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Dataplot Vol 2 Vol 1


    Library Function
    Compute the Borel-Tanner cumulative distribution function.
    Given a single queue with random arrival times of customers at constant rate l, constant service time beta, and k initial customers, the Borel-Tanner distribution is the distribution of the total number of customers served before the queue vanishes.

    The probability mass function of the Borel-Tanner distribution is

      p(x;lambda,k) = k*EXP(-lambda*x)*(lambda*x)**(x-k)/(x*(x-k)!),
   x >= k, 0 < lambda < 1

    with lambda and k denoting the shape parameters. The k shape parameter is a positive integer and lambda = l beta.

    The cumulative distribution function is computed by summing the probability mass function.

    LET <y> = BTACDF(<x>,<lambda>,<k>)
                            <SUBSET/EXCEPT/FOR qualification>
    where <x> is a positive integer variable, number, or parameter;
                <lambda> is a number or parameter in the range (0,1) that specifies the first shape parameter;
                <k> is a number or parameter denoting a positive integer that specifies the first shape parameter;
                <y> is a variable or a parameter where the computed Borel-Tanner cdf value is stored;
    and where the <SUBSET/EXCEPT/FOR qualification> is optional.
    LET A = BTACDF(3,0.5,3)
    LET Y = BTACDF(X1,0.3,2)
    PLOT BTACDF(X,0.3,2) FOR X = 2 1 20
Related Commands:
    BTAPDF = Compute the Borel-Tanner probability mass function.
    BTAPPF = Compute the Borel-Tanner percent point function.
    POIPDF = Compute the Poisson probability mass function.
    HERPDF = Compute the Hermite probability mass function.
    BINPDF = Compute the binomial probability mass function.
    NBPDF = Compute the negative binomial mass density function.
    GEOPDF = Compute the geometric probability mass function.
    Haight and Breuer (1960), "The Borel-Tanner Distribution", Biometrika, 47, pp. 143-150.

    Johnson, Kotz, and Kemp (1992), "Univariate Discrete Distributions", Second Edition, Wiley, pp. 394-396.

    Distributional Modeling
Implementation Date:
    title size 3
    tic label size 3
    label size 3
    legend size 3
    height 3
    multiplot scale factor 1.5
    x1label displacement 12
    y1label displacement 17
    multiplot corner coordinates 0 0 100 95
    multiplot scale factor 2
    label case asis
    title case asis
    case asis
    tic offset units screen
    tic offset 3 3
    title displacement 2
    y1label Probability
    x1label X
    ylimits 0 1
    major ytic mark number 6
    minor ytic mark number 3
    xlimits 0 20
    line blank
    spike on
    multiplot 2 2
    title Lambda = 0.3
    plot btacdf(x,0.3,1) for x = 1 1 20
    title Lambda = 0.5
    plot btacdf(x,0.5,1) for x = 1 1 20
    title Lambda = 0.7
    plot btacdf(x,0.7,1) for x = 1 1 20
    title Lambda = 0.9
    plot btacdf(x,0.9,1) for x = 1 1 20
    end of multiplot
    justification center
    move 50 97
    text Cumulative Distribution for Borel-Tanner
    plot generated by sample program

Date created: 6/5/2006
Last updated: 6/5/2006
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