446652763353796`*^9}, { 3.676002861312434*^9, 3.676002873795472*^9}, 3.7409988232661543`*^9}, TextAlignment->Center], Cell[CellGroupData[{ Cell["Simulation", "Section", CellChangeTimes->{{3.6767265393723297`*^9, 3.6767265675160203`*^9}, { 3.740998801618332*^9, 3.7409988046317225`*^9}}], Cell["\<\ Simulation is a tool to help us making a decision. It has been used for \ simulating a complex system by mimicking an uncertain system with random \ variables of our defined distribution.\ \>", "DisplayText", CellChangeTimes->{{3.6770760391711044`*^9, 3.6770761946144943`*^9}}], Cell[BoxData[ RowBox[{ RowBox[{"pdist", "=", " ", RowBox[{"PoissonDistribution", "[", "20", "]"}]}], ";"}]], "Input"], Cell[BoxData[ RowBox[{"plot1", "=", RowBox[{"ListPlot", "[", RowBox[{ RowBox[{"Table", "[", RowBox[{ RowBox[{"PDF", "[", RowBox[{"pdist", ",", "x"}], "]"}], ",", " ", RowBox[{"{", RowBox[{"x", ",", "0", ",", "50"}], "}"}]}], "]"}], ",", " ", RowBox[{"Joined", "\[Rule]", " ", "True"}]}], "]"}]}]], "Input", CellChangeTimes->{{3.5769000405625*^9, 3.576900063609375*^9}}], Cell[BoxData[ RowBox[{"data1", "=", RowBox[{"RandomInteger", "[", RowBox[{"pdist", ",", "1000"}], "]"}]}]], "Input", CellChangeTimes->{{3.514151686171875*^9, 3.514151687609375*^9}}], Cell[BoxData[ RowBox[{"(*", RowBox[{ RowBox[{"<<", "\"\\""}], ";"}], " ", "*)"}]], "Input", CellChangeTimes->{{3.5141564101875*^9, 3.514156413703125*^9}, { 3.576899661296875*^9, 3.576899664390625*^9}}], Cell[BoxData[ RowBox[{"plot2", " ", "=", " ", RowBox[{"Histogram", "[", RowBox[{"data1", ",", " ", StyleBox[ RowBox[{ StyleBox["HistogramScale", "MR"], " ", "->", " ", "1"}], GridBoxOptions->{ GridBoxDividers->{ "Columns" -> {{False}}, "ColumnsIndexed" -> {}, "Rows" -> {False, True, {False}, False}, "RowsIndexed" -> {}}, GridBoxSpacings->{"Columns" -> { Offset[0.27999999999999997`], { Offset[0.5599999999999999]}, Offset[0.27999999999999997`]}, "ColumnsIndexed" -> {}, "Rows" -> { Offset[0.2], { Offset[0.16]}, Offset[0.2]}, "RowsIndexed" -> {}}}]}], "]"}], " ", RowBox[{"(*", " ", RowBox[{"Pre", "-", RowBox[{"Version", " ", "8"}]}], "*)"}]}]], "Input", CellChangeTimes->{{3.576899668*^9, 3.576899676890625*^9}, 3.576899901171875*^9, 3.576900124359375*^9}], Cell[BoxData[ RowBox[{"plot2", " ", "=", " ", RowBox[{"Histogram", "[", RowBox[{"data1", ",", " ", RowBox[{"{", "1", "}"}], ",", " ", "\"\\""}], "]"}], " ", RowBox[{"(*", " ", RowBox[{"Version", " ", "8"}], " ", "*)"}]}]], "Input", CellChangeTimes->{ 3.57689968921875*^9, {3.576899860125*^9, 3.576899861*^9}, {3.5768998955*^9, 3.57689989603125*^9}, {3.57690002121875*^9, 3.576900029875*^9}}], Cell[BoxData[ RowBox[{"Show", "[", RowBox[{"plot2", ",", " ", "plot1"}], "]"}]], "Input"], Cell["\<\ Example 1 Assume that the number of customers coming to a shop for each one-minute \ interval folows a Poisson distribution with a mean of 15. Simulate the number \ of customers coming from the first minute until the 30th minute. \ \>", "DisplayText", CellChangeTimes->{{3.576901578390625*^9, 3.57690157934375*^9}, { 3.57690187365625*^9, 3.576901891703125*^9}, {3.576902222921875*^9, 3.576902231625*^9}, {3.677076202970314*^9, 3.67707641236287*^9}}], Cell[BoxData[ RowBox[{ RowBox[{"pdist2", "=", " ", RowBox[{"PoissonDistribution", "[", "15", "]"}]}], ";"}]], "Input"], Cell[BoxData[ RowBox[{"simdat2", "=", RowBox[{"RandomInteger", "[", RowBox[{"pdist2", ",", "30"}], "]"}]}]], "Input", CellChangeTimes->{{3.51415173753125*^9, 3.51415176234375*^9}}], Cell["\<\ Example 2 At a bank, let x_i be an interarrival time between customer i and custumer \ i+1. x_i follows an exponential distribution with an average at three \ minutes. Simulate the interarrival time of the first customer until the 30th \ custumer.\ \>", "DisplayText", CellChangeTimes->{{3.576901578390625*^9, 3.57690157934375*^9}, { 3.57690187365625*^9, 3.576901891703125*^9}, 3.5769022078125*^9, { 3.57690227309375*^9, 3.576902454625*^9}, {3.576902553*^9, 3.576902685140625*^9}, {3.57690427625*^9, 3.57690428159375*^9}, { 3.6770764173812723`*^9, 3.6770764408403034`*^9}, {3.677076501630068*^9, 3.677076652598542*^9}, {3.6770766866658306`*^9, 3.677076750056828*^9}}], Cell[BoxData[ RowBox[{ RowBox[{"edist2", "=", " ", RowBox[{"ExponentialDistribution", "[", RowBox[{"1", "/", "3"}], "]"}]}], ";"}]], "Input", CellChangeTimes->{{3.576902465203125*^9, 3.5769024975*^9}}], Cell[BoxData[ RowBox[{"interarrivaltimes", "=", RowBox[{"RandomReal", "[", RowBox[{"edist2", ",", "30"}], "]"}]}]], "Input", CellChangeTimes->{{3.576902511421875*^9, 3.57690251478125*^9}, { 3.576903413765625*^9, 3.576903419109375*^9}, 3.576903783265625*^9, { 3.57690406778125*^9, 3.5769040679375*^9}, {3.57690417634375*^9, 3.57690417653125*^9}, 3.576904221015625*^9}], Cell["\<\ Assume that this bank had only one counter service and the service time for \ each customer took X minutes. If X has an exponential distribution with an \ average 2 minutes, simulate the service times of the first customer until the \ 30th custumer. \ \>", "DisplayText", CellChangeTimes->{{3.576901578390625*^9, 3.576901594796875*^9}, { 3.57690163940625*^9, 3.5769019801875*^9}, {3.576902900375*^9, 3.5769029165625*^9}, {3.576904285765625*^9, 3.57690428934375*^9}, { 3.6770767652994733`*^9, 3.677077004694584*^9}}], Cell[BoxData[ RowBox[{ RowBox[{"edist3", "=", " ", RowBox[{"ExponentialDistribution", "[", RowBox[{"1", "/", "2"}], "]"}]}], ";"}]], "Input", CellChangeTimes->{{3.576901997375*^9, 3.57690200840625*^9}, { 3.57690205196875*^9, 3.576902071890625*^9}, {3.576902466703125*^9, 3.576902487953125*^9}}], Cell[BoxData[ RowBox[{"servicetimes", "=", RowBox[{"RandomReal", "[", RowBox[{"edist3", ",", "30"}], "]"}]}]], "Input", CellChangeTimes->{{3.576902093890625*^9, 3.576902105296875*^9}, 3.576902491015625*^9, {3.576903403828125*^9, 3.576903422359375*^9}, 3.576903792546875*^9, {3.57690407190625*^9, 3.5769040720625*^9}, { 3.576904181828125*^9, 3.576904182015625*^9}, 3.57690422734375*^9}], Cell["\<\ Assume that the first customer came at minute 0 and there was no any customer \ at the counter service. When a customer came and found that there was another \ customer at the counter service, he/she need to be queued. Simulate the \ waiting time of each customer in a queue before getting a service and \ simulate the departure time of each customer. \ \>", "DisplayText", CellChangeTimes->{{3.576901578390625*^9, 3.576901594796875*^9}, { 3.57690163940625*^9, 3.5769019801875*^9}, {3.57690304415625*^9, 3.57690310975*^9}, {3.5769031448125*^9, 3.5769031575*^9}, { 3.57690459809375*^9, 3.576904637359375*^9}, {3.677077009645014*^9, 3.6770770716043386`*^9}, {3.6770771025847607`*^9, 3.6770773313895836`*^9}}], Cell["\<\ Method 1: Simulate the problem with a recursive function (take a long time to \ compute if simulating such a high number of custumers)\ \>", "DisplayText", CellChangeTimes->{{3.576901578390625*^9, 3.576901594796875*^9}, { 3.57690163940625*^9, 3.5769019801875*^9}, {3.57690304415625*^9, 3.57690310975*^9}, {3.5769031448125*^9, 3.5769031575*^9}, { 3.57690459809375*^9, 3.576904637359375*^9}, {3.5769048389375*^9, 3.576904855484375*^9}, {3.576905167328125*^9, 3.576905169609375*^9}, { 3.5769053573125*^9, 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StyleBox["body", "TI"]}], "]"}]], "InlineFormula"], " executes ", Cell[BoxData[ StyleBox["start", "TI"]], "InlineFormula"], ", then repeatedly evaluates ", Cell[BoxData[ StyleBox["body", "TI"]], "InlineFormula"], " and ", Cell[BoxData[ StyleBox["incr", "TI"]], "InlineFormula"], " until ", Cell[BoxData[ StyleBox["test", "TI"]], "InlineFormula"], " fails to give ", Cell[BoxData[ ButtonBox["True", BaseStyle->"Link", ButtonData->"paclet:ref/True"]], "InlineFormula"] }], "DisplayText", CellChangeTimes->{{3.576901578390625*^9, 3.576901594796875*^9}, { 3.57690163940625*^9, 3.5769019801875*^9}, {3.57690304415625*^9, 3.57690310975*^9}, {3.5769031448125*^9, 3.5769031575*^9}, { 3.57690459809375*^9, 3.576904637359375*^9}, {3.5769048389375*^9, 3.576904855484375*^9}, {3.576905167328125*^9, 3.576905169609375*^9}, { 3.576905388109375*^9, 3.576905391046875*^9}, {3.576905807171875*^9, 3.576905855515625*^9}, {3.6770775692495484`*^9, 3.677077607574809*^9}}], Cell[BoxData[ RowBox[{ RowBox[{"arrivaltimes", 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"0", ",", " ", RowBox[{ RowBox[{"departuretimes", "[", RowBox[{"[", RowBox[{"i", "-", "1"}], "]"}], "]"}], "-", " ", RowBox[{"arrivaltimes", "[", RowBox[{"[", "i", "]"}], "]"}]}]}], "]"}]}], ";"}], "\[IndentingNewLine]", ")"}]}], "]"}], ";"}]], "Input", CellChangeTimes->{{3.5769049451875*^9, 3.5769050306875*^9}, { 3.576905064640625*^9, 3.576905187296875*^9}, {3.57690522396875*^9, 3.5769052675*^9}, 3.57690531396875*^9, {3.576906012296875*^9, 3.576906022109375*^9}}], Cell[BoxData["waitingtimes"], "Input"], Cell[BoxData[ RowBox[{"Mean", "[", "%", "]"}]], "Input"], Cell[BoxData["departuretimes"], "Input"], Cell["\<\ Example 2 Simulate the event of drawing one ball out of a box 10 times with \ replacement if there were 10 balls in the box (5 blue balls and 5 white \ balls) and each ball in each color has a letter on it from A to E.\ \>", "DisplayText", CellChangeTimes->{{3.576901581390625*^9, 3.576901581921875*^9}, { 3.6770776321727777`*^9, 3.677077737742561*^9}, {3.6770778834702897`*^9, 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